180 S. Levantesi and M. Menzietti
Fig. 2.Expected value of annual insurance and patrimonial profit with safety loading = 10%
reduction of death probabilities, initial capitalK( 0 )=RBC 99 .5%( 0 , 1 )
Ta b le 4 .Moments ofu(t)and the finite time ruin probability with initial capitalK( 0 )=0,
safety loading = 10% reduction of death probabilities
u(T) T= 1 T= 5 T= 10 T= 20 T= 30
Mean (%) 0. 10 0. 73 1. 85 5. 40 9. 31
Std Dev (%) 0. 37 1. 26 4. 58 5. 27 6. 36
Skew 0. 4495 0. 2867 0. 0010 − 0. 0194 − 0. 0002
!u( 0 ,T)(%) 42. 06 58. 42 62. 37 67. 72 67. 79Figure 2 shows the expected values of annual profit components as stated in
Section 5. The insurance profit line shows greater variability, being affected by de-
mographic risks. Meanwhile, the patrimonial profit line is more regular due to the
absence of financial risk, and increases with time, depending on investments of risk
reserve (return produced by the investment of risk reserve).
In order to evaluate the effect of different initial capital provisions, we fix
K( 0 )=0. Further, according to ISVAP (the Italian insurance supervisory author-
ity), which shares the minimum solvency margin in life insurance to face demo-
graphic and financial risk in 1% and 3% of technical provisions, respectively, we fix
K( 0 )=1%V( 0 +). The moments ofu(t)distribution and the ruin probability are
reported in Tables 4 and 5. They can be compared with the results of Table 2.
Note thatK( 0 )values do not affect the standard deviation and skewness ofu(t)
distribution, while they do influence theu(t)expected value, which increases when
K( 0 )rises. Now, let us consider the highest safety loading given by a 20% reduction
of death probabilities for both healthy and disabled people. Values of the moments
ofu(t)and finite time ruin probabilities are reported in Table 6. If we compare these
values with the ones in Table 2 (where the safety loading is equal to a 10% reduction
of death probabilities), we find that safety loading strongly affects the expected values
ofu(t), but does not significantly affect the standard deviation and skewness. In other