Investment portfolio optimization for a dynamic life insurance product by using stochastic control tools

Vahabi, S.; Payandeh Najafabadi, A.T.

doi:10.22056/ijir.2023.03.05

Document Type : Original Research Paper

Authors

Department of Actuarial Science, Faculty of Mathematical Sciences, Shahid Behshti University, Tehran, Iran

https://doi.org/10.22056/ijir.2023.03.05

Abstract

BACKGROUND AND OBJECTIVES: In this article, a life insurance product is designed with the help of stochastic control approach. These products are defined in such a way that in exchange for receiving an amount as insurance premium that is paid at specified times, the insurer undertakes to pay insurance benefits when the insured is alive at the end of the contract.
Methodology: This research is an analytical study in terms of developmental-applicative purpose. In the literature of life insurance, there are various products that are not the same in the type of benefit payment and the timing of their implementation. Among these examples, term life insurance, term life insurance, and mixed life insurance can be mentioned. Traditional insurance products with fixed benefits are quickly losing their appeal due to inflationary markets. In this research, it is focused on the design of a life insurance product on the condition of life, which is connected to the investment markets. Stochastic differential calculus models have been used to simulate capital markets assets. All the numerical results of this research have been calculated with the help of Matlab and Maple software.
FINDINGS: To achieve the best choice of investment type, with the help of stochastic optimal control tool, the best investment strategy was calculated for a person who has the CRRA utility function and buys this product, so that the most benefits are paid at the end of the contract. To invest in this contract, modeling was done in a non-risky market such as a bank and a risky market such as stocks, which have price jumps. In addition, to model the risk asset, the Merton model, which is a representative of the models with finite activity, was used, and at the end, a comparison was made for several mortality functions.
CONCLUSION: The main purpose of this article is investment for the insured who bought this product. In the product designed in this article, the insurer undertakes to pay the premiums received at a guaranteed rate at the end of the contract. Also, the insured will share the profit from the investment based on a certain percentage that is determined at the beginning of each year. The simulations show that the behavior of the optimal consumption rate is the same as the Merton model with the approach that the behavior of full price jumps is transparent in the optimal consumption rate designed in this article. Investment results for several mortality functions are reported in the Numerical Results section.

Keywords

Main Subjects

Insurance pricing

References

Ait-Sahalia, Y.; Cacho-Diaz, J.; Hurd, T.R., (2009). Portfolio choice with jumps: A closed-form solution. Annals Appl. Probability., 19(2): 556-584 (29 Pages).

Barigou, K.; Delong, L., (2022). Pricing equity-linked life insurance contracts with multiple risk factors by neural networks. J. Comput. Appl. Math., 404(1).

Ceci, C.; Colaneri, K.; Cretarola, A., (2020). Indifference pricing of pure endowments via BSDEs under partial information. Scand. Actuarial. J., 2020(10): 904-933 (30 Pages).

Ceci, C.; Colaneri, K.; Cretarola, A., (2017). Unit-linked life insurance policies: Optimal hedging in partially observable market models. Insur. Math. Econ., 76(1): 149-163 (15 Pages).

Choulli, T.; Hurd, T.R., (2001). The role of hellinger processes in mathematical finance. Entropy., 3(3): 150-161 (12 Pages).

Cox, J.C.; Huang, C.F., (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econ. Theory., 49(1): 33-83 (51 Pages).

Deelstra, G.; Grasselli, M.; Koehl, P.F., (2004). Optimal design of the guarantee for defined contribution funds. J. Econ. Dyn. Control., 28(11): 2239-2260 (22 Pages).

Dong, Y.; Zheng, H., (2020). Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to defined contribution pension plan. Eur. J. Oper. Res., 281(2): 341-356 (16 Pages).

Emmer, S.; Klüppelberg, C., (2004). Optimal portfolios when stock prices follow an exponential Lévy process. Finance. Stochastics., 8(1): 17-44 (28 Pages).

Gao, J., (2008). Stochastic optimal control of DC pension funds. Insur. Math. Econ., 42(3): 1159-1164 (6 Pages).

Gerber, H.U., (1997). Life insurance mathematics. Springer. Sci. Bus. Media

Gradshteyn, I.S.; Ryzhik, I.M., (2014). Table of integrals, series, and products. Acad. Press

Huertas, K.E.A.; Alfonso, K.E., (2021). Mathematical reserves of unit-linked insurance policies with fractional volatility.

Kallsen, J., (2000). Optimal portfolios for exponential Lévy processes. Math. Methods. Oper. Res., 51(1): 357-374 (18 Pages).

Kou, S.G., (2002). A jump-diffusion model for option pricing. Manage. Sci., 48(8): 1086-1101 (16 Pages).

Kung, K.L.; Yang, S.Y., (2019). Optimal consumption and investment problem incorporating housing and life insurance decisions: The continuous time case. J. Risk. Insur., 87(1): 143-171 (29 Pages).

Li, D.; Rong, X.; Zhao, H., (2014). Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model. Comput. Appl. Mathematics ., 35(2): 533-557 (25 Pages).

Liang, Z.; Ma, M., (2015). Optimal dynamic asset allocation of pension fund in mortality and salary risks framework. Insur. Math. Econ., 64(1): 151-161 (11 Pages).

Liu, J.; Longstaff, F.A.; Pan, J., (2002). Dynamic asset allocation with event risk. J. Finance., 58(1): 1-45 (45 Pages).

Merton, R.C., (1969). Lifetime portfolio selection under uncertainty: The continuous-timecCase. Rev. Econ. Stat., 51(3): 247-257 (11 Pages).

Merton, R.C., (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory., 3(4): 373-413 (41 Pages).

Merton, R.C., (1976). Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 3(1/2): 125-144 (20 Pages).

Øksendal, B., (2013). Stochastic differential equations: An introduction with applications. Springer. Sci. Bus. Media

Øksendal, B.; Sulem, A., (2019). Applied stochastic control of jump diffusions. Springer. Berlin., 498(1).

Pasin, L.; Vargiolu, T., (2010). Optimal portfolio for CRRA utility functions when risky assets are exponential additive processes. Econ. Notes., 39(1/2): 65-90 (26 Pages).

Tankov, P.; Cont, R., (2004). Financial modelling with jump processes. Chapman. AMP

Wang, Y.; Zhang, Z.; Yu, W., (2021). Pricing equity-linked death benefits by complex Fourier series expansion in a regime-switching jump diffusion model. Appl. Math. Comput., 399(1).

Xu, W.; Gao, J., (2020). An optimal portfolio problem of DC pension with Input-Delay and Jump-Diffusion process. Math. Probl. Eng., 1-9 (9 Pages).

Yaari, M.E., (1965). Uncertain lifetime, life insurance, and the theory of the consumer. Rev. Econ. Stud., 32(2): 137-150 (14 Pages).

Yao, H.; Chen, P.; Zhang, M.; Li, X., (2020). Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. J. Ind. Manage. Optim., 18(1): 511-540 (30 Pages).

Letters to Editor

IJIR Journal welcomes letters to the editor for the post-publication discussions and corrections which allows debate post publication on its site, through the Letters to Editor. Letters pertaining to manuscript published in IJIR should be sent to the editorial office of IJIR within three months of either online publication or before printed publication, except for critiques of original research. Following points are to be considering before sending the letters (comments) to the editor.

[1] Letters that include statements of statistics, facts, research, or theories should include appropriate references, although more than three are discouraged.

[2] Letters that are personal attacks on an author rather than thoughtful criticism of the author’s ideas will not be considered for publication.

[3] Letters can be no more than 300 words in length.

[4] Letter writers should include a statement at the beginning of the letter stating that it is being submitted either for publication or not.

[5] Anonymous letters will not be considered.

[6] Letter writers must include their city and state of residence or work.

[7] Letters will be edited for clarity and length.

Name *

Email Address *

Affiliation *

Comments *

Security Code *

Iranian Journal of Insurance Research

Investment portfolio optimization for a dynamic life insurance product by using stochastic control tools

References

References

Letters to Editor

Send comment about this article

Volume 12, Issue 3 - Serial Number 45
May 2023
Pages 225-238

Investment portfolio optimization for a dynamic life insurance product by using stochastic control tools

References

References

Letters to Editor

Send comment about this article

Volume 12, Issue 3 - Serial Number 45May 2023Pages 225-238

Volume 12, Issue 3 - Serial Number 45
May 2023
Pages 225-238