Mathematical and Statistical Methods for Actuarial Sciences and Finance

Managing demographic risk in enhanced pensions 181

Ta b le 5 .Moments ofu(t)and the finite time ruin probability with initial capitalK( 0 )=
1%V( 0 +), safety loading = 10% reduction of death probabilities

u(T) T= 1 T= 5 T= 10 T= 20 T= 30

Mean (%) 1. 13 1. 89 3. 20 7. 21 11. 74

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)(%) 0. 00 7. 02 24. 18 36. 69 36. 82

Ta b le 6 .Moments of capital ratio and the finite time ruin probability with initial capitalK( 0 )=
RBC 99 .5%( 0 , 1 ), safety loading = 20% reduction of death probabilities

u(T) T= 1 T= 5 T= 10 T= 20 T= 30

Mean (%) 0. 78 2. 07 4. 49 12. 01 20. 39

Std Dev (%) 0. 37 1. 26 4. 50 5. 25 6. 20

Skew 0. 4544 0. 2892 0. 0120 − 0. 0197 − 0. 0038

!u( 0 ,T)(%) 0. 50 7. 07 22. 21 30. 56 30. 56

Ta b le 7 .Risk-based capital with safety loading = 20% reduction of death probabilities, initial
capitalK( 0 )=RBC 99 .5%( 0 , 1 )

rbc 1 −( 0 ,T) T= 1 T= 5 T= 10 T= 20 T= 30

= 0 .5% 0 .57% 1 .23% 5 .39% 5 .39% 5 .39%

= 1 .0% 0 .52% 1 .09% 5 .14% 5 .14% 5 .14%

= 2 .0% 0 .46% 0 .92% 4 .88% 4 .88% 4 .88%

rbc 1 VaR−( 0 ,T) T= 1 T= 5 T= 10 T= 20 T= 30

= 0 .5% 0 .57% 1 .18% 4 .52% 0 .97% − 1 .52%

= 1 .0% 0 .52% 1 .02% 4 .24% 0 .49% − 2 .02%

= 2 .0% 0 .46% 0 .81% 3 .78% − 0 .12% − 2 .65%

words, safety loading reduces the probability of risk reserve to become negative (as
provedbythe!u( 0 ,T)values), but does not lower its variability. Moreover, required
capital decreases with safety loading increase (see Table 7 compared with Table 3).
Note that in the long term therbcVaRare negative, therefore an initial capital is not
necessary to guarantee the insurance solvency. Nonetheless, the requirement reduction
is financed by the policyholders through a premium increase – due to a higher safety
loading – making the insurance company less competitive on the market. Therefore,
it is important to combine a solvency target with commercial policies.
Now, let us consider the safety loading impact onY( 0 ,T)discounted at a rate
ρ=5%>i(see (11)). As expected,Y( 0 ,T)rises with an increase of safety loading:
the values – expressed as a ratio to total single premium income – move from 6.21%
to 11.85% when the safety loading rises. It is worth noting that looking at the profit