6. Numerical Illustrations
In this paper, endowment policies maturing at 10 years after the issuance are issued to 10,000 policyholders under the following conditions respectively: the initial interest rate r0 is 6% when the policy is issued; the force of reversion
is 0.1; the long-term mean interest rate is 8%; the volatility parameter of interest rate is 1%; the leveled assumed interest rate r announced to the policyholders is 6%; the death t benefit is 1000 dollar paid to each dead policyholders who die before the insurance matures; the endowment benefit is 1000 paid to those policyholders survival to the maturity date; the parameter corresponding to the surrender cash value is 0.9; the parameter for the liquidity risk is 1%.The above situation is considered to be the parameters used by the insurance company when computing the premium and reserves by Monte Carlo method.
Sometimes things do not act like what we feel like. We would like to see what will happen is these situation changes as followed right after the insurance company issue the contracts to 30-year-old policyholders and 80-year-old policyholders:
1) Long-term interest rate : 8%;
2) volatility parameter of interest rate : 1%;
3) force of reversion
: 0.1;4) parameter for surrender cash value : 0.8;
5) parameter for liquidity risk : 1%.
6.1. Sensitivity to Long-Term Interest Rate
Figure 1 shows that the surplus will be enormously effected result from the
deviation of the long-term interest rate. As the long-term interest rate is set to be 8%
when the endowment contract is issued, the premium and the reserve is computed under
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this long-term mean. If the long-term mean level goes to 10% right after the issue date, the interest rate will grow higher than it was expected, which will lead to the increased number of surrender. Since the surrender cash value is usually set to be less than the reserve at the time policyholders surrender, greater number of surrender will make the insurance company more profitable. Thus, the simulated surplus under 𝜃=10% will be greater than that of 𝜃=8%. On the other hand, if the long-term mean decrease to 6%, then the interest rate will tend to decrease more than expected. Therefore the surplus also decreases as time passes. These phenomena appear no matter when the endowment policies are issue to 30-year-old policyholders or 80-year-old ones.
As the long-term interest rate increases with other parameters fixed, the reinvestment income of the insurance company also increases. Therefore, under such circumstances will the value of the total asset held by the insurer more possibly be greater than its liabilities. This will lead to the decrease of probability of insolvency shown in Figure 2.
We consider the situation that the long-term mean level of the interest rate increase to 10%. The increased growth of interest rate will lead to the unexpected increasing number of surrender, since exercising the surrender options is a rational decision if the market interest rate grows above the assumed interest rate. As a result, more surrender paid will be announced and the insurance company will be necessary to sell the immature assets for these unexpected call for cash.
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Figure 1. Sensitivity of Expectation of Surplus per Policy to Long-Term Interest Rate.
Figure 2. Sensitivity of Probability of Insolvency to Long-Term Interest Rate.
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Figure 3. Sensitivity to Probability of Illiquidity to Long-Term Interest Rate
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6.2. Sensitivity to Volatility Parameter of Interest Rate
As the parameter of volatility increases, the interest rate will be more volatile than expected. For the analysis of surplus from the four risks, we find the evidence that the surplus form interest risk and surrender risk decreases largely when the volatility of interest rate grows high. This is because the policyholder may surrender more severely than expected in the first few years. This will lead to the increasing cash outflow for surrender in these years and thus result to the increase in loss of liquidity risk for the insurance company. This is shown in Figure 4.
In the case that the endowment policies are issued to 30-year-old policyholders, as the volatility of interest rate rises, the path of interest rate is more unlikely act as it is expect when the policies were issued. Thus, the investing decision that the insurance company made become unsuitable, which will make the probability of insolvency increase. On the other hand, if the volatility is not so high as it was expected, the path
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of interest rate is more likely to stay in the smaller interval. Therefore, although there is a little decrease in investment income, the investing decision still works, leading to the decrease in the probability of insolvency in Figure 5.
The surrender rate cannot be determined directly from the volatility of the interest rate, however, its asymmetric property lead the increasing surrender paid to policyholders when the interest rate is more volatile. Thus, the overall impact on the probability of illiquidity is higher when the volatility is higher for both cases.
Figure 4. Sensitivity of Expectation of Surplus per Policy to Volatility Parameter of Interest Rate.
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Figure 5.Sensitivity of Probability of Insolvency to Volatility Parameter of Interest Rate.
Figure 6. Sensitivity of Probability of Illiquidity to Volatility Parameter of Interest Rate.
6.3. Sensitivity to Force of Reversion
If the force of reversion rises, it cost shorter time for the interest rate reach the
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policies are issued to 30-year-old policyholders and 80-year-old-policyholders shown in Figure 7.Figure 8 shows that the probability of insolvency rises when the force of reversion
of interest rate increases in both cases. On the other hand, if the force of reversion is lower than it is expected, the number of surrender is also lower. Thus, the surrender payment which is paid to policyholders decreases in both cases that the policies are issued to 30-year-old policyholders and 80-year-old-policyholders.
Similar to the reason above, when the force of reversion of interest rate increases, the possibility that the insurance company is short in cash rises. This is illustrated in Figure 9.
Figure 7. Sensitivity of Expectation of Surplus per Policy to Force of Reversion.
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Figure 8. Sensitivity of Probability of Insolvency to Force of Reversion.
Figure 9. Sensitivity of Probability of Illiquidity to Force of Reversion.
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6.4. Sensitivity to Parameter for Surrender Cash Value
The parameter is associated with how much proportion of reserve can a surrendering policyholder get back when he or she decides to surrender at the end of a certain policy year. The more proportion the insurance company needs to pay back, the less profitable is. In Figure 10, we can easily find out that the expectation of surplus decreases when the parameter increases in both cases that the policies are issued to 30-year-old policyholders and 80-30-year-old-policyholders.
Clearly, the probability of insolvency increases when the proportion of reserve paid back to the surrendered policyholders increase. This phenomenom happens in both cases and is illustrated in Figure 11.
We can find out from Figure 12 that the parameter of surrender cash value effect mainly at the beginning few policy years after the issue date. This is because that the surrender options are exercised mostly during this period.
Figure 10. Sensitivity of Expectation of Surplus per Policy to Parameter for Surrender Cash Value.
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Figure 11. Sensitivity of Probability of Insolvency to Parameter for Surrender Cash Value.
Figure 12. Sensitivity of Probability of Illiquidity to Parameter for Surrender Cash Value.
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6.5. Sensitivity to Parameter for Liquidity Risk
The parameter decides the discount level of the zero coupon bond price when the asset is necessarily be sold in a short period. Clearly, the insurance company may loss much more value under larger gamma. Thus, the decreasing expectation of surplus when the parameter rises in both cases are shown in Figure 13.
As the loss from liquidity risk rises, which is resulted from the increased parameter for liquidity risk, the increasing probability of insolvency of the insurance company for both cases is illustrated in Figure 14.
Since the parameter effect mainly on the level of discount of the assets sold, the probability of illiquidity do not change largely in Figure 15.
Figure 13. Sensitivity of Expectation of Surplus per Policy to Parameter for Liquidity Risk.
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Figure 14. Sensitivity of Probability of Insolvency to Parameter for Liquidity Risk.
Figure 15. Sensitivity of Probability of Illiquidity to Parameter for Liquidity Risk.
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6.6. Sensitivity to Long-term Interest rate at Different Initial Interest rate
Additional to the interest rate starting from 8%, we would also like to know how the sensitivity of the surplus distribution of the endowment contracts to the long-term interest rate at lower initial interest rate: 3%.
When the interest rate increases more than it is expected, the rate of surrender will grow higher, which may cause the endowment contract gain from the surrender penalty given to surrendered policyholders. From Figure 16 we can clearly find out that the expectation of surplus rises as the long-term interest rate increases, which acts likely as when the initial interest rate is 6%. The expectation of surplus is similar when the endowment contracts are issued to 80-year-old policyholders.
If the long-term interest rate increases, the probability that occurring negative surplus for the endowment contracts decreases. This is shown in Figure 17. According to Figure 18, the probability that occurring liquidity problem rises in the second policy year, and then decreases in the following policy years.
Figure 16. Sensitivity of Expectation of Surplus per Policy to Long-term Interest Rate at Different Initial Interest Rate.
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Figure 17. Sensitivity of Probability of Insolvency to Long-term Interest Rate at Different Initial Interest Rate.
Figure 18. Sensitivity of Probability of Illiquidity to Long-term Interest Rate at Different Initial Interest Rate.
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