Managing demographic risk in enhanced pensions 179Ta b le 2 .Moments ofu(t)and the finite time ruin probability with initial capitalK( 0 )=
RBC 99 .5%( 0 , 1 ), safety loading = 10% reduction of death probabilities
u(T) T= 1 T= 5 T= 10 T= 20 T= 30
Mean (%) 0. 79 1. 51 2. 75 6. 61 10. 94
Std Dev (%) 0. 37 1. 26 4. 58 5. 27 6. 36
Coeff Var 0. 4673 0. 8372 1. 6614 0. 7974 0. 5817
Skew 0. 4495 0. 2867 0. 0010 − 0. 0194 − 0. 0002
!u( 0 ,T)(%) 0. 50 15. 06 28. 98 40. 67 40. 84relevant for the insurer solvency and will be quantified through solvency requirements.
In Table 2 we report the values of theu(t)moments and the coefficient of variation as
well as the finite time ruin probability. It can be noticed that expected values ofu(t)
are always positive and increase with time as well as the standard deviation. Looking
at the coefficient of variation we observe an increase of relative variability up to
t=10; thereafter it decreases. Such a behaviour demonstrates that demographic risk is
mainly caused by the scenario changes (perceived with a delay of 5 years) affecting the
evaluation of technical provisions. When technical provisions decrease, the coefficient
of variation ofu(t)becomes steady. The risk tendency to become stable is confirmed
by the finite time ruin probability values that increase with time. As expected,!u( 0 , 1 )
is consistent with the initial capital provision,K( 0 )=RBC 99 .5%( 0 , 1 ).
Table 3 shows the values of RBC requirements for three different confidence
levels: 98%, 99% and 99.5%. RBC values rise with time and become steady inT= 20
only if RBC is computed on a time horizon( 0 ,T), rather than at timeT. On the other
hand, if we look at RBC computed according to VaR, we obtain lower values with
respect to the previous ones, especially forT>10. Results show that the initial
capital should be increased by about 6% of the single premium income to guarantee
the insurance solvency on the portfolio time horizon.
Ta b le 3 .Risk-based capital with safety loading = 10% 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 .67% 1 .78% 6 .52% 6 .57% 6 .57%
= 1 .0% 0 .62% 1 .62% 6 .20% 6 .24% 6 .24%
= 2 .0% 0 .55% 1 .43% 5 .91% 5 .94% 5 .94%
rbcVaR 1 −( 0 ,T) T= 1 T= 5 T= 10 T= 20 T= 30
= 0 .5% 0 .67% 1 .77% 6 .03% 4 .07% 2 .63%
= 1 .0% 0 .62% 1 .60% 5 .75% 3 .61% 2 .12%
= 2 .0% 0 .55% 1 .39% 5 .30% 3 .01% 1 .48%