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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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The Design of Securitization for Longevity Risk
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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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.
20
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
average force of mortality, age 85-89
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
average force of mortality, age 90-94
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
average force of mortality, age 65-69
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
average force of mortality, age 70-74
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
average force of mortality, age 75-79
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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The Design of Securitization for Longevity Risk
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:
22
/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.
34
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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