parameters. Given the utility parameter is equal to 0.5, the average weight will decrease when the time closes to maturity. It meets the life cycle allocation; which represent the policyholder will put his own asset on high risk asset when close to maturity. However, in the other hand, the average weight will increase when the time closes to maturity given the utility parameters are equal to or greater than 0.75. This is due to the influence of withdraw; the policyholder will receive more times of future withdraw in the first few periods, so the policyholder will have more investment risk and put less money on high risk asset which we mention before.
5.2 sensitivity analysis of model
In order to explore whether our approach in different asset model will have similar results, we compare the difference between lognormal model and ARIMA-GARCH model-.
The result is listed in Table 5.4.
Table 5.4 (The difference of the cost of GMWB between lognormal model and ARIMA-GARCH model)
Lognormal 0.5 0.9087 2.9293 58.2773 13.6649 9.7644 0.1779
ARIMA-GARCH 0.5 0.6902 5.8229 195.9438 28.6194 19.4097 0.1960
Percentage change -24.05% 98.78% 236.23% 109.44% 98.78% 10.17%
Lognormal 0.75 0.3762 1.6874 33.2665 3.0507 5.6247 0.1120
ARIMA-GARCH 0.75 0.1279 2.2134 69.8356 0.0000 7.3782 0.0945
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Lognormal 1 0.2069 1.2949 24.1715 0.0000 4.3164 0.0933
ARIMA-GARCH 1 0.0358 1.6295 49.1840 0.0000 5.4317 0.0754
Percentage change -82.70% 25.84% 103.48% - 25.84% -19.19%
Lognormal 2 0.0893 1.0123 17.7468 0.0000 3.3745 0.0825
ARIMA-GARCH 2 0.0100 1.3179 36.3768 0.0000 4.3931 0.0710
Percentage change -88.80% 30.19% 104.98% - 30.19% -13.94%
We can find that the expected cost in ARIMA-GARCH model is greater than the expected cost in lognormal model. This is because that the ARIMA-GARCH model is more fat-tailed than lognormal model. When the market goes down, the insurance company needs to pay more guaranteed costs in ARIMA-GARCH model.
Table 5.5 The difference of expected cost between different asset models Low risk
asset
Gamma=2 Benchmark (Gamma=1)
Gamma=0.75 Gamma=0.5 high risk asset
Lognormal 0.8503 1.0123 1.2949 1.6874 2.9293 3.3704
ARIMA-GARCH
0.9755 1.3179 1.6295 2.2134 5.8229 10.3051
Percentage change
14.71% 30.19% 25.84% 31.18% 98.78% 205.75%
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The main goal of this study is to explore whether the guaranteed cost will increase when the policyholders have the right to rebalance his own account in every period. We can get the following conclusions. First, The guaranteed cost of variable annuities with GMWB when the policyholder can rebalance his own asset allocation in every year is about 12% more than the guaranteed cost of the fixed weight which is calculated by the average of the weights in each year of our optimal allocation. Second, the optimal allocation is affected by the asset model and the risk averse attitude of the policyholder in our model. When the asset model is a fat-tailed model, it needs more guaranteed cost; when the policyholder is more aggressive, the more guaranteed cost the insurance companies need to afford .
In our study, there are still several restrictions when we price the guaranteed products.
First, The main restriction in our study is that we assume that the policyholder will hold the policy until the end of the policy, and the policyholder doesn‟t have the right to terminate the policy in any time in the light of actual conditions. However, it will be involved in the issues of the inter-temporal utility and the discount of utility, and the issue is not in our study.
Second, we use backward method to decide the optimal allocation of the policyholder.
However, if we want to price products such as VA with GMAB, it does not work. The reason is that we don‟t know the renew guaranteed value at the last period, and there is no enough information to calculate the optimal allocation.
The future studies may have the following issues. First, we assume there are only two linked assets in our study, but the dynamic programming method can apply to three assets, four assets, or even more. The future studies can attempt more linked assets. Second, we propose the risk-free interest are fixed, but the risk-free interest rate model can transfer to stochastic interest rate model, and the future research can consider the withdrawal behavior of the policyholder. Furthermore, there are many different utility functions such as exponential utility function and logarithm utility function in financial economic theory, and the different
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analyst can compare the different utility functions.
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1. Aase, K.K., and Persson, S.A., 1994, Pricing of Unit-linked Life Insurance Policies.
Scandinavian Actuarial Journal 1, 26-52.
2. Aase, K.K., and Persson, S.A., 1997, Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Contracts. Journal of Risk and Insurance 64 (4), 599-617.
3. Bacinello, A. R., Biffis, E., and Millossovich, P., 2009, Regression-Based Algorithm for Life Insurance Contracts with Surrender Guarantees, to appear in Quantitative Finance, Version as of April 7, 2009.
4. Boyle, P.P., Schwartz, E., 1977. Equilibrium Prices of Guarantees under Equity-linked Contracts. Journal of Risk and Insurance 44(2), 639–680.
5. Boyle, P.P. and Schwartz, E.S., 1997, Equilibrium Prices of Guarantees under Equity-linked Contracts. Journal of Risk and Insurance 44, 639-680.
6. Boyle, P.P. and Hardy, M.R., 1997, Reserving for Maturity Guarantees: Two Approaches.
Insurance: Mathematics and Economics 21, 113-127.
7. Boyle, P.P. and Hardy, M.R. 2003, Guaranteed Annuity Options. Astin Bulletin 33 (2), 125-152
8. Brennan, M.J., and Schwartz, E.S., 1976. The Pricing of Equity-linked Life Insurance Policies with an Asset Value Guarantee. Journal of Financial Economics 3 (1), 195–213.
9. Brennan, M.J. and Schwartz, E.S., 1979, Alternative Investment Strategies for the Issuers of Equity-linked Life Insurance Policies with an Asset Value Guarantee, Journal of Business 52, 63-93.
10. Coleman, T.F. ,Kim, Y., and Patron, M. (2005), Hedging Guaranteed Variable Annuities Under Both Equity and Interest Rate Risks, Cornell University, New York.
11. Coleman, T.F. ,Kim, Y., and Patron, M. (2005), Robustly Hedging Variable Annuities With Guaranteed Under Jump and Volatility Risk, The Journal of Risk and Insurance, 2007, Vol. 74, No,2, 347-376.
12. Chen, K. Verzal, and P. Forsyth. The effect of modeling parameters on the value of GMWB guarantees. Insurance: Mathematics and Economics, 43(1):165-173, 2008 13. Delbaen, F., and M. Yor, 2002, Passport Options, Mathematical Finance, 12(4): 299-328.
14. Hardy, M.R. 2000, Hedging and Reserving for Single-premium Segregated Fund Contracts. North American Actuarial Journal 4 (2), 63-74.
15. Hardy, M.R., 2003, Investment Guarantees: Modeling and Risk Management for
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16. Holz, D., Kling, A., and Ru, J. , 2007, GMWB For Life An Analysis of Lifetime Withdrawal Guarantees. Working Paper, Ulm University, 2007.
17. Hung,D.Y.,2009,The numerical solution of optimal asset allocation dynamic programming. Cheng-Chi University master degree paper.
18. Liu,Y,2006, 2010Pricing and Hedging the Guaranteed Minimum Withdrawal Benefits in Variable Annuities. Working Paper.
19. Milevsky, M.A., and Posner, S.E., 2001, The Titanic option: Valuation of Guaranteed Minimum DEATH Benefit in Variable Annuities and Mutual Fund. The Journal of Risk and Insurance, Vol. 68, No. 1,91-216,2001.
20. Milevsky, M.A., Salisbury, T.S., 2006. Financial Valuation of Guaranteed Minimum Withdrawal Benefits. Insurance: Mathematics and Economics 38, 21-38
21. Nielsen, J.A., K. Sandmann, 1995, Equity-Linked Life Insurance: a Model with Stochastic Interest Rates. Insurances: Mathematics and Economics, 16,225-253.
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23. J. Peng, K. Leung, and Y. Kwong. Pricing Guaranteed Minimal Withdrawal Benefit under the stochastic interest rate. Technical report, Jan 2009.
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Appendix A. The data of the main five indexes in U.S.
Table Appendix A.1 The normal test of main five indexes in U.S ( Expire at 2010/12)
Russell 2000 P value S&P500 P value Dow Jones P value NASDAQ P value S&P100 P value
Table Appendix A.2 Annualized return and volatility of five main indexes on monthly information
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Table Appendix A.3 Correlation table of five main indexes on monthly information
Russell 2000 S&P500
Table Appendix A.4 The order of ARIMA in NASDAQ index
c p q R-square AIC SBIC LR
1 1 0 0.000616 -3.421534 -3.393443 431.4025
1 0 1 0.000653 -3.425473 -3.397461 433.6096
1 1 1 0.013978 -3.427027 -3.384890 433.0919
1 1 2 0.015278 -3.417477 -3.394800 431.1846
1 2 1 0.014014 -3.419096 -3.396486 433.0965
1 2 2 0.014854 -3.409046 -3.380701 431.1308
Table Appendix A.5 The order of ARIMA in Doe Jones index
c p q R-square AIC SBIC LR
1 1 0 0.014735 -2.471756 -2.460451 312.2053
1 0 1 0.014858 -2.475704 -2.447692 313.9387
1 1 1 0.014825 -2.463879 -2.421742 312.2168
1 1 2 0.014877 -2.455964 -2.399782 312.2235
1 2 1 0.078479 -2.518772 -2.462429 318.8465
1 2 2 0.071195 -2.502898 -2.432469 317.8623
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Appendix B. The decision table of GMMB
At the section 4.4, we show the result of VA with GMMB, and the below table is the decision table of GMMB. 104.3687 0.8649 110.2776 0.8655 105.3292 0.9356 109.9459 0.9378 108.8753 0.8412 117.0591 0.8287 111.5283 0.9028 117.3902 0.8884 113.7488 0.8144 124.6159 0.7887 117.8442 0.8637 125.0781 0.8373 119.4148 0.7890 133.4243 0.7514 124.4178 0.8274 133.0205 0.7919 126.7116 0.7562 145.1522 0.7143 131.3956 0.7934 141.5099 0.7509 138.4424 0.7222 164.4849 0.6638 139.1076 0.7542 151.1070 0.7158 242.1262 0.5913 367.8705 0.5657 148.2108 0.7288 162.5776 0.6882 159.5394 0.6979 177.0545 0.6536 175.4649 0.6674 197.2879 0.6194 202.8585 0.6255 233.9415 0.5909 501.1539 0.5698 653.2512 0.5544
6th year Decision table 7th year Decision table 8th year Decision table 9th year Decision table
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105.0113 0.9862 109.0165 0.9900 113.2491 0.9997 118.2201 0.9916 110.9620 0.9594 115.6604 0.9781 120.8728 0.9832 126.6335 0.9642 116.7214 0.9300 122.2629 0.9504 128.3732 0.9529 135.0334 0.9102 122.5025 0.8951 129.0653 0.9048 135.9402 0.9068 143.5446 0.8392 128.3390 0.8564 135.8147 0.8587 143.6636 0.8566 152.2427 0.7829 134.2484 0.8235 142.6894 0.8178 151.4669 0.8081 161.0882 0.7382 140.3847 0.7847 149.8818 0.7775 159.8543 0.7653 170.5872 0.6985 146.8360 0.7563 157.5107 0.7401 168.7438 0.7249 180.4739 0.6670 153.7039 0.7342 165.7620 0.7101 178.3536 0.6947 191.4300 0.6368 161.2835 0.7119 174.8710 0.6838 188.9058 0.6703 203.7048 0.6155 169.6727 0.6860 184.8237 0.6601 200.6492 0.6499 217.3511 0.5974 179.3382 0.6621 196.3120 0.6416 214.1369 0.6278 233.1626 0.5805 190.6550 0.6428 209.8943 0.6191 229.9791 0.6108 251.8587 0.5696 204.5037 0.6203 226.6311 0.6031 250.1702 0.5979 275.3404 0.5596 222.7196 0.6040 248.8497 0.5884 277.1045 0.5885 306.7487 0.5523 248.6439 0.5877 281.3096 0.5783 316.4414 0.5788 354.7241 0.5470 298.7948 0.5730 342.7591 0.5667 393.5771 0.5740 445.6414 0.5457 858.8344 0.5604 1093.5610 0.5608 1362.4171 0.5716 1631.8749 0.5441
10th year Decision table 11th year Decision table 12th year Decision table 13th year Decision table Account
124.7092 0.9984 130.0035 0.9979 135.7629 0.9997 142.3342 0.9955 138.9119 0.9618 145.9905 0.9253 153.2537 0.8984 161.3492 0.8101 153.3263 0.8554 162.0244 0.7782 171.1028 0.7489 181.0513 0.6683 168.4052 0.7578 179.1015 0.6904 190.0669 0.6606 201.9280 0.6042
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184.9951 0.6883 197.6330 0.6324 210.5052 0.6146 225.1562 0.5782 203.3699 0.6427 218.4123 0.5953 234.1120 0.5886 251.1703 0.5703 224.6581 0.6119 242.6935 0.5750 261.9043 0.5772 282.0647 0.5673 250.6091 0.5911 272.5393 0.5653 295.9922 0.5721 320.3329 0.5666 285.0832 0.5802 312.0279 0.5585 341.3551 0.5700 370.9376 0.5664 335.8311 0.5745 371.3070 0.5562 409.3814 0.5696 450.1335 0.5663 434.6593 0.5720 486.4801 0.5553 543.5612 0.5696 605.9494 0.5663 2105.1230 0.5713 2459.2426 0.5541 2915.8501 0.5690 3573.7358 0.5659
14th year Decision table 15th year Decision table Account
148.9906 0.9655 156.0086 0.5824 169.7268 0.6756 178.6510 0.5502 191.4258 0.6027 202.1420 0.5475 214.5025 0.5823 227.4812 0.5474 240.1117 0.5774 255.7465 0.5474 269.1236 0.5766 288.0422 0.5474 303.3431 0.5765 326.4945 0.5474 346.2646 0.5764 374.8867 0.5474 404.3939 0.5764 439.8087 0.5474 492.8482 0.5764 540.4006 0.5474 671.3695 0.5764 744.8425 0.5474 4299.0267 0.5761 5457.2780 0.5474
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The figure is the average weight ranked by weight in each period of VA with GMMB.
0 2 4 6 8 10 12 14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time
high risk weight
The average weight ranked by weight in each period of VA with GMMB
5th 25th 50th 75th 95th