Fair costs of guaranteed minimum death benefit contracts 2894.1 Impact of jumps and interest rates
The GMDB contract expiry is set at age 75. A guaranty cap of 200 % of the initial
investment is also among the terms of the contract.
The Gompertz mortality model is used in this subsection. The Gompertz param-
eters used in this subsection and the next one are those calibrated to the 1994 Group
Annuity Mortality Basic table in Milevsky and Posner [8]. They are recalled in Table 1.
Ta b le 1 .Gompertz distribution parametersFemale Male
Age (years) m b m b
30 88.8379 9.213 84.4409 9.888
40 88.8599 9.160 84.4729 9.831
50 88.8725 9.136 84.4535 9.922
60 88.8261 9.211 84.2693 10.179
65 88.8403 9.183 84.1811 10.282A purely diffusive model with a volatility of 20 % serves as a benchmark through-
out the study. It corresponds to the model used by Milevsky and Posner [8].
The particular jump diffusion model used in the following study is the one pro-
posed by Kou [5]. Another application in life insurance can be seen in Le Courtois
and Quittard-Pinon [6]. In this model, jump sizesJ=ln(Y)are i.i.d. and follow a
double exponential law:
fJ(y)=pλ 1 e−λ^1 y (^1) y> 0 +qλ 2 eλ^2 y (^1) y≤ 0 , (22)
withp≥0,q≥0,p+q=1,λ 1 >0andλ 2 >0.
The following Kou model parameters are set as follows:p= 0 .4,λ 1 =10 and
λ 2 =5. The jump arrival rate is set toλ= 0 .5. The diffusive part is set so that the
overall quadratic variation is 1.5 times the variation of the no-jump case.
Table 2 shows the percentage of premium versus the annual insurance risk charge
in the no-jump case and the Kou jump diffusion model case for a female policyholder.
A flat interest rate term structure was taken into account in this table and set atr=6%.
The initial yield curvey( 0 ,t)is supposed to obey the following parametric equa-
tion:y( 0 ,t)=α−βe−γtwhereα,βandγare positive numbers. The yield is also
supposed to converge towardsrfor longer maturities. The initial yield curve equation
is set as follows:
y( 0 ,t)= 0. 0595 − 0 .0195 exp(− 0. 2933 t). (23)
As stated earlier, the interest rate volatility structure is supposed to be of expo-
nential form. Technically, it writes as follows:
σP(s,T)=σaP
(
1 −e−a(T−s)