# American Institute of Mathematical Sciences

January  2020, 16(1): 71-101. doi: 10.3934/jimo.2018141

## Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework

 1 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China 2 Department of Mathematics and Statistics, Curtin University, Bentley Campus, Perth, Western Australia 6845, Australia

* Corresponding author: Yan Zhang, Yonghong Wu

Received  April 2016 Revised  November 2017 Published  January 2020 Early access  September 2018

This paper investigates the asset liability management problem for an ordinary insurance system incorporating the standard concept of proportional reinsurance coverage in a stochastic interest rate and stochastic volatility framework. The goal of the insurer is to maximize the expectation of the constant relative risk aversion (CRRA) of the terminal value of the wealth, while the goal of the reinsurer is to maximize the expected exponential utility (CARA) of the terminal wealth held by the reinsurer. We assume that the financial market consists of risk-free assets and risky assets, and both the insurer and the reinsurer invest on one risk-free asset and one risky asset. By using the stochastic optimal control method, analytical expressions are derived for the optimal reinsurance control strategy and the optimal investment strategies for both the insurer and the reinsurer in terms of the solutions to the underlying Hamilton-Jacobi-Bellman equations and stochastic differential equations for the wealths. Subsequently, a semi-analytical method has been developed to solve the Hamilton-Jacobi-Bellman equation. Finally, we present numerical examples to illustrate the theoretical results obtained in this paper, followed by sensitivity tests to investigate the impact of reinsurance, risk aversion, and the key parameters on the optimal strategies.

Citation: Yan Zhang, Yonghong Wu, Benchawan Wiwatanapataphee, Francisca Angkola. Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. Journal of Industrial and Management Optimization, 2020, 16 (1) : 71-101. doi: 10.3934/jimo.2018141
##### References:
 [1] L. H. Bai and J. Y. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [2] T. R. Bielecki, S. Pliska and S. J. Sheu, Risk sensitive portfolio management with Cox-Ingersoll-Ross interest rates: The HJB equation, SIAM Journal on Control and Optimization, 44 (2005), 1811-1843.  doi: 10.1137/S0363012903437952. [3] N. Bj$ä$uerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1. [4] Y. Cao and N. Wan, Optimal proportional reinsurance and investment based on Hailton-Jacobi-Bellman equation, Insurance: Mathematics and Economics, 45 (2009), 157-162.  doi: 10.1016/j.insmatheco.2009.05.006. [5] G. Chacko and L. M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, Review of Financial Studies, 18 (2005), 1369-1402. [6] H. Chang and X. M. Rong, An investment and consumption problem with cir interest rate and stochastic volatility, Abstract and Applied Analysis, 2013 (2013), Art. ID 219397, 12 pp. doi: 10.1155/2013/219397. [7] S. M. Chen, Z. F. Li and K. M. Li, Optimal investment-reinsurance for an insurance company with VaR constraint, Insurance: Mathematics and Economics, 47 (2010), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002. [8] M. C. Chiu and H. Y. Wong, Optimal investment for insurer with cointegrated assets: CRRA utility, Insurance: Mathematics and Economics, 52 (2013), 52-64.  doi: 10.1016/j.insmatheco.2012.11.004. [9] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [10] A. Dassios and J. Nagaradjasarma, Pricing of Asian Options on Interest Rates in the CIR model, LSE Research Online, 2011. Available from: http://eprints.lse.ac.uk/32084. [11] G. Deelstra, M. Grasselli and P. F. Koehl, Optimal Investment Strategies in a CIR Framework, Journal of Applied Probability, 37 (2000), 936-946.  doi: 10.1239/jap/1014843074. [12] J. W. Gao, Optimal portfolio for dc pension plans under a CEV model, Insurance: Mathematics and Economics, 44 (2009), 479-490.  doi: 10.1016/j.insmatheco.2009.01.005. [13] J. W. Gao, An extended CEV model and the legendre transform-dual-asymptotic solutions for annuity contracts, Insurance: Mathematics and Economics, 46 (2010), 511-530.  doi: 10.1016/j.insmatheco.2010.01.009. [14] M. Grasselli, A stability result for the HARA class with stochastic interest rates, Insurance: Mathematics and Economics, 33 (2003), 611-627.  doi: 10.1016/j.insmatheco.2003.09.003. [15] L. Grzelak and K. Oosterlee, On the heston model with stochastic interest rate, SIAM Journal on Financial Mathematics, 2 (2011), 255-286.  doi: 10.1137/090756119. [16] M. D. Gu, Y. P. Yang, S. D. Li and J. Y. Zhang, Consistant elasticity of variance model for proportional reinsurance and invesment stategies, Insurance: Mathematics and Economics, 46 (2010), 580-587.  doi: 10.1016/j.insmatheco.2010.03.001. [17] A. Gu, X. Guo, Z. F. Li and Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance: Mathematics and Economics, 51 (2012), 674-684.  doi: 10.1016/j.insmatheco.2012.09.003. [18] G. Guan and Z. Liang, Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework, Insurance: Mathematics and Economics, 57 (2014), 58-66.  doi: 10.1016/j.insmatheco.2014.05.004. [19] V. Henderson, Analytical comparisons of option prices in stochastic volatility models, Mathematical Finance, 15 (2005), 49-59.  doi: 10.1111/j.0960-1627.2005.00210.x. [20] H. Huang, M. A. Milevsky and J. Wang, Portfolio choice and life insurance: The CRRA case, Journal of Risk and Insurance, 75 (2008), 847-872. [21] J. Kallsen and J. Muhle-Jarbe, Utility maximization in affine stochastic volatility models, International Journal of Theoretical and Applied Finance, 13 (2010), 459-477.  doi: 10.1142/S0219024910005851. [22] A. Kell and H. M$ü$ller, Efficient portfolio in the asset liability context, Astin Bulletin, 25 (1995), 33-48. [23] H. Kraft, Optimal portfolio and heston's stochastic volatility model: an explicit solution for power utility, Quantitative Finance, 5 (2005), 303-313.  doi: 10.1080/14697680500149503. [24] D. Li, X. Rong and H. Zhao, Optimal investment problem for an insurer and a reinsurer, Journal of Systems Science and Complexity, 28 (2015), 1326-1343.  doi: 10.1007/s11424-015-3065-9. [25] Z. F. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002. [26] S. Z. Luo, M. Taksar and A. Tsoi, On Reinsurance and Investment for Large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444.  doi: 10.1016/j.insmatheco.2007.04.002. [27] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257. [28] R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [29] R. C. Merton, An analytical derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872. [30] R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [31] A. Sepp, Pricing options on realized variance in the heston model with jumps in returns and volatility, Journal of Computational Finance, 11 (2008), 33-70. [32] W. F. Sharpe and L. G. Tint, Liabilities-a new approach, Journal of Portfolio Management, 16 (1990), 5-10. [33] M. Taksar and X. D. Zeng, A General Stochastic Volatility Model and Optimal Portfolio with Explicit Solutions, Working Paper, (2009). [34] M. Taksar and X. D. Zeng, A stochastic volatility model and optimal portfolio selection, Quant. Finance, 13 (2013), 1547-1558.  doi: 10.1080/14697688.2012.740568. [35] B. Yi, Z. F. Li, F. G. Viens and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under heston's stochastic volatility model, Insurance: Mathematics and Economics, 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011. [36] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.  doi: 10.1007/978-1-4612-1466-3. [37] Y. Zeng and Z. F. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance: Mathematics and Economics, 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001. [38] Y. Zeng and Z. F. Li, Optimal reinsurance-investment strategies for insurers under mean-CaR criteria, Journal of Industry and Management Optimization, 8 (2012), 673-690.  doi: 10.3934/jimo.2012.8.673. [39] Y. Zeng, Z. F. Li and Y. Lai, Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps, Insurance: Mathematics and Economics, 52 (2013), 498-507.  doi: 10.1016/j.insmatheco.2013.02.007. [40] H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance: Mathematics and Economics, 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004. [41] H. Zhao, C. Weng and Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, SSRN 2432207, 2014. [42] A. A. Zimbidis, Premium and reinsurance control of an ordinary insurance system with liabilities driven by a fractional brownian motion, Scandinavian Actuarial Journal, 1 (2008), 16-33.  doi: 10.1080/03461230701722810.

show all references

##### References:
 [1] L. H. Bai and J. Y. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [2] T. R. Bielecki, S. Pliska and S. J. Sheu, Risk sensitive portfolio management with Cox-Ingersoll-Ross interest rates: The HJB equation, SIAM Journal on Control and Optimization, 44 (2005), 1811-1843.  doi: 10.1137/S0363012903437952. [3] N. Bj$ä$uerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1. [4] Y. Cao and N. Wan, Optimal proportional reinsurance and investment based on Hailton-Jacobi-Bellman equation, Insurance: Mathematics and Economics, 45 (2009), 157-162.  doi: 10.1016/j.insmatheco.2009.05.006. [5] G. Chacko and L. M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, Review of Financial Studies, 18 (2005), 1369-1402. [6] H. Chang and X. M. Rong, An investment and consumption problem with cir interest rate and stochastic volatility, Abstract and Applied Analysis, 2013 (2013), Art. ID 219397, 12 pp. doi: 10.1155/2013/219397. [7] S. M. Chen, Z. F. Li and K. M. Li, Optimal investment-reinsurance for an insurance company with VaR constraint, Insurance: Mathematics and Economics, 47 (2010), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002. [8] M. C. Chiu and H. Y. Wong, Optimal investment for insurer with cointegrated assets: CRRA utility, Insurance: Mathematics and Economics, 52 (2013), 52-64.  doi: 10.1016/j.insmatheco.2012.11.004. [9] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [10] A. Dassios and J. Nagaradjasarma, Pricing of Asian Options on Interest Rates in the CIR model, LSE Research Online, 2011. Available from: http://eprints.lse.ac.uk/32084. [11] G. Deelstra, M. Grasselli and P. F. Koehl, Optimal Investment Strategies in a CIR Framework, Journal of Applied Probability, 37 (2000), 936-946.  doi: 10.1239/jap/1014843074. [12] J. W. Gao, Optimal portfolio for dc pension plans under a CEV model, Insurance: Mathematics and Economics, 44 (2009), 479-490.  doi: 10.1016/j.insmatheco.2009.01.005. [13] J. W. Gao, An extended CEV model and the legendre transform-dual-asymptotic solutions for annuity contracts, Insurance: Mathematics and Economics, 46 (2010), 511-530.  doi: 10.1016/j.insmatheco.2010.01.009. [14] M. Grasselli, A stability result for the HARA class with stochastic interest rates, Insurance: Mathematics and Economics, 33 (2003), 611-627.  doi: 10.1016/j.insmatheco.2003.09.003. [15] L. Grzelak and K. Oosterlee, On the heston model with stochastic interest rate, SIAM Journal on Financial Mathematics, 2 (2011), 255-286.  doi: 10.1137/090756119. [16] M. D. Gu, Y. P. Yang, S. D. Li and J. Y. Zhang, Consistant elasticity of variance model for proportional reinsurance and invesment stategies, Insurance: Mathematics and Economics, 46 (2010), 580-587.  doi: 10.1016/j.insmatheco.2010.03.001. [17] A. Gu, X. Guo, Z. F. Li and Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance: Mathematics and Economics, 51 (2012), 674-684.  doi: 10.1016/j.insmatheco.2012.09.003. [18] G. Guan and Z. Liang, Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework, Insurance: Mathematics and Economics, 57 (2014), 58-66.  doi: 10.1016/j.insmatheco.2014.05.004. [19] V. Henderson, Analytical comparisons of option prices in stochastic volatility models, Mathematical Finance, 15 (2005), 49-59.  doi: 10.1111/j.0960-1627.2005.00210.x. [20] H. Huang, M. A. Milevsky and J. Wang, Portfolio choice and life insurance: The CRRA case, Journal of Risk and Insurance, 75 (2008), 847-872. [21] J. Kallsen and J. Muhle-Jarbe, Utility maximization in affine stochastic volatility models, International Journal of Theoretical and Applied Finance, 13 (2010), 459-477.  doi: 10.1142/S0219024910005851. [22] A. Kell and H. M$ü$ller, Efficient portfolio in the asset liability context, Astin Bulletin, 25 (1995), 33-48. [23] H. Kraft, Optimal portfolio and heston's stochastic volatility model: an explicit solution for power utility, Quantitative Finance, 5 (2005), 303-313.  doi: 10.1080/14697680500149503. [24] D. Li, X. Rong and H. Zhao, Optimal investment problem for an insurer and a reinsurer, Journal of Systems Science and Complexity, 28 (2015), 1326-1343.  doi: 10.1007/s11424-015-3065-9. [25] Z. F. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002. [26] S. Z. Luo, M. Taksar and A. Tsoi, On Reinsurance and Investment for Large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444.  doi: 10.1016/j.insmatheco.2007.04.002. [27] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257. [28] R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [29] R. C. Merton, An analytical derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 7 (1972), 1851-1872. [30] R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. [31] A. Sepp, Pricing options on realized variance in the heston model with jumps in returns and volatility, Journal of Computational Finance, 11 (2008), 33-70. [32] W. F. Sharpe and L. G. Tint, Liabilities-a new approach, Journal of Portfolio Management, 16 (1990), 5-10. [33] M. Taksar and X. D. Zeng, A General Stochastic Volatility Model and Optimal Portfolio with Explicit Solutions, Working Paper, (2009). [34] M. Taksar and X. D. Zeng, A stochastic volatility model and optimal portfolio selection, Quant. Finance, 13 (2013), 1547-1558.  doi: 10.1080/14697688.2012.740568. [35] B. Yi, Z. F. Li, F. G. Viens and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under heston's stochastic volatility model, Insurance: Mathematics and Economics, 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011. [36] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.  doi: 10.1007/978-1-4612-1466-3. [37] Y. Zeng and Z. F. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance: Mathematics and Economics, 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001. [38] Y. Zeng and Z. F. Li, Optimal reinsurance-investment strategies for insurers under mean-CaR criteria, Journal of Industry and Management Optimization, 8 (2012), 673-690.  doi: 10.3934/jimo.2012.8.673. [39] Y. Zeng, Z. F. Li and Y. Lai, Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps, Insurance: Mathematics and Economics, 52 (2013), 498-507.  doi: 10.1016/j.insmatheco.2013.02.007. [40] H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance: Mathematics and Economics, 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004. [41] H. Zhao, C. Weng and Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, SSRN 2432207, 2014. [42] A. A. Zimbidis, Premium and reinsurance control of an ordinary insurance system with liabilities driven by a fractional brownian motion, Scandinavian Actuarial Journal, 1 (2008), 16-33.  doi: 10.1080/03461230701722810.
Evolutions of the CIR stochastic interest rate $r(t)$ and Heston stochastic volatility $\sigma(t)$ within the investment horizon $[0,\ T]$
Evolutions of the risky assets' prices for the insurer and the reinsurer
Evolutions of the wealth processes for the insurer and the reinsurer
The dynamic behaviour of (a) the optimal reinsurance control strategy $\psi^{*}(t)$, (b) the optimal investment strategy for the insurer $\pi^{*}(t)$ and (c) the optimal investment strategy for the reinsurer $u^{*}(t)$
Sensitivities of $\psi^{*}(t)$ with respect to $\zeta(t)$
Sensitivities of $\psi^{*}(t)$ with respect to $\gamma$
Sensitivities of $\pi^{*}(t)$ with respect to $\gamma$
Sensitivities of $\pi^{*}(t)$ with respect to the parameter $\nu$
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the interest rate $\mu$
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the risk aversion coefficient $q$
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the positive correlation coefficient
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the negative correlation coefficient.
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the mean reversion speed $\kappa$ when $q = 4$ and $\rho>0$
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to the mean reversion speed $\kappa$ when $q = 2$ and $\rho <0$
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to "volatility of volatility" $\xi$ when $q = 2$ and $\rho <0$
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to "volatility of volatility" $\xi$ when $q = 2$ and $\rho>0$
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to $a$
Sensitivities of the optimal investment strategy $u^{*}(t)$ for the reinsurer with respect to $b$
Parameter values for the original model
 Symbol Value Symbol Value Symbol Value $T$ 5 $\nu$ 1 $r_{0}$ 0.05 $\phi(t)$ 1.2 $\theta$ 0.06 $b_{0}$ 1 $m(t)$ 0.6 $\kappa$ 2 $b^{re}_{0}$ 1 $\zeta(t)$ 0.8 $\xi$ 0.1 $\sigma_{0}$ 0.04 $\alpha$ 0.1 $a$ 1 $s_{0}$ 1 $\beta$ 0.1 $b$ 1 $s^{re}_{0}$ 1 $K$ 0.15 $\mu$ 0.1 $l_{0}$ 2 $\tau$ 1.5 $\rho$ 0.5 $x_{0}$ 5 $\gamma$ 4 $q$ 0.5 $y_{0}$ 5
 Symbol Value Symbol Value Symbol Value $T$ 5 $\nu$ 1 $r_{0}$ 0.05 $\phi(t)$ 1.2 $\theta$ 0.06 $b_{0}$ 1 $m(t)$ 0.6 $\kappa$ 2 $b^{re}_{0}$ 1 $\zeta(t)$ 0.8 $\xi$ 0.1 $\sigma_{0}$ 0.04 $\alpha$ 0.1 $a$ 1 $s_{0}$ 1 $\beta$ 0.1 $b$ 1 $s^{re}_{0}$ 1 $K$ 0.15 $\mu$ 0.1 $l_{0}$ 2 $\tau$ 1.5 $\rho$ 0.5 $x_{0}$ 5 $\gamma$ 4 $q$ 0.5 $y_{0}$ 5
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