Mathematical and Statistical Methods for Actuarial Sciences and Finance

Fair costs of guaranteed minimum death benefit contracts 287

A lengthy calculation shows that the characteristic exponentφT(u)ofXTunder
theT-forward measure, defined byEQT

[

eiuXT

]

=eφT(u), writes:

φT(u)=−iuT−iu 2 T^2 −u

2

2 

2

T+λT

(

φJ(u)−φJ(−i)

)

, (14)

whereφJ(u)denotes the characteristic function of the i.i.d. r.v.’sJi=ln

(

(Y)i

)

and

^2 T=

∫T

0

(

σ^2 − 2 ρσσP(s,T)+σP^2 (s,T)

)

ds. (15)

3.2 Present value of fees

Using the definition ofFtand (6), it can be shown that:

ME()= 1 −

∫∞

0

e−tfx(t)dt, (16)

wherefxis the p.d.f. of the r.v.T. A very interesting fact is that only the mortality
model plays a role in the computation of the present value of fees as seen in (16).
Taking into account the time to contract expiry date",wehave:

ME()= 1 −

∫"

0

e−tfx(t)dt−

(

1 −Fx(")

)

e−". (17)

3.3 Mortality models

Two mortality models are taken intoaccount, namely the Gompertz model and the
Makeham model. Another approach could be to use the Lee-Carter model, or introduce
a mortality hazard rate as in Ballotta and Haberman (2006). In the case of the Gompertz
mortality model, the force of mortality at agexfollows

λ(x)=B.Cx, (18)

whereB>0andC>1. It can also be written asλ(x)=^1 bexp

(

x−m

b

)

,wherem> 0

is the modal value of the Gompertz distribution andb>0 is a dispersion parameter.
Starting from (2), it can be shown that the present value of fees^1 in the case of a
Gompertz-type mortality model amounts to:

ME()= 1 −ebλ(x)e(x−m)

[



(

1 −b,bλ(x)

)

−

(

1 −b,bλ(x)e

"

b

)]

−ebλ(x)

(

1 −e

"

b

)

e−",

(19)

(^1) It is to be noted that formula (19) corrects typos in Milevsky and Posner’s (2001) original
article.