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TRANSCRIPT
Methods of pooling longevity risk
Catherine Donnelly
Risk Insight Lab Heriot-Watt University
22 May 2018
The lsquoMinimising Longevity and Investment Risk while Optimising
Future Pension Plansrsquo research programme is being funded by the
Actuarial Research Centre
wwwactuariesorgukarc
httprisk-insight-labcom
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 2
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 3
I Motivation
bull Background
bull Focus on life annuity
bull Example of a tontine in action
22 May 2018 4
Setting
22 May 2018 5
Value of pension
savings
Time
Setting
22 May 2018 6
Value of pension
savings
Time
Contribution plan
Setting
22 May 2018 7
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 8
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 2
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 3
I Motivation
bull Background
bull Focus on life annuity
bull Example of a tontine in action
22 May 2018 4
Setting
22 May 2018 5
Value of pension
savings
Time
Setting
22 May 2018 6
Value of pension
savings
Time
Contribution plan
Setting
22 May 2018 7
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 8
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 3
I Motivation
bull Background
bull Focus on life annuity
bull Example of a tontine in action
22 May 2018 4
Setting
22 May 2018 5
Value of pension
savings
Time
Setting
22 May 2018 6
Value of pension
savings
Time
Contribution plan
Setting
22 May 2018 7
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 8
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
I Motivation
bull Background
bull Focus on life annuity
bull Example of a tontine in action
22 May 2018 4
Setting
22 May 2018 5
Value of pension
savings
Time
Setting
22 May 2018 6
Value of pension
savings
Time
Contribution plan
Setting
22 May 2018 7
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 8
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Setting
22 May 2018 5
Value of pension
savings
Time
Setting
22 May 2018 6
Value of pension
savings
Time
Contribution plan
Setting
22 May 2018 7
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 8
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Setting
22 May 2018 6
Value of pension
savings
Time
Contribution plan
Setting
22 May 2018 7
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 8
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Setting
22 May 2018 7
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 8
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Setting
22 May 2018 8
Value of pension
savings
Time
Contribution plan
Investment
strategy
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Setting
22 May 2018 9
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
Drawdown
Something else
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
The present in the UK ndash DC on the rise
bull Defined benefit plans are closing (87 are closed in 2016 in
UK)
bull Most people are now actively in defined contribution plans or
similar arrangement (97 of new hires in FTSE350)
bull Contribution rates are much lower in defined contribution plans
22 May 2018 10
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Size of pension fund assets in 2016[Willis Towers Watson]
Country Value of
pension
fund assets
(USD billion)
As percentage
of GDP
Of which DC
asset value
(USD billion)
USA 22rsquo480 1211 13rsquo488
UK 2rsquo868 1082 516
Japan 2rsquo808 594 112
Australia 1rsquo583 1260 1rsquo377
Canada 1rsquo575 1028 79
Netherlands 1rsquo296 1683 78
22 May 2018 11
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Drawdown
22 May 2018 12
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Drawdown
22 May 2018 13
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Drawdown
22 May 2018 14
Value of pension
savings
Time
Contribution plan
Investment
strategyInvestment
strategy II
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Drawdown
22 May 2018 15
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Investment
strategy II
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Life insurance mathematics 101
bull PV(annuity paid from age 65) = 119886119879|
bull Expected value of the PV is
11988665 = 11990711990165 + 1199072211990165 + 1199073311990165 + 1199074411990165 +⋯
bull To use as the price
bull Law of Large Numbers holds
bull Same investment strategy
bull Known investment returns and future lifetime distribution
12 June 2018 16
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Life annuity contract
22 May 2018 17
Insurance company
Purchase of the
annuity contract
Insurance company
Annuity income
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Life annuity contract
22 May 2018 18
Insurance company
Annuity income
Insurance company
Annuity income
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Life annuity
22 May 2018 19
Value of pension
savings
Time
Contribution plan
Investment
strategy
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Life annuity
22 May 2018 20
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Life annuity
22 May 2018 21
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Longevity pooling
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Life annuity
22 May 2018 22
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
+ investment guarantees
+ longevity guarantees
Longevity pooling
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Life annuity contract
bull Income drawdown vs life annuity if follow same investment
strategy then life annuity gives higher income
ignoring fees costs taxes etc
bull Pooling longevity risk gives a higher income
bull Everyone in the group becomes the beneficiaries of each
other indirectly
22 May 2018 23
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Annuity puzzle
bull Why donrsquot people annuitize
bull Can we get the benefits of life annuities without the full
contract
bull Example showing income withdrawal from a tontine
22 May 2018 24
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Drawdown
22 May 2018 25
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Drawdown
22 May 2018 26
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Drawdown
22 May 2018 27
Value of pension
savings
Time
Contribution plan
Investment
strategyLongevity
risk
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Drawdown
22 May 2018 28
Value of pension
savings
Time
Contribution plan
Investment
strategy
Longevity
risk
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Aim of modern tontines
bull Aim is to provide an income for life
bull It is not about gambling on your death or the deaths of others in the
pool
bull It should look like a life annuity
bull With more flexibility in structure
bull Example is based on an explicitly-paid longevity credit
22 May 2018 29
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 0 Simple setting of 4 Rule
bull Pension savings = euro100000 at age 65
bull Withdraw euro4000 per annum at start of each year until funds exhausted
bull Investment returns = Price inflation + 0
bull No longevity pooling
22 May 2018 30
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 0 income drawdown (4 Rule)
22 May 2018 31
0
1000
2000
3000
4000
5000
6000
7000
8000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling)
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 1 Join a tontine
bull Same setup excepthellippool all of asset value in a tontine for rest of life
bull Withdraw a maximum real income of euroX per annum for life
(we show X on charts to follow)
bull Mortality table S1PMA
bull Assume a perfect pool longevity credit=its expected value
bull Longevity credit paid at start of each year
22 May 2018 32
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
UK mortality table S1PMA
22 May 2018 33
00
01
02
03
04
05
06
07
08
09
10
65 75 85 95 105 115
qx
Age x (years)
Annual probability of death for table S1MPA
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 1i
0 investment returns above inflation
22 May 2018 34
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+0 pa
4 Rule (no pooling) 100 pooling
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 1ii
+2 pa investment returns above inflation
22 May 2018 35
0
1000
2000
3000
4000
5000
6000
7000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 1iii Inv Returns = Inflation ndash 2
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 36
0
1000
2000
3000
4000
5000
6000
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-2 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 1iv Inv Returns = Inflation ndash 5
pa from age 65 to 75 then Inflation +2 pa
22 May 2018 37
0
500
1000
1500
2000
2500
3000
3500
4000
4500
65 75 85 95 105 115
Real in
com
e w
ithdra
wn a
t age
Age (years)
Investment returns = inflation-5 pa from age 65 to 75 then inflation+2 pa
4 Rule (no pooling) 100 pooling
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 38
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
II One way of pooling longevity risk
bull Aim of pooling retirement income not a life-death gamble
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
22 May 2018 39
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling
22 May 2018 40
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling
22 May 2018 41
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling rule [DGN]
bull λ(i) = Force of mortality of ith member at time T
bull W(i) = Fund value of ith member at time T
bull Payment (longevity credit) to ith member∶
120582(119894)times119882(119894)
σ119896isin119866119903119900119906119901 120582(119896)times119882(119896) times Bobprimes remaining fund value
22 May 2018 42
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example I(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 10 10 = 100-100+10
B 001 200 2 20 220 = 200+20
C 001 300 3 30 330 = 300+30
D 001 400 4 40 440 = 400+40
Total 1000 10 100 1000
22 May 2018 43
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example I(ii) D dies
Member Force of
mortality
Fund
value
before
D dies
Force of
mortality
x Fund value
Longevity
credit from
Drsquos fund value
= 400 x
(4)Sum of (4)
Fund value afer D
dies
(1) (2) (3) (4) (5) (6)
A 001 100 1 40 140 = 100+40
B 001 200 2 80 280 = 200+80
C 001 300 3 120 420 = 300+120
D 001 400 4 160 160 = 400-400+160
Total 1000 10 400 1000
22 May 2018 44
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 2(i) A dies
Member Force of
mortality
Fund
value
before
A dies
Force of
mortality
x Fund value
Longevity
credit from Arsquos
fund value =
100 x
(4)Sum of (4)
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 100 4 20 20 = 100-100+20
B 003 200 6 30 230 = 200+30
C 002 300 6 30 330 = 300+30
D 001 400 4 20 420 = 400+20
Total 1000 20 100 1000
22 May 2018 45
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling rule
bull q(i) = Probability of death of ith member from time T to T+1
bull Unit time period could be 112 year 14 year 12 yearhellip
bull Longevity credit paid to ith member∶
119902(119894)times119882(119894)
σ119896isin119866119903119900119906119901 119902(119896)times119882(119896) times Total fund value of members dying
between time 119879 and 119879 + 1
22 May 2018 46
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
(1) (2) (3) (4)
75 0035378 euro100000 100
76 0039732 euro96500 96
77 0044589 euro93000 92
78 0049992 euro89500 88
100 036992 euro12500 1
Total (S1MPA) 1121
22 May 2018 47
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Prob of death
multiplied by
Fund value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 48
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age x
Observed
number of
deaths from
age x to x+1
Total funds
released by
deaths
= (3)x(7)
(1) (2) (3) (4) (7) (8)
75 0035378 euro100000 100 2 euro200000
76 0039732 euro96500 96 2 euro193000
77 0044589 euro93000 92 0 euro0
78 0049992 euro89500 88 5 euro447500
100 036992 euro12500 1 0 euro0
Total (S1MPA) 1121 97 euro5818500
22 May 2018 49
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
5818500
22 May 2018 50
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 3 larger group total assets of
group euro85461500Total funds
released by
deaths
= (3)x(7)
(8)
euro5818500
22 May 2018 51
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Per member
share of funds of
deceased
members =
(5)sum of (4)x(5)
(1) (2) (3) (4) (5) (6)
75 0035378 euro100000 100 353780 000056
76 0039732 euro96500 96 383414 000060
77 0044589 euro93000 92 414678 000065
78 0049992 euro89500 88 447428 000070
100 036992 euro12500 1 462400 000073
Total (S1MPA) 1121
22 May 2018 52
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Number of
members at
age
Prob of death
times Fund
value
= (2)x(3)
Longevity credit
per member
= (6) x sum of (8)
(1) (2) (3) (4) (5) (9)
75 0035378 euro100000 100 353780 euro323733
76 0039732 euro96500 96 383414 euro350850
77 0044589 euro93000 92 414678 euro379458
78 0049992 euro89500 88 447428 euro409428
100 036992 euro12500 1 462400 euro423128
Total (S1MPA) 1121
22 May 2018 53
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 3 larger group total assets of
group euro85461500Age x of
member
Prob of
death
from age
x to x+1
Fund
value of
each
member
Longevity
credit per
member
= (6) x sum
of (8)
Fund value of
survivor at
age x+1
Fund value of
deceased at age
x+1
(1) (2) (3) (9) (10) (11)
75 0035378 euro100000 euro323733 euro10323733 euro323733
76 0039732 euro96500 euro350850 euro10000850 NA
77 0044589 euro93000 euro379458 euro9679458 euro379458
78 0049992 euro89500 euro409428 euro9359428 euro409428
100 036992 euro12500 euro423128 euro1673128 NA
22 May 2018 54
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] - features
bull Total asset value of group is unchanged by pooling
bull Individual values are re-arranged between the members
bull Expected actuarial gain = 0 for all members at all times
bull Actuarial gain of member (x) from time T to T+1
=
+ Longevity credits gained by (x) from deaths (including (x)rsquos own death)
between time T and T+1
- Loss of (x)rsquos fund value if (x) dies between times T and T+1
ie the pool is actuarially fair at all times no-one expects to gain
from pooling
22 May 2018 55
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] - features
bull Expected longevity credit =
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 119909 times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
times 1 minus119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
σ119910isin119866119903119900119906119901 119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119910) times 119865119906119899119889 119907119886119897119906119890 119900119891 119910
bull Expected longevity credit tends to
119875119903119900119887 119900119891 119889119890119886119905ℎ 119900119891 (119909) times 119865119906119899119889 119907119886119897119906119890 119900119891 119909
as group gets bigger
22 May 2018 56
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] - features
bull There will always be some volatility in the longevity credit
bull Actual value ne expected value (no guarantees)
bull But longevity credit ge 0 ie never negative
bull Loss occurs only upon death
bull Volatility in longevity credit can replace investment return
volatility
22 May 2018 57
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] - features
bull Scheme works for any group
bull Actuarial fairness holds for any group composition but
bull Requires a payment to estate of recently deceased
bull Sabin [see Part IV] proposes a survivor-only payment However it
requires restrictions on membership
bull Should it matter Not if group is well-diversified (Law of Large Numbers
holds) ndash then schemes should be equivalent
22 May 2018 58
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] - features
bull Increase expected lifetime income
bull Reduce risk of running out of money before death
bull Non-negative return except on death
bull Update force of mortality periodically
22 May 2018 59
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] - features
bull ``Costrsquorsquo is paid upon death not upfront like life annuity
bull Mitigates longevity risk but does not eliminate it
bull Anti-selection risk remains as for life annuity Waiting period
22 May 2018 60
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] - features
bull Splits investment return from longevity credit to enable
bull Fee transparency
bull Product innovation
22 May 2018 61
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] ndashanalysis
bull Compare
a) Longevity risk pooling versus
b) Equity-linked life annuity paying actuarial return (λ(i) ndash Fees) x W(i)
Fees have to be lt05 for b) to have higher expected return in a
moderately-sized (600 members) heterogeneous group [DGN]
22 May 2018 62
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] ndash some ideas
bull Insurer removes some of the longevity credit volatility eg
guarantees a minimum payment for a fee [DY]
bull Allow house as an asset ndash monetize without having to sell it
before death [DY]
22 May 2018 63
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [DGN] ndash some ideas
bull Pay out a regular income with the features
bull Each customer has a ring-fenced fund value
bull Explicitly show investment returns and longevity credits on annual
statements
bull Long waiting period before customerrsquos assets are pooled to reduce
adverse selection risk eg 10 years
bull More income flexibility
bull Opportunity to withdraw a lumpsum from asset value
bull Update forces of mortality periodically
22 May 2018 64
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
II One way of pooling longevity risk -
Summary
bull DGN method of pooling longevity risk
ndash Explicit scheme
ndash Everything can be different member characteristics investment
strategy
bull Can provide a higher income in retirement
bull Reduces chance of running out of money in retirement
bull May also result in a higher bequest
bull Transparency may encourage more people to ldquoannuitizerdquo
22 May 2018 65
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 66
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Classification of methods
bull Explicit tontines eg [DGN] (Part II) and Sabin (Part IV)
bull Individual customer accounts
bull Customer chooses investment strategy
bull Customer chooses how much to allocate to tontine
bull Initially
22 May 2018 67
Tontine part
of customer
account
Non-tontine part of
customer account
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Explicit tontines
bull Add in returns and credits
22 May 2018 68
Tontine part
of customer
account
Non-tontine part of
customer account
Investment returns credited
Longevity
credits
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Explicit tontines
bull Subtract income withdrawn by customer chosen by customer
subject to limitations (avoid anti-selectionmoral hazard)
22 May 2018 69
Withdrawal
by customer
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Explicit tontines
bull Either re-balance customer account to maintain constant
percentage in tontine or
bull Keep track of money in and out of each sub-account
22 May 2018 70
Tontine part
of customer
account
Non-tontine part of
customer account
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Implicit tontines
bull Implicit tontines eg GSA (Part V)
bull Works like a life annuity
bull Likely to assume that idiosyncratic longevity risk is zero
bull Customers are promised an income in exchange for upfront
payment
bull Income adjusted for investment and mortality experience
bull The explicit tontines can be operated as implicit tontines
22 May 2018 71
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Implicit methods
bull Same investment strategy for all customers
bull Less clear how to allow flexible withdrawals (eg GSA not
actuarially fair except for perfect pool)
bull Might be easier to implement from a legalregulatory viewpoint
22 May 2018 72
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 73
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
A second explicit scheme [Sabin] - overview
bull [DGN] scheme works for any heterogeneous group
bull Simple rule for calculating longevity credits
bull Requires payment to the estate of recently deceased to be
actuarially fair
bull [Sabin] shares out deceasedrsquos wealth only among the
survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation in [Sabin] is more complicated
22 May 2018 74
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [Sabin]
22 May 2018 75
Pool risk over lifetime
Individuals withdraw
income from their own
funds
However when someone
dies at time Thellip
Individuals make their
own investment
decisions
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling [Sabin]
22 May 2018 76
Share out remaining
funds of Bob
Bob
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling rule [Sabin]
bull Longevity credit paid to ith member is
120572119894119861119900119887 times Bobprimes remaining fund value
bull 120572119894119861119900119887= Share of Bobrsquos fund value received by ith member with
120572119894119861119900119887 isin 01
bull Payment to survivors only so 120572119861119900119887119861119900119887 = minus1
bull No more and no less than Bobrsquos fund is shared out so
119894ne119861119900119887
120572119894119861119900119887 = 1
22 May 2018 77
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling rule [Sabin]
bull Impose actuarial fairness Expected gain from tontine is zero
bull 120572119894119889= Share of deceased drsquos fund value received by ith
member
bull 120582 119894 = Force of mortality of ith member at time T
bull 119882 119894 = Fund value of ith member at time T
bull Expected gain of ith member from tontine is
119889ne119894
120582119889120572119894119889119882119889 minus 120582119894119882119894 = 0
22 May 2018 78
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling rule [Sabin]
Simple setting of 3 members
Then we must solve for 120572119894119895 119894119895=123the system of equations
1205822120572121198822 + 1205823120572131198823 minus 12058211198821 = 01205821120572211198821 + 1205823120572231198823 minus 12058221198822 = 01205821120572311198821 + 1205822120572321198822 minus 12058231198823 = 0
subject to the constraints
119894ne119895
120572119894119895 = 1 for 119895 = 123
120572119894119895 isin 01 for all 119894 ne 119895
22 May 2018 79
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Longevity risk pooling rule [Sabin]
bull Does a solution exist [Sabin] proves that for each member i
in the group
119896isin119892119903119900119906119901
120582119896119882119896 ge 2120582119894119882119894
is a necessary and sufficient condition for 120572119894119895 119894119895isin119866119903119900119906119901to exist
bull In general there is no unique solution
bull [Sabin] and [Sabin2011b] contain algorithms to solve the
system of equations
22 May 2018 80
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 4(i) [Sabin Example 1] A dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119912 x 2
Fund value afer
A dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 -1 -2 0
B 015983 6 061302 122604 722604
C 014447 3 023766 047532 347532
D 014107 2 014932 029864 229864
Total 10000 13 000000 000000 1300000
22 May 2018 81
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 4(ii) [Sabin Example 1] B dies
Member
i
120640119946 σ119948isin119912119913119914119915120640119948
Fund
value
before
B dies
120630119946119913 Longevity
credit from Brsquos
fund value =
120630119946119913 x 6
Fund value afer
B dies = (3) + (5)
(1) (2) (3) (4) (5) (6)
A 055464 2 075754 454524 654524
B 015983 6 -1 -6 0
C 014447 3 014814 088884 388884
D 014107 2 009432 056592 256592
Total 10000 13 000000 000000 1300000
22 May 2018 82
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 5(i) A dies ndash one solution
Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 13 50 250
C 002 300 13 50 350
D 001 600 13 50 650
Total 1250 0 0 1250
22 May 2018 83
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 5(i) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 13 13 13
B 13 -1 13 13
C 13 13 -1 13
D 13 13 13 -1
Total 0 0 0 0
22 May 2018 84
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 5(ii) A dies ndash another solution
(not so nice)Member Force of
mortality
Fund
value
before
A dies
120630119946119912 Longevity
credit from Arsquos
fund value =
120630119946119913 x 150
Fund value afer
A dies
(1) (2) (3) (4) (5) (6)
A 004 150 -1 -150 0
B 003 200 0 0 200
C 002 300 0 0 300
D 001 600 1 150 750
Total 1250 0 0 1250
22 May 2018 85
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Example 5(ii) Full solution
Member 120630119946119912 120630119946119913 120630119946119914 120630119946119915
(1) (2) (3) (4) (5)
A -1 0 0 1
B 0 -1 1 0
C 0 1 -1 0
D 1 0 0 -1
Total 0 0 0 0
22 May 2018 86
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Choosing a solution [Sabin]
bull [Sabin] suggests minimizing the variance of 120572119894119895 among
other possibilities However for M group members the
algorithm has run-time 119978 1198723
bull He suggests another approach (called Separable Fair Transfer
Plan) which has run-time 119978 119872
22 May 2018 87
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
A second explicit scheme [Sabin] - summary
bull Shares out deceasedrsquos wealth only among the survivors
bull Restrictions on the group composition to maintain actuarial
fairness
bull Longevity credit allocation is more complicated
bull No unique solution but a desired solution can be chosen
bull For implementation [Sabin] can operate like [DGN]
22 May 2018 88
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 89
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
An implicit scheme [GSA] ndash Group Self-
Annuitisation
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Income calculation assumes Law of Large Numbers holds
bull Works for heterogeneous membership
bull But assume homogeneous example next
22 May 2018 90
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Homogeneous membership
bull Group of 119872 homogeneous members all age 65 initially
bull Track total fund value 119865119899
bull Each receives a payment at start of first year
1198610 =1
119872
1198650ሷ11988665
=1
11989765lowast
1198650ሷ11988665
with 11989765lowast = 119872 (actual number alive at age 65)
and
ሷ11988665 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990165
22 May 2018 91
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Homogeneous membership
bull End of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 119877
where 119877 is the actual investment return in the first year (assume
it equals its expected return 119877)
bull 11989766lowast members alive (expected number was 11989765
lowast times 11990165)
bull Each survivor receives a payment at start of second year
1198611 =1
11989766lowast
1198651ሷ11988666
ሷ11988666 = 1 +
119896=1
infin
(1 + 119877)minus119896 times119896 11990166
22 May 2018 92
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Homogeneous membership
bull Straightforward to show
1198611 = 1198610 times1199016511990165lowast
where
bull 11990165lowast is the empirical probability of one-year survival and
bull 11990165 is the estimated probability of one-year survival
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast
22 May 2018 93
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Homogeneous membership
bull Allow for actual annual investment returns 1198771lowast 1198772
lowast hellip in year
12hellip
bull Then end of first year total fund value in GSA is
1198651 = 1198650 minus 1198970lowast1198610 times 1 + 1198771
lowast
bull Benefit paid to each survivor at start of second year is
1198611 = 1198610 times1199016511990165lowast times
1 + 1198771lowast
1 + 119877
22 May 2018 94
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Homogeneous membership
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus111990165+119899minus1lowast times
1 + 119877119899lowast
1 + 119877
bull Or
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where
119872119864119860119899= Mortality Experience Adjustment
119868119877119860119899=Interest Rate Adjustment
22 May 2018 95
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Different initial contributions
bull Group of 119872 members all age 65 initially
bull Member i pays in amount 1198650(119894)
bull Total fund value 1198650 = σ119894=1119872 1198650
(119894)
bull Member i receives a payment at start of first year
1198610(119894)
=1198650(119894)
ሷ11988665
with ሷ11988665 = 1 + σ119896=1infin (1 + 119877)minus119896 times119896 11990165
22 May 2018 96
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Different initial contributions
bull At end of first year fund value of member i is
1198651 = 1198650 minus119894=1
119872
1198610(119894)
times 1 + 1198771lowast
where 1198771lowast is the actual investment return in the first year
bull Fund value of member i is
1198651(119894)
= 1198650(119894)minus 1198610
(119894)times 1 + 1198771
lowast
bull Fund value of members dying over first year is distributed
among survivors in proportion to fund values
22 May 2018 97
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Different initial contributions
bull If member i is alive at start of second year they get a benefit
payment
1198611(119894)
=1
ሷ119886661198651(119894)+
1198651(119894)
σ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
times
119889isin119863119890119886119889
1198651(119889)
bull Can show that
1198611(119894)
= 1198610(119894)times
11990165
ൗσ119904isin1198781199061199031199071198941199071199001199031199041198651(119904)
1198651times1 + 1198771
lowast
1 + 119877
22 May 2018 98
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Different initial contributions
bull More generally
119861119899 = 119861119899minus1 times11990165+119899minus1
ൗσ119904isin119878119906119903119907119894119907119900119903119904119865119899(119904)
119865119899times1 + 119877119899
lowast
1 + 119877
bull which has the form
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899
where 119872119864119860119899= Mortality Experience Adjustment and
119868119877119860119899=Interest Rate Adjustment
22 May 2018 99
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash Different initial contributions
bull [GSA] extend to members of different ages
bull Further allow for updates to future mortality
119861119899 = 119861119899minus1 times119872119864119860119899 times 119868119877119860119899 times 119862119864119860119899
where 119862119864119860119899= Changed Expectation Adjustment = ሷ11988665+119899minus1old
ሷ11988665+119899minus1new
22 May 2018 100
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
[GSA] ndash analysis
bull Same investment strategy for all members strategy for 65 year
old = strategy for 80 year old Are all 65 year olds the same
bull Fixed benefit calculation - no choice
bull Not actuarially fair 1198650(119894)
ne 120124 Discounted future benefits
bull Two finite groups with different wealth otherwise identical
bull Higher wealth group lose 1198650(119894)
gt 120124(Discounted future benefits)
bull Higher wealth group expect higher benefits if groups had same wealth
bull Only significant in small or highly heterogeneous groups
[Donnelly2015]
22 May 2018 101
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
GSA ndash analysis [QiaoSherris] Figure 1
22 May 2018 102
bull $100 paid on entry at
age 65
bull Max age 105
bull Single cohort
bull Interest rate 5 pa
bull Allow for systemic
mortality changes
120583119909119905 = 119884119905(1)
+ 119884119905(2)
10966119909
119889119884119905(1)
= 1198861119889119905 + 1205901119889119882119905(1)
119889119884119905(2)
= 1198862119889119905 + 1205902119889119882119905(2)
119889 1198821199051119882119905
2= 0929119889119905
with 120583119909119905 = 0 119894119891120583119909119905 lt 0
bull Donrsquot allow for future
expected
improvements in
annuity factor
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
GSA ndash analysis [QiaoSherris] Figure 2
22 May 2018 103
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
GSA ndash analysis [QiaoSherris] Figure 3
22 May 2018 104
bull 1000 members age 65 join every 5 years
bull Update annuity factor to allow for mortality improvements
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Group Self-Annuitisation - Summary
bull Group Self-Annuitisation (GSA) pays out an income to its
members
bull Collective fund one investment strategy
bull Income is adjusted for mortality and investment experience
bull Works for heterogeneous membership
22 May 2018 105
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Overview of entire session
I Motivation
II One way of pooling longevity risk
III Classification of methods amp discussion
IV A second explicit scheme
V An implicit scheme
VI Summary and discussion
22 May 2018 106
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Summary and discussion
bull Reduce risk of running out of money
bull Provide a higher income than living off investment returns
alone
bull Should be structured to provide a stable fairly constant income
(not increasing exponentially with the longevity credit)
bull Two types of tontine
bull Explicit Longevity credit payment
bull Implicit Income implicitly includes longevity credit
22 May 2018 107
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
Summary and discussion
bull Looked at two actuarially fair explicit tontines [DGN] [Sabin]
bull Enable tailored solution eg individual investment strategy
bull Easier to add product innovation eg partial guarantees
bull Others have been proposed not necessarily actuarially fair
bull In practice Mercer Australia LifetimePlus appears to be an
explicit tontine (though income profile unattractive)
bull [GSA] is an implicit tontine
bull Isnrsquot actuarially fair but shouldnrsquot matter if enough members
bull In practice TIAA-CREF annuities are similar
22 May 2018 108
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
22 May 2018 109
The views expressed in this presentation are those of the presenter
Questions Comments
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111
The Actuarial Research Centre (ARC)
A gateway to global actuarial research
The Actuarial Research Centre (ARC) is the Institute and Faculty of Actuariesrsquo (IFoA)
network of actuarial researchers around the world
The ARC seeks to deliver cutting-edge research programmes that address some of the
significant global challenges in actuarial science through a partnership of the actuarial
profession the academic community and practitioners
The lsquoMinimising Longevity and Investment Risk while Optimising Future Pension Plansrsquo
research programme is being funded by the ARC
wwwactuariesorgukarc
Bibliography
bull [DGN] Donnelly C Guilleacuten M and Nielsen JP (2014) Bringing cost transparency to the life annuity
market Insurance Mathematics and Economics 56 pp14-27
bull [DY] Donnelly C and Young (2017) J Product options for enhanced retirement income British Actuarial
Journal 22(3)
bull [Donnelly2015] C Donnelly (2015) Actuarial Fairness and Solidarity in Pooled Annuity Funds ASTIN
Bulletin 45(1) pp 49-74
bull [GSA] J Piggott E A Valdez and B Detzel (2005) The Simple Analytics of a Pooled Annuity Fund Journal
of Risk and Insurance 72(3) pp 497-520
bull [QiaoSherris] C Qiao and M Sherris (2013) Managing Systematic Mortality Risk with Group Self-Pooling
and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
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of Risk and Insurance 72(3) pp 497-520
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and Annuitization Schemes Journal of Risk and Insurance 80(4) pp 949-974
bull [Sabin] MJ Sabin (2010) Fair Tontine Annuity Available at SSRN or at httpsagedrivecomfta
bull [Sabin2011a] MJ Sabin (2011) Fair Tontine Annuity Presentation at httpsagedrivecomfta11_05_19pdf
bull [Sabin2011b] MJ Sabin (2011) A fast bipartite algorithm for fair tontines Available at
httpsagedrivecomfta
bull [Willis Towers Watson] Global Pensions Assets Study 2017
22 May 2018 111