Mathematical and Statistical Methods for Actuarial Sciences and Finance

Fair costs of guaranteed minimum death benefit contracts 291

Ta b le 3 .Stochastic interest rates impact, male policyholder;g=5 %, 200 % cap

Purchase age Kou model (flat rate) Kou model (stochastic rates)

(years) (%) (bp) (%) (bp)

30 2.01 4.86 8.87 22.27

40 3.46 10.99 11.38 37.81

50 5.35 24.46 13.38 64.07

60 5.81 44.82 11.14 88.65

65 4.08 46.31 6.82 78.55

Gompertz mortality model. Ineach case, the left column displays the relative importance of the
M&E charges given by the ratioME()/S 0. The right column displays the annual insurance
risk charge.

Ta b le 4 .Gompertz (G) and Makeham (M) mortality model parameters for the USA [7]

ABC

GUS 6.148× 10 −^5 1.09159

MUS 9.566× 10 −^4 5.162× 10 −^5 1.09369

Ta b le 5 .Mortality impact on the annual insurance risk charge (bp), USA;g=5 %, 200 % cap

Gompertz Makeham

Age No jumps Kou (flat) Kou (stoch.) Kou (stoch.)

30 4.79 6.99 30.23 32.20

40 11.16 15.15 50.86 52.34

50 24.88 31.50 82.50 83.03

60 44.45 52.97 105.27 104.77

65 45.20 53.18 90.41 89.78

stochastic interest rates but changes the mortality model to a Makeham one. Figure 1
displays the annual risk insurance charge with respect to the purchase age in the USA.
From 30 years old to around 60 years old, the risk charge is steadily rising across all
models. It decreases sharply afterwards as the contract expiry approaches.
The two lower curves correspond strikingly to the flat term structure of the interest
rate setting. The jump effect is less pronounced than the stochastic interest rate effect
as represented by the two upper curves. The thin band in which these upper curves
lie shows that the change of mortality model has also much less impact than the
stochastic nature of interest rates.
As is reported in Table 5, and displayed in Figure 1, the behaviour of the insurance
risk charge with respect to age is of the same type whatever the considered model.
However, within this type, differences can be seen. First, the jump effect alone does not
change the fees very much but there are more differences when stochastic interest rates