Mathematical and Statistical Methods for Actuarial Sciences and Finance

178 S. Levantesi and M. Menzietti

whileYI( 0 ,T)=

∑T

t= 1 Y

I(t)vρ( 0 ,t)andYP( 0 ,T)=∑T

t= 1 Y

P(t)vρ( 0 ,t)are the

present value of the future insurance and patrimonial profits, respectively.

6 Portfolio simulation results

Let us consider a cohort of 1000 policyholders, males, with the same age at policy
issue,x =65, same year of entry (time 0), a maximum policy durationω=49,
expense loadingsα=5%,β=2%,γ= 0 .7% and a constant interest ratei( 0 ,t)=
i=3%∀t. The annual benefit amounts are distributed as in Table 1.

Ta b le 1 .Annual benefit amounts distribution (euros)

b 1 b 2 fr(%)

6,000 12,000 40

9,000 18,000 30

12,000 24,000 15

15,000 27,000 10

18,000 30,000 5

Results of 100,000 simulations are reported in the following tables, assuming a
safety loading on demographic pricing bases given by a 10% reduction of healthy and
disabled death probabilities and an initial capitalK( 0 )=RBC 99 .5%( 0 , 1 ). Simulated
values ofu(t)are shown in Figure 1. The figure highlights the strong variability of the
risk reserve distribution, especially whent>5, as a consequence of demographic
scenario changes. Even though the risk reserve has a positive trend due to safety
loading, lower percentiles are negative. Economic consequences of such an aspect are

Fig. 1.u(t)with safety loading = 10% reduction of death probabilities, initial capitalK( 0 )=
RBC 99 .5%( 0 , 1 )