14th year Decision table 15th year Decision table Account
The optimal weight of the high risk asset at the beginning of the first year is 20.69% of the account value.
Table 4.2 The initial weight of 100 times of 10000 different senarios.
High weight Min max mean Std. Confidence Interval
T=0 weight 0.0782 0.3188 0.2069 0.0533 (0.1982,0.2157)
4.2 Analyst of result
The number in row 5 of the 15th year decision table is 133.2606 | 0.5474. This represents that the account value is 133.2606 in the beginning of year 15, and the optimal utility weight of high risk asset is 0.5474; The number in row 10 in the 10th year decision table is 123.0092 | 0.3862. This represents that the account value is 123.0090 in the beginning of year 10, and the optimal utility weight of high risk asset is 0.3862.
In the 15th year decision table, the only circumstances that the weight is 1 is when the
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value 2.0027 is very low, and it has very large probability to fall below the guaranteed value (two dollars) in the end of year 15. But even if the account value dropped below 2 dollars, the policyholder can still get guaranteed value. In this guaranteed condition, the policyholder will invest higher proportion in high-risk asset. In other account values of 15th year decision table, the weights of high-risk asset are the same. This is due to the power utility function assumption. In a almost no guaranteed situation, the weight of any account value will be the same.
In the other year decision tables, the weight of high risk asset goes down first and goes up as the account value goes up. When the account value becomes smaller, and it could be less than the guaranteed value in the future. In this situation, the weight of high risk asset will be weighted higher. As the asset value becomes smaller, the weight of high risk asset will close to 1. Because the investment account is guaranteed, the policyholder can tolerance asset value fell below the guaranteed value. However, we cannot find a definite reason to explain the phenomena the weight of high risk asset increases as the asset value increases.
One of the explanations comes from the fixed withdrawal of the policyholder at the beginning of each year. When the account value is low, the ratio of withdraw amount to the account value is big; When the account value is high, the ratio of withdraw to the account value is small. We assume three different withdraw amounts which are equal to 1, 7, and 10, and examine the two period strategy from year 13 to year 15. Table 4.2.1 displays results.
The account values at the beginning of year 14 are assumed to range from 21.47 to 248.16, and the policyholder will withdraw now and one year later. Based on the scenario, we then simulate 10000 scenarios and calculate the account value at the beginning of year 15 and the
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optimal weight of high risk asset. We find that the optimal weight is related to the volatility of the ratio of withdraw to the account value. If volatilities of the ratio of withdraw to the account value among different cases are close, then the corresponding weights are close and similar. When the withdraw amount is equal to 1 and the volatility of the ratio is 0.00809, the weight is 0.4026 (in row 1). This set of values (0.00809, 0.4026) is between row 11 and row 12 when the withdraw amount is 7 and between row 15 and row 16 when the withdraw amount is 10. The volatility of the account asset will become larger when the ratio of withdraw to the account value increases. The ratio:
To the policyholder‟s point of view, the volatility of the account value will go up as the volatility of the ratio of withdrawal to his own account value goes up. As a result, the policyholder will put lower proportion on high-risk asset. This can explain why weights of high-risk asset will go up as account values go up in the case of high asset values.
Table 4.3 The explain of decision table
Account value at t=14
Withdraw=1 Withdraw=7 Withdraw=10
Mean
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9 113.02 0.0083 0.00149 0.4225 0.0617 0.0109 0.3940 0.0907 0.0159 0.3784 10 125.99 0.0075 0.00134 0.4230 0.0550 0.0097 0.3976 0.0806 0.0142 0.3839 11 140.01 0.0067 0.00120 0.4234 0.0492 0.0087 0.4007 0.0719 0.0127 0.3886 12 156.29 0.0060 0.00108 0.4238 0.0438 0.0078 0.4036 0.0639 0.0113 0.3928 13 172.52 0.0054 0.00098 0.4241 0.0395 0.0070 0.4059 0.0575 0.0102 0.3962 14 192.14 0.0049 0.00088 0.4244 0.0353 0.0063 0.4081 0.0513 0.0091 0.3996 15 213.65 0.0044 0.00079 0.4246 0.0317 0.0056 0.4101 0.0459 0.0081 0.4025 16 238.16 0.0039 0.00071 0.4249 0.0283 0.0050 0.4119 0.0410 0.0073 0.4051
In addition, we calculate the guaranteed cost by the decision table. We use the same asset assumption with our benchmark. The discount rate is 0.0308, which is the interpolation3 of the 10-year bond yield and 30-year bond yield at 2011/1/1. We simulate 10000 scenarios and calculate the cost of each scenarios. The average cost is about 1.29 and the other results are listed in Table 4.4
Table 4.4 The cost in 10000 different scenarios
initial Weight Avg. of cost var. of cost VaR.90% CTE70 Prob. Of ruin
0.2069 1.2949 24.1715 0 4.3164 0.0933
When the insurance companies price the VA with GMWB, they almost assume that the policyholders put all their money on the high-risk because they can avoid the deficit.
However, not all of the policyholders will allocate all their money in risky asset. In this condition, the guaranteed cost will be overestimated, and it is even 383% greater than our benchmark (see table 4.5). We believe that this approach is not reasonable because the policyholders will consider their own utility function and decide the asset allocation.
3 The interpolation equation : 0.75* yield of 10-year bond in U.S + 0.25*yield of 30-year bond in U.S at 2011/1/1
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equal to our benchmark‟s initial weight and average weight of each period. The result is listed in table 4.5. From this table, we can find that the average cost (1.2949) of our benchmark is much larger than the average cost of fixed weight in 0.2069 (0.9446) and the average cost of fixed weight in 0.3639 (1.1548). According to this result, we conclude that the expected cost of variable annuities with guaranteed minimum withdrawal benefits will increase when the policyholders have the right to choose his own portfolio. However, the insurance companies almost assume that the policyholders puts all their money on the high-risk asset when price the VA with GMWB. As a result, the guaranteed cost will be overestimated and the policyholders will be charged unreasonable insurance feeTable 4.5 The comparison between our benchmark and fixed weights
Benchmark Fixed
In order to know the change of the weight as the account value increases in each periods, we depict it in figure 4.1. We observe that the 5th line is higher than other lines, and this result is consistent with intuition because the policyholder for optimal own utility will put more high
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than the 50th,75th and 95th lines in the first few periods and lower than the 50th, 75th and 95th lines after the 6th period? We see the probability of ruin is about 9.33% at the end of the policy in table 4.4. The probability of ruin of 25th line in the periods closed to maturity is very low and the effect of the ratio of withdraw to the account value is big, and therefore the 25th line is lower than other lines except the 95th line. But for the 25th line in the first few periods, the effect of guaranteed is still big because there is still some possibility that the account value touch the guaranteed value. As a result, the twenty-fifty line is higher than the 50th,75th and 95th lines in the first few periods
The figure 4.2 is the 5th, 25th, median, 75th and 95th lines of the weight of high risk asset in each period in our benchmark. A place worthy of observation is that the 25th line is close to the 5th line in the first few periods, but the 25th line is close to the 50th line in the last few periods. This is because the curve of the weight related to the account value is upward, and the account value of the minimum weight is closed to the expected value in the first few period. In contract, the account value of the minimum weight is far from the expected value in the last period.
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Figure 4.1 The weights of high-risk asset rank by the account value in each period
Figure 4.2 The weight of 5th, 25th, 50th , 75th, 95th rank by the weights in each period
The weight in each period ,rank by account value
5th
1990/1~2010/12 NASDAQ and Dow Jones allocation(Gamma=1)lognormal model
5th 25th 50th 75th 95th