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Policyholder Exercise Behavior for Variable
Annuities including Guaranteed Minimum
Withdrawal Benefits1
Thorsten Moenig2
Department of Risk Management and Insurance, Georgia State University35 Broad Street, 11th Floor; Atlanta, GA 30303; USA
Email: [email protected]
June 2011
1We gratefully acknowledge sponsorship by the Society of Actuaries.2Joint work with Dr. Daniel Bauer, Georgia State University
Page 2 / 23 Overview
1 Introduction
2 A Lifetime Utility Model for Variable Annuities
3 Results
4 Conclusions and Future Research
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 3 / 23 Introduction
1 Introduction
Motivation
Risk-Neutral Valuation Approach
2 A Lifetime Utility Model for Variable Annuities
3 Results
4 Conclusions and Future Research
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 4 / 23 Introduction
Motivation
• Variable Annuities:
I Popular long-term investment vehicles
I Tax-deferred growth
I Investment evolves according to underlying (risky) portfolio
I Uncertain payout
• Guaranteed Minimum (Death / Income / Accumulation /Withdrawal) Benefits
I Insurers offer guaranteed payments
I Policyholders can purchase security
I Similar to (combination of) financial options
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 5 / 23 Introduction
Motivation
• Withdrawal uncertainty
I Could mitigate or intensify insurer’s exposure to investment and/ormortality risks
I Interactions non-trivial
I Affects pricing and risk management
• Insurers in trouble
I Disintermediation in 1970s
I Equitable Life closed to new business in 2000
I The Hartford accepted $3.4B in TARP money in June 2009 afterlosing $2.75B in 2008, hurt by investment losses and the cost of VAguarantees
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 6 / 23 IntroductionRisk-Neutral Valuation Approach
• Used in actuarial literature to price variety of options:I Milevsky and Posner (2001): GMDBI Ulm (2006): “Real option to transfer”I Zaglauer and Bauer (2008): Participating life insurance contracts
• To analyze withdrawal behavior for GMWBs:I Milevsky and Salisbury (2006)I Bauer, Kling and Russ (2008)I Optimal stopping problem, akin to pricing American put optionI Exercise / Withdraw if exercise value exceeds continuation valueI Worst-case scenario, calculate correct upper bound
• VA market incomplete: cannot sell – or repurchase – policy at itsrisk-neutral valueI Withdrawing means giving up possible guarantees and tax benefits
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 7 / 23A Lifetime Utility Model for Variable Annuities
1 Introduction
2 A Lifetime Utility Model for Variable Annuities
The Model
Bellman Equation
Implementation in a Black-Scholes Framework
Parameter Assumptions
3 Results
4 Conclusions and Future Research
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 8 / 23A Lifetime Utility Model for Variable Annuities
The Model
• Consider withdrawal decisions in life-cycle model with outside
investment
• PH maximizes expected lifetime utilityI Consumption and bequestsI Initial wealth W0
I Annual (deterministic) income It
• Invests P0 in VA with finite maturity T, remainder in outsideaccountI Includes GMWB, possibly other guaranteesI Return-of-investment guaranteesI Other types possible, at cost of larger state spaceI All guarantee accounts identical to benefits base, G·
tI Annual guarantee fee φ as percentage of concurrent account value
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 9 / 23A Lifetime Utility Model for Variable Annuities
The Model
• VAs grow tax-deferredI Withdrawals taxed on last-in first-out basisI Early withdrawal tax (10%) if PH withdraws prior to age 59.5
• Restrict all actions to policy anniversary datesI Four state variables
? VA account X−t? Outside account A−t? Benefits base G·t? Tax base Ht
I Three choice variables? Withdrawal amount wt
? Consumption Ct
? Risk allocation in outside account νt
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 10 / 23A Lifetime Utility Model for Variable Annuities
Bellman Equation
Vt(yt) = maxCt ,wt ,νt
uC(Ct)+e−β ·Et [qx+t · uB(bt+1|St+1) + px+t · Vt+1(yt+1|St+1)] , (1)
yt ≡ (A−t ,X−t ,G
·t ,Ht ),
X+t =
(X−t − wt
)+,
A+t = A−t + It + wt − Ct − feeI − feeG − taxes,
feeI = s ·max{
wt −min(gWt ,GW
t )},
feeG = sg · (wt − feeI) · I{x+t<59.5},
taxes = τ ·min{wt − feeI − feeG, (X−t − Ht )
+},
G·t+1 =
(G·t − w
)+: w ≤ gW
t(min
{G·t − w , G·t ·
X+t
X−t
} )+
: w > gWt
,
Ht+1 = Ht −(
wt −(
X−t − Ht
)+)+
,
A−t+1 = A+t ·[νt ·
St+1St− κ ·
(νt ·
St+1St− 1)+]
,
X−t+1 = X+t · e
−φ ·[νX · St+1
St
],
bt+1 = A−t+1 + max{X−t+1,GDt+1},
νt ≥ 0 ,∑
i νt (i) = 1,
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 11 / 23A Lifetime Utility Model for Variable Annuities
Implementation in a Black-Scholes Framework
• Solve by recursive dynamic programming:
(I) Create appropriate state space grid
(II) For t = T : for all grid points (A,X ,G·,H), compute V−T (A,X ,G·,H).
(III) For t = T − 1,T − 2, . . . ,1: Given V−t+1, calculate V−
t (A,X ,G·,H)
recursively for each (A,X ,G·,H) on the grid using anapproximation of the integral in (1)
I Discretize return space and evaluate via Green’s function
I Gauss-Hermite quadrature
(IV) For t = 0: For the given starting values A0 = W0 − P0, X0 = P0,G·
0 = G·1 = P0 and H0 = H1 = P0, compute
V−0 (W0 − P0,P0,P0,P0) recursively from Equation (1)
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 12 / 23A Lifetime Utility Model for Variable Annuities
Parameter Assumptions
• Policyholder is 55 years old, T = 15 years to maturity
• P0 = 100K ; W0 = 2 · P0 = 200K ; It = 40K
• CRRA(γ = 3) utilities; B = 1; β = r
• τ = 30%, κ = 15%
• Guarantee fee φ = 50 bps
• Surrender fee s = 5%,
gWt =
{0 : t ≤ 5
20,000 : t > 5
• r = 4%, µ = 8%, σ = 15%I Merton ratio: µ−r
σ2·γ = 0.08−0.040.152·3 ≈ 0.5926
• νX = 100% equity exposure in VA
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 13 / 23 Results
1 Introduction
2 A Lifetime Utility Model for Variable Annuities
3 Results
Withdrawal Behavior
Pricing and Sensitivities
4 Conclusions and Future Research
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 14 / 23 Results
Withdrawal Behavior
• Little withdrawal activity (approx. 12K per PH on average)I No withdrawals during accumulation periodI No premature withdrawals in 67% of casesI PH empties guarantee account in 6% of casesI < 1% chance of excessive withdrawal during contract phase
• Two main reasons to withdraw prematurely:I VA account below tax base (approx. 7K on average)
? Nuanced patterns? Interaction of in-the-moneyness of guarantee, tax considerations and
excess withdrawal charge
I VA account much greater than outside account (approx. 5K onaverage)? To reduce overall risk exposure (Merton ratio)
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 15 / 23 Results
Withdrawal Behavior
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
x 104
0
2
4
6
8
10
12
14
15x 10
4
Fig. 1: t=14, At = 180K , G.
t = 100K, H
t = 100K.
Xt−
wt
wt
max(Guarantee,Xt−)
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 16 / 23 Results
Withdrawal Behavior
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
x 104
0
2
4
6
8
10
12
14
15x 10
4 Fig. 3: t=10, At = 180K , G.
t = 100K, H
t = 100K.
Xt−
wt
wt
max(Guarantee,Xt−)
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 17 / 23 Results
Withdrawal Behavior
0 1 2 3 4 5 6 7 8 9 10 11 12 13
x 105
0
1
2
3
4
5
6
7
8x 10
4 Fig. 7: t=10, At = 20K , G.
t = 100K, H
t = 100K.
Xt−
wt,c
t
wt
Ct A
t− + I
t = 60K
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 18 / 23 ResultsPricing and Sensitivities
• Guarantee fee of φ = 50 bps sufficient to cover expected costs• In-the-moneyness appears to be OK proxy for pricing
I Different source to withdrawals
• Eliminating excess withdrawal fee increases net profits (win-win)
Base Case w/d if X−t ≤ G·t s = 0%
EQ[Fees] 6, 252 5, 925 5, 907EQ[Excess-Fee] 19 0 0EQ[Guarantee] 4, 558 4, 761 2, 136
%(Guarantee > 0) 24.34% 33.97% 31.94%
E[agg. w/d] 12, 084 14, 180 16, 374E[agg. w/d & t ≤ 6] 0 0 4, 374
E[agg. w/d & X−t ≤ Ht ] 6, 953 14, 119 6, 047E[agg. w/d & X−t > Ht ] 5, 030 0 5, 816
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 19 / 23 ResultsPricing and Sensitivities
• Withdrawal patterns highly sensitive to volatility
• Considering taxes important
BC: σ = 15% σ = 20% σ = 25% No Taxes
EQ[Fees] 6, 252 6, 047 5, 152 4, 734EQ[Excess-Fee] 19 62 1, 006 64EQ[Guarantee] 4, 558 8, 384 11, 533 4, 746
%(Guarantee > 0) 24.34% 36.48% 45.57% 28.16%
E[agg. w/d] 12, 084 29, 914 85, 879 88, 791E[agg. w/d & t ≤ 6] 0 54 13, 356 82
E[agg. w/d & X−t ≤ Ht ] 6, 953 17, 615 27, 877 19, 500E[agg. w/d & X−t > Ht ] 5, 030 10, 674 31, 221 69, 033
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 20 / 23 Conclusions and Future Research
1 Introduction
2 A Lifetime Utility Model for Variable Annuities
3 Results
4 Conclusions and Future Research
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 21 / 23 Conclusions and Future Research
• Develop lifetime-utility model to analyze withdrawal behavior for
VA with guarantees
• Numerically solve policyholder’s decision making problem inBlack-Scholes frameworkI Return-of-investment GMWB
• Infrequent withdrawals
• PH withdraws when VA account is below tax baseI Interaction of in-the-moneyness of guarantee, tax considerations
and excess w/d fee
• PH withdraws when VA account is largeI To lower overall risk exposure
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs
Page 22 / 23 Conclusions and Future Research
• Extend policyholder environment
I Unemployment Risk
I Subjective mortalities
• Withdrawal patterns highly sensitive w.r.t. volatility σ
I Stochastic volatility framework
• Alternatives to EUT
I Epstein-Zin preferences
I Correlation Aversion
Thorsten Moenig Policyholder Exercise Behavior for Variable Annuities including GMWBs