the design of securitization for longevity risk

8/14/2019 The Design of Securitization for Longevity Risk

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The Design of Securitization for Longevity Risk

longevity risk is modeling under a non mean-reverting feller process introduced in

Luciana and Vigna(2005). We value the longevity risk and calculate the transformed

distribution under Wangs method to consider the market price of longevity risk. A

securitization tranching example is illustrated and the mortality information is based

on the US mortality data observed in Human mortality data base.

Key words: Securitization, Credit Default Swap,

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I. Introduction

A. Longevity Risk

The term longevity risk has an opposite meaning of the term mortality risk. Both

of them are often used interchangeably. Mortality refers to the rate of death, while

longevity refers to the length of life. For insurers providing life contracts, sudden

mortality shocks will cause capital insufficiency. On the other hand, unexpected

decline in mortality may bring serious financial loss to pension plans. As mortality

has been improving for several decades, risk management of longevity for annuity

insurer becomes more and more important. In addition, social security reform and the

shift of pension plan from defined benefit (DB) to defined contribution (DC) pension

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plans increase the demand for commercial annuity products. It also makes longevity

risk a more significant issue consequently. Such risk is non-diversifiable. Therefore,

dealing with this potential mortality improvement risk exposure is crucial for either

social security system or private annuity providers. In this study, we introduce how to

manage longevity risk through securitization techniques.

B. Advantages of Securitization

Mortality based securities can be viewed as an alternative method to reinsurance.

However, the former has more advantages than the latter does. Dowd (2003) argued

that purchase of survivor bonds would enable insurance companies to lay off

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mortality improvement risk to a much wider range of counterparties at a lower

arrangement cost than traditional reinsurance. Denuit, Devolder, Godernaiaux (2007)

mentioned that many life insurance companies are less willing to buy reinsurance

covering for longevity risk because of its expensive price and potential credit risk of

the counterparty. Securitization can transfer mortality risk to capital market, which

has enough capital to absorb catastrophe mortality risk and a low correlation to

mortality risk. Lin and Cox(2005) also mentioned that holding capital by insurance

companies to meet regulation needs is very expensive. Therefore, insurance

companies will transfer that cost to policy holders, which will make them loss

competitive advantage. Transferring mortality risk through securitization can not

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only reduce large capital costs but also introduce a new investment opportunity to the

capital market.

C. Mortality Linked Securities in The Market

The first mortality based security was the mortality bond, with the face amount of

$400 million, issued by Swiss Re in Dec.2003 and matured on Jan. 2007. The bonds

principal is at risk and only if no catastrophe mortality risks happen, investors can get

back full amount at maturity. Lin and Cox (2005) used geometric Brownian motion

with jump to estimate parameters based on U.S. mortality data from 1900 to 1998 and

simulated future mortality index to price the Swiss Re. deal. They found that the

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Swiss Re. deal over-compensates investors because insurer behaves like a risk averter

when it faces unhedgeable risks. After the issuance of Swiss Re. mortality bond, in

Nov, 2004 BNP Paribas announced a long-term longevity bond for both U.K. pension

plans and other annuity providers to hedge their longevity risk. The bond had a 25-

year maturity with the face amount of 540 million. Its coupon payments were

linked to a survivor index based on the realized mortality rates of 65-aged English and

Welsh males in 2002. Although the issue attracted much public attention, it didnt

bring enough demand and was later withdrawn for redesign. Blake, Cairns, Dowd

(2006) noted some problems in this security. They found that a bond with a 25-year

maturity provides less effective hedge than a bond with a longer duration and that

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previously projected one, coupon payments to investors will reduce a portion equal to

the benefit paid to the insurer. On the contrary, investors may have more coupons if

annuitants die sooner than expected. As a result, the aggregate cash flow out of the

SPV annually is the same. If the insurance premium and proceeds from sale of the

longevity bond are sufficient, the SPV can buy a straight bond and have exactly the

required coupon cash flow it needs to meet its obligation to the insurer and the

investors. Denuit, Devolder, Godernaiaux (2007) used Lee-Carter framework to

model stochastic dynamic mortality for pricing the survivor bond. They supposed

that an insurer issues an index-linked floating coupon bond and collects these

proceeds to buy a fixed-rate coupon bond. The difference between fixed-rate coupons

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received and floating rate coupons paid out is set to be the same amount as the loss

portion of the insurer when annuitants live longer. Therefore, the additional amount

paid to annuitants can be simply covered via this transaction. Dowd, Blake, Cairns,

Dawson4 (2006) suggested survivor swaps as a more advantageous survivor derivative

than survivor bonds. They argued that survivor swaps are more tailor-made securities

which can be arranged at lower transaction costs and are more easily cancelled than

traditional bond contracts. Survivor swaps need only the counterparties, usually life

insurance companies, to transfer their death exposure without requirement of the

existence of a liquid market.

4 Readers can refer to D. Blake, A. J. G. Cairns and K. Dowd (2006), Living With Mortality:Longevity Bonds And Other Mortality-Linked Securities for more types of survivor derivatives.

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E. Mortality Model

Before going on to securitization, we must know first the trends of the underlying

risk. Since mortality has improved in an unexpected way in the past decades, we can

not project future mortality only based on the static mortality table. Besides Lee-

Carter (1992) mortality model, recent researches have proposed many other mortality

models. Blake, Cairns, Dowd(2005) use a two factor model for stochastic mortality to

fit English and Welsh males aged 65 from 1961 to 2002 and from 1982 to 2002

respectively. They found that the level of survivor index based on 1982-2002 data is

higher than that based on 1961-2002 data, which means mortality improves much

more significant in recent years. Luciano and Vigna (2005) used stochastic processes

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to model the random evolution of the force of mortality. They adopted both mean-

reverting and non mean-reverting processes to test the accuracy of the fit to the U.K.

population, and found that the mean-reverting processes were not adequate. It is true

that mortality wont improve to a certain degree and then turn to be worse.

F. Agenda

In this research, we first project future survivor probability of US mortality data under

the non mean-reverting Feller process introduced in Luciano and Vigna (2005). We

then value the longevity risk and calculate the transformed distribution under Wangs

method to consider the market price of longevity risk. Hereafter, the tranching

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technique is utilized to design a security for longevity risk. The methodology used in

this research is introduced below.

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)(1)()(00tFtTPtS T=>=

(1)

where0T

is the random variable that describes the life duration of an individual aged

0, and0T

F is its distribution function. )(tS means the probability that an individual

aged 0 will still survive to time t. So the survivor function can be rewritten as the

survivor probability,xt p

, where 0=x . The relationship between the survivor

function and the survivor probability is shown as follows:

)(

)()(

tS

txStTPp xxt

+=>= (2)

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Then, we propose a stochastic process to model the dynamic of force of mortality in

order to calculate the survivor probability. The force of mortality,x

, is the

instantaneous rate of change of mortality at agex , whichis defined as follows:

)(log xSdx

dx =

(3)

In other words,

)exp()exp(

)exp(

)(

)(

)exp()(

0

0

0

0

dtdt

dt

xS

txSp

dtxS

t

txx

t

tx

t

xt

x

t

+

+

=

=

+=

=

(4)

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The dynamic of term-structure force of mortality to forecast future survivor

probability is shown as follows:

FEL process )()()()( tdWtdttatd xxx +=(5)

where )(txin Luciano and Vigna (2005) indicates the force of mortality for a

specific cohort age x+tof an individual aged x at time t=0. However, we explain it

another way. We define that )(txis the force of mortality for any agex over the

time period t. That is, we look at the rows of mortality table. So the parametera,

which depicts the trend of mortality path, is negative. W(t) is a standard Brownian

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motion and0> . This process has an advantage that it will not generate the

negative value of, given that the initial value is non-negative.

With parameters, we can simulate different paths by the following equation:

+=

t

x

staat

xx sdWseet0

)( )()()0()( (6)

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B. Calibration to the US Mortality Table

Average Force of Mortality 1965-2003

1965 1970 1975 1980 1985 1990 1995 20000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

year t

forceofmortality

x=65-69

x=70-74

x=75-79

x=80-84

x=85-89

x=90-94

Figure 1: Average force of mortality for US males of agex=65-69, 70-74, 75-79, and so on.

The US mortality table is selected from Human Morality Database (HMD)5. HMD

has US mortality tables from 1946 to 2003, from which the trend of mortality

improvement under each age can be observed. We choose data after 1965 because

mortality rates before 1965 are more volatile and do not show an obvious trend of

5 Human Morality Database: http://www.mortality.org/

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http://www.mortality.org/http://www.mortality.org/


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improvement. For simplification, an assumption is made that the mortality of any age

improves at the same rate for every interval between age 65 and 69, age 70 and 74,

and so on. Therefore, we take average values of force of mortality at each interval

and estimate parameters in Eq.(5) based on these values. The method of least squares

is used to estimate the parameters of FEL process in fitting the rows of mortality table

from 1965 to 1999. Estimation result is shown in Figure 2 and Table 1.

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1965 1970 1975 1980 1985 1990 19950.09

0.1

0.11

0.12

0.13

0.14

0.15

year t

forceofmortality

average force of mortality, age 80-84

simulated value

observed value

1965 1970 1975 1980 1985 1990 19950.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

year t

forceofmortality


simulated value

observed value

1965 1970 1975 1980 1985 1990 19950.27

0.28

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.36

year t

forceofmortality


simulated value

observed value

1965 1970 1975 1980 1985 1990 19950.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042

0.044

year t

forceofmortality


simulated value

observed value

1965 1970 1975 1980 1985 1990 1995

0.04

0.045

0.05

0.055

0.06

0.065

year t

forceofmortality


simulated value

observed value

1965 1970 1975 1980 1985 1990 19950.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

year t

forceofmortality


simulated value

observed value


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Figure 2: Fitness of FEL process to US mortality table, 1965-1999. The broken line is the observed

average force of mortality between age intervals, and the continuous line is force of mortality generated

from FEL process. FEL process is best fit to younger ages because the observed data are smoother

curves. Whereas the observed data for older ages are more volatile, FEL process is not fit well to these

ages.

Table 1: Estimate of parameters for US males 1965-1999.

age 65-69 70-74 75-79 80-84 85-89 90-94

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a -0.01500 -0.01230 -0.01150 -0.01140 -0.00950 -0.00644

0.00860 0.01100 0.01387 0.01750 0.02340 0.03030

SSE 0.00005 0.00029 0.00022 0.00034 0.00099 0.00575

With parameters above, the projection of future force of mortality can be easily

obtained via Eq.(6). We take the force of mortality for ages 65-94 in 2000 mortality

table (t=0) as initial values in FEL process and simulate 100,000 paths for each age

x=65,,94 through time t=0,,29. Then, the expected values of ),2000(65

),2001(66, )2029(94

are chosen and calculated by Eq.(4) to represent the

expected survivor probability of an individual aged 65+twith an initial age x=65 at

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t=0. It is considered a good approximation for survivor probabilities of a specific

generation at any age.

Forecast of Force of Mortality

2000 2005 2010 2015 2020 2025 2030

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

year t

forceofmortality

observed value

95% confidence interval bound

forecasting value

95% confidence interval bound

Figure 3: Forecasting force of mortality of US males aged 65 in 2000 to aged 94 in 2029.

MAPE=0.011966984.

Future Survivor Probability of US Males aged 65 in 2000

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2000 2005 2010 2015 2020 20250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

year t

survivorprobability

projection of survivor probability

survivor probability based on Wang's Transformation

Figure 4: Future survivor probability of US males aged 65 in 2000. Broken line plots future survivor

probability calculated by Eq.(4). Continuous line plots survivor probability based on Wangs

Transformation at =20% (market price of risk).

III. Securitization of Longevity Risk

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As mortality improves significantly, annuity providers would suffer a great loss when

the expected death rates of annuitants are lower than the realized ones. In this section,

we will first value the longevity risk and then introduce a long-term longevity bond

utilizing structure financing technique to transfer the risk to the capital market.

A. Valuation of Longevity Risk

An insurer (or annuity provider) suffers losses if the realized death rates of annuitants

improve. Those losses can be described by Eq.(7).

=0

)(* ttt

XXBL , if

tt

tt

XX

XX

(7)

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wheretL: The loss (extra payment to annuitants) the insurer suffers at time t.

B

: Level annuity payment to each annuitant.

tX: Actual number of survivors at time t.

tX: Expected number of survivors at time t.

t: time =1,2,,30.

We can calculatetX

by multiplying the number of annuitants aged 65 at time 0 (that

is,0X) with the survivor probability,

65pt, projected under FEL process. In order to

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manage its longevity risk, the insurer may want to transfer this potential loss to an

SPV via reinsurance. The reinsurance premium is equal to the present value of

expected loss under FEL process and is calculated as follows:

=

=T

t

t tdLEELPV

1

),0()()( (8)

TaELPVP /)(= (9)

where )(ELPV : Present value of expected loss.

),0( td : Default free zero coupon price with face value of $1 at maturity time t.

P: Level premium paid to SPV annually.

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Ta : Present value of T-period $1 annuity-due.

T: A horizon for 30 years.

In our example, we define that the SPV collects reinsurance premium annually and

distributes them to investors as coupon payments. We will come back to this point

later, when presenting the design of longevity bond. We assume the number of initial

cohort0X is 10,000 and annuity payout per person is $1,000 each year.

B. Wang Transform

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Sharpe ratio, describing excess return per unit of risk, is a measure of risk-adjusted

performance for assets in complete market. It is also called market price of risk.

However, Sharpe ratio can not be applied to measuring market price of insurance risk

because insurance products are not traded in the market. Fortunately, Wang (2000,

2001) has proposed a method of pricing risks that unifies financial and insurance

pricing theories. The method is called Wang transform which is widely used in

insurance applications. Given thatF

is the distribution function of survivor

probability, and its transformed distribution function based on Wang method is

described as following:

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)))((()(1* =

xFxF (10)

where is the standard normal cumulative distribution function and is the market

price of risk for insurance products. We set equal to 20%6 and calculate the

expected loss )( tLEin Eq.(8) under the transformed distribution. The transformed

survivor probability is denoted as *65pt

which is shown in Figure 4. Based on

transformed distribution, the risk premium for investors to bear longevity risk is

6 can be calculated given the market price of annuity. As we can not find market price of annuity, isset exogenously and is a reasonable estimate of because Lin and Cox (2005) estimated to be 0.1792for male annuitants and 0.2312 for female annuitants by using US mortality data.

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considered in the expected loss, so we can discount it at risk-free interest rate and

calculate reinsurance premium in Eq.(9). Assuming that the risk-free rate follows

CIR stochastic interest rate model:

dztrdttrbatdr )())(()( += (11)

under risk-neutral probability transition, we have zero coupon bond price at t=0:

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/2

2/)(

),0(

2

2)1)((

)1(2),0(

]2)1)((

2[),0(

*),0(),0(),0(

2

+=

++

=

++=

==

+

a

ea

etB

ea

etA

etAtztz

t

t

abt

ta

rtBQ

(12)

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Eq.(12) is the discount factor in our simulation with parameters 0.2339=a ,

0.01889=b , 0.0854= from Chan, et al. (1992).

C. Design of Longevity Bond

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Structure of Securitization of Longevity Risk

Figure 5: Structure of Securitization of Longevity Risk. The SPV receives reinsurance premium and

issues several bonds with different ratings to investors. The premium is transferred to investors as

coupon payments and the principal is responsible for the loss of an insurer.

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ContingentPayment (L

t)

Ct

ProceedsHigh qualitybond

Aa1Aa3A3Equity

SPVInsurer

Premium


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The structure of this securitization is the same as other asset-based securities. For an

insurer and SPV (Special Purpose Vehicle), the insurer pays a level premium annually

to the SPV and purchases reinsurance from it. It is similar to set a credit default swap

(CDS) between two counterparties. The SPV then issues securities to investors,

depositing proceeds in a high quality bond collateral, and distributes the premium

paid by the insurer to them as coupon payments, denoted astC

in Figure 5. In our

example, we calculate the reinsurance premium via Eq. (8), Eq.(9) under Wangs

transformed survivor rate distribution. For investors of Aa1, Aa3, and A3 bond, they

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can receive a coupon rate of LIBOR+5bp, LIBOR+10bp, and LIBOR+15bp

respectively.

The different design from other mortality-based securities in our research is that the

credit tranche technique is applied in pricing the longevity bond. Credit tranching is

usually used in asset securitization such as Collateralized Bonds Obligation (CBO)

and Collateralized Debt Obligation (CDO). Banks can sell receivables, bonds or

debts, and issue CBO or CDO to transfer credit risk and enhance liquidity of their

assets. The concept of credit tranching is to divide a portfolio of asset into several

bonds of different ratings according to their expected loss rates. The most junior

tranche, usually equity tranche, takes losses first when the portfolio starts to lose.

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Then, A3 tranche, in our example, takes loss in turn when equity tranche is retired,

and so forth. We will explain this method by the following example.

Assuming that an insurer must pay immediate life annuities to 10,000 annuitants all

aged 65 at time 0. The payment rate is $1,000 per annuitant per year. The cumulative

premium is equal to the present value of future annuity payout, which is calculated

under expected survivor rate from FEL process. Therefore,tL

is denoted as the

additional annuity payout of an insurer at each time t when actual survivor rate is

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higher than expected one.0L

is zero because the initial value of force of mortality in

FEL process is given in 2000 mortality table and consequently each simulation path

yields the same value at t=0. The expected loss distribution is plotted in Figure 6.

Expected Loss Distribution

2000 2005 2010 2015 2020 2025 20300

0.5

1

1.5

2

2.5

3x 10

5

year t

expectedloss

expected loss

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Figure 6: Expected loss from 2000 to 2029. The standard deviation of survivor rate goes up as time

passes but then goes down after time t=27; as a result, losses at older ages are larger and then become

smaller. Sum of expected loss over 30 years is $4,951,800.

Loss Rate Probability Distribution

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

loss rate

probabilitydensity

loss rate distribution

Figure 7: Loss rate probability distribution. The expected loss rate of the portfolio is 9.9037% by

integrating this distribution. Critical points can be calculated under this distribution to divide the

portfolio into four tranches. Integrating within the range between critical points yields sustainable loss

responsible by each tranche.

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For the longevity bond, we assume total face value of this contract is $50,000,000,

and the SPV plans to divide the portfolio into four tranches to issue three 30-year

bonds of different-rating and holds the equity tranche. We can calculate the loss rate

by dividing the sum of loss amount under each simulation path by the face value. The

loss rate probability distribution is shown in Figure 7. By integrating this probability

distribution, we can get the expected loss rate of the portfolio, which is 9.9037%. In

Table 2, the tranche weights of the portfolio and their rating and are exogenously

given according to hypothetical market demands. The expected loss rates of Aa1,

Aa3, and A3 tranches for a horizon 30 years are 0.2066%, 0.7876%, and 3.3440%

respectively, which are extended from Moodys idealized expected loss rate table7.

7 As we can not find the Moodys expected loss rate table for 30 years, we approximate it via linearinterpolation.

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Sustainable loss is the portion of total expected loss responsible by each tranche. It is

calculated by multiplying the total expected loss rate by each tranche weight. The

tranche weight of A3 and equity tranche is unknown, but we can solve them under

two constraints. First, the sum of total tranche weight is equal to 1. Second, the total

sustainable loss must equal to the expected loss rate under the portfolio. The

solutions are highlighted in bold in Table 2 below.

Table 2:Example of tranching longevity bond.

Tranches A B C Equity Total

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Rating Aa1 Aa3 A3 N.A.

Tranche Weight 50% 20% 21.0614% 8.9386% 100%

Expected Loss Rate 0.2066% 0.7876% 3.3440% 100%

Sustainable Loss 0.1033% 0.1575% 0.7043% 8.9386% 9.9037%

Example of Tranching Longevity Bond

Figure 8: Example of tranching longevity bond. Critical points for tranche A, B, and C are 84.7890%,

49.4660%, and 43.6770% respectively.

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Then, we can calculate critical points that divide the portfolio into four tranches by

considering together the required expected loss rate under each tranche and the loss

rate distribution. Critical points for A, B, and C tranche are 84.7890%, 49.4660%,

and 43.6770% respectively. Therefore, equity tranche, tranche C, tranche B, and

tranche A are responsible for portfolio loss under first 43.6770%, between 43.6770%

and 49.4660%, between 49.4660% and 84.7890%, and above 84.7890% respectively.

The concept is diagramed in Figure 8 to give a more clear explanation.

IV. Conclusion and Discussions

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Mortality has been improving around a long time but the importance of the issue has

only been fully emphasized recently. The trend of privatizing social security systems

is increasing the market demand for annuities and consequently increasing the

longevity risk that insurance companies bear. It is not only a question of risk

management to insurance companies but also a critical problem for policyholders if

insurance companies fail to make annuity payments. Securitization is recently

proposed to be a good risk management tool to mitigate longevity or mortality risks.

It helps companies reduce the cost of holding capital for regulation needs and also

enhance their competitive advantages.

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Our study shows how to project survivor rate by Feller process considering the rows

of US population mortality table rather than the diagonal of it. It can be more tailor-

made for insurance companies to project survivor rate based on their own experience

mortality tables. Our projection result and historical data both indicate that the force

of mortality is more volatile for people at older ages who account for a large portion

of portfolio loss. Besides, the accuracy of estimating force of mortality for older ages

does not perform as well as it does for younger ages. For lack of information, the

market price of risk is approximated as 20% on which Wangs transform survivor

distribution is based when we consider pricing securities in an incomplete market.

What we concern here is the longevity risk inherent in only one policy of a specific

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generation. Further studies may consider managing risk over a portfolio of annuity

products which may encompass several survivor distributions of different generations

and different annuity payment schemes. The credit tranche technique is first applied

here to longevity risk securitization. It has an advantage for a portfolio originally

rated under investment degree to be issued for higher ratings. Besides, it has a

feature of providing bonds of different yields and risks to meet the needs of various

investors. Tranches in our example have the same maturity of 30 years, but this

design can be extended to consider different maturities, for instance the most senior

tranche can be repaid earlier than junior tranches. Another assumption made here is

that we ignore credit risk of insurance companies which means the risk that

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companies fail to pay the reinsurance premium. If companies do not pay the

reinsurance premium, investors can not receive a portion of coupons. Therefore,

investors should require additional risk premium for bearing such risks. Further

studies may take this issue into consideration.

Securitization of longevity risk can be of many forms. They can be survivor swaps,

longevity futures or longevity options. A portfolio of annuity products securitized can

encompass policies over different nations, types and generations. Besides insurance

companies, longevity bonds can also be issued by other institutes or governments with

coupons linked to realized survivor rate. This kind of longevity bonds are designed to

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be purchased by insurance companies or pension plans. Securitization of longevity

risk is not only a good method for risk diversifying, but also provides low beta

investment assets to the capital market. To date, mortality or longevity derivatives are

in its very early stages, but with its attractive features we can expect that this

innovation will be a popular instrument in the future.

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Bonds And Other Mortality-Linked Securities, Presented to the Institute of

Actuaries, 27 Feb 2006.

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the design of securitization for longevity risk

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