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July  2021, 17(4): 2139-2159. doi: 10.3934/jimo.2020062

Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform

1. 

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

2. 

Department of General Education, Army Engineering University of PLA, Nanjing 211101, China

* Corresponding author: Peibiao Zhao

Received  March 2019 Revised  December 2019 Published  July 2021 Early access  March 2020

Fund Project: This work was supported by NNSF of China (No.11871275; No.11371194)

The present paper investigates an optimal reinsurance-investment problem with Hyperbolic Absolute Risk Aversion (HARA) utility. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer. The insurer is allowed to purchase reinsurance from the reinsurer. Both the insurer and the reinsurer are assumed to invest in one risk-free asset and one risky asset whose price follows Heston's SV model. Our aim is to seek optimal investment-reinsurance strategies to maximize the expected HARA utility of the insurer's and the reinsurer's terminal wealth. In the utility theory, HARA utility consists of power utility, exponential utility and logarithmic utility as special cases. In addition, HARA utility is seldom studied in the optimal investment and reinsurance problem due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the insurer. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solutions of optimal investment-reinsurance strategies. Moreover, some special cases are also discussed in detail. Finally, some numerical examples are presented to illustrate the impacts of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategies.

Citation: Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2139-2159. doi: 10.3934/jimo.2020062
References:
[1]

C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005.

[2]

L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.

[3]

K. Borch, The optimal reinsurance treaty, ASTIN Bulletin, 5 (1969), 293-297. 

[4]

G. Chacko and L. M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, The Review of Financial Studies, 18 (2005), 1369-1402. 

[5]

H. Chang and K. Chang, Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform, Insurance Math. Econom., 72 (2017), 215-227.  doi: 10.1016/j.insmatheco.2016.10.014.

[6]

J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance Math. Econom., 45 (2009), 9-18.  doi: 10.1016/j.insmatheco.2009.02.006.

[7]

H. U. Gerber, An Introduction to Mathematical Risk Theory, in S. S. Huebner Foundation Monograph Series, 8, Richard D. Irwin, Inc., Homewood, Ⅲ., 1979.

[8]

J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9058-9.

[9]

M. Grasselli, A stability result for the HARA class with stochastic interest rates, Insurance Math. Econom., 33 (2003), 611-627.  doi: 10.1016/j.insmatheco.2003.09.003.

[10]

M. GuY. YangS. Li and J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46 (2010), 580-587.  doi: 10.1016/j.insmatheco.2010.03.001.

[11]

E. J. Jung and J. H. Kim, Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance Math. Econom., 51 (2012), 667-673.  doi: 10.1016/j.insmatheco.2012.09.009.

[12]

V. Henderson, Explicit solutions to an optimal portfolio choice problem with stochastic income, J Econ. Dyn. Control, 29 (2005), 1237-1266.  doi: 10.1016/j.jedc.2004.07.004.

[13]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.

[14]

Y. HuangX. Yang and J. Zhou, Robust optimal investment and reinsurance problem for a general insurance company under Heston model, Math. Methods Oper. Res., 85 (2017), 305-326.  doi: 10.1007/s00186-017-0570-8.

[15]

Y. HuangY. OuyangL. Tang and J. Zhou, Robust optimal investment and reinsurance problem for the product of the insurer's and the reinsurer's utilities, J. Comput. Appl. Math., 344 (2018), 532-552.  doi: 10.1016/j.cam.2018.05.060.

[16]

D. LiX. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, J. Comput. Appl. Math., 283 (2015), 142-162.  doi: 10.1016/j.cam.2015.01.038.

[17]

D. LiX. Rong and H. Zhao, Optimal reinsurance-investment problem for maximizing the product of the insurer's and the reinsurer's utilities under a CEV model, J. Comput. Appl. Math., 255 (2014), 671-683.  doi: 10.1016/j.cam.2013.06.033.

[18]

D. LiX. Rong and H. Zhao, Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model, Comp. Appl. Math., 35 (2016), 533-557.  doi: 10.1007/s40314-014-0204-1.

[19]

Z. LiY. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.

[20]

Z. Liang and K. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899.

[21]

X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the CEV mode, N. Am. Actuar. J., 15 (2011), 417-431.  doi: 10.1080/10920277.2011.10597628.

[22]

J. Liu, Portfolio selection in stochastic environments, The Review of Financial Studies, 20 (2007), 1-39.  doi: 10.1093/rfs/hhl001.

[23]

S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.

[24]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.

[25]

D.-L. Sheng, Explicit solution of reinsurance-investment problem for an insurer with dynamic income under Vasicek model, Adv. Math. Phys., 2016 (2016), Art. ID 1967872, 13 pp. doi: 10.1155/2016/1967872.

[26]

Z. SunX. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, J Math. Anal. Appl., 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053.

[27]

Y. WangX. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 328 (2018), 414-431.  doi: 10.1016/j.cam.2017.08.001.

[28]

J. XiaoZ. Hong and C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302-310.  doi: 10.1016/j.insmatheco.2006.04.007.

[29]

B. YiZ. LiF. G. Viens and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model, Insurance Math. Econom., 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011.

[30]

H. ZhaoX. 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 Math. Econom., 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004.

[31]

X. ZhengJ. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model, Insurance Math. Econom., 67 (2016), 77-87.  doi: 10.1016/j.insmatheco.2015.12.008.

[32]

B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance Math. Econom., 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006.

[33]

Y. Zhang and P. Zhao, Optimal reinsurance-investment problem with dependent risks based on Legendre transform, Journal of Industrial & Management Optimization, (2019). doi: 10.3934/jimo.2019011.

[34]

H. ZhaoC. WengY. Shen and Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, Sci. China Math., 60 (2017), 317-344.  doi: 10.1007/s11425-015-0542-7.

show all references

References:
[1]

C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005.

[2]

L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4.

[3]

K. Borch, The optimal reinsurance treaty, ASTIN Bulletin, 5 (1969), 293-297. 

[4]

G. Chacko and L. M. Viceira, Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, The Review of Financial Studies, 18 (2005), 1369-1402. 

[5]

H. Chang and K. Chang, Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform, Insurance Math. Econom., 72 (2017), 215-227.  doi: 10.1016/j.insmatheco.2016.10.014.

[6]

J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance Math. Econom., 45 (2009), 9-18.  doi: 10.1016/j.insmatheco.2009.02.006.

[7]

H. U. Gerber, An Introduction to Mathematical Risk Theory, in S. S. Huebner Foundation Monograph Series, 8, Richard D. Irwin, Inc., Homewood, Ⅲ., 1979.

[8]

J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4613-9058-9.

[9]

M. Grasselli, A stability result for the HARA class with stochastic interest rates, Insurance Math. Econom., 33 (2003), 611-627.  doi: 10.1016/j.insmatheco.2003.09.003.

[10]

M. GuY. YangS. Li and J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance Math. Econom., 46 (2010), 580-587.  doi: 10.1016/j.insmatheco.2010.03.001.

[11]

E. J. Jung and J. H. Kim, Optimal investment strategies for the HARA utility under the constant elasticity of variance model, Insurance Math. Econom., 51 (2012), 667-673.  doi: 10.1016/j.insmatheco.2012.09.009.

[12]

V. Henderson, Explicit solutions to an optimal portfolio choice problem with stochastic income, J Econ. Dyn. Control, 29 (2005), 1237-1266.  doi: 10.1016/j.jedc.2004.07.004.

[13]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.

[14]

Y. HuangX. Yang and J. Zhou, Robust optimal investment and reinsurance problem for a general insurance company under Heston model, Math. Methods Oper. Res., 85 (2017), 305-326.  doi: 10.1007/s00186-017-0570-8.

[15]

Y. HuangY. OuyangL. Tang and J. Zhou, Robust optimal investment and reinsurance problem for the product of the insurer's and the reinsurer's utilities, J. Comput. Appl. Math., 344 (2018), 532-552.  doi: 10.1016/j.cam.2018.05.060.

[16]

D. LiX. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, J. Comput. Appl. Math., 283 (2015), 142-162.  doi: 10.1016/j.cam.2015.01.038.

[17]

D. LiX. Rong and H. Zhao, Optimal reinsurance-investment problem for maximizing the product of the insurer's and the reinsurer's utilities under a CEV model, J. Comput. Appl. Math., 255 (2014), 671-683.  doi: 10.1016/j.cam.2013.06.033.

[18]

D. LiX. Rong and H. Zhao, Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model, Comp. Appl. Math., 35 (2016), 533-557.  doi: 10.1007/s40314-014-0204-1.

[19]

Z. LiY. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.

[20]

Z. Liang and K. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899.

[21]

X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the CEV mode, N. Am. Actuar. J., 15 (2011), 417-431.  doi: 10.1080/10920277.2011.10597628.

[22]

J. Liu, Portfolio selection in stochastic environments, The Review of Financial Studies, 20 (2007), 1-39.  doi: 10.1093/rfs/hhl001.

[23]

S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.

[24]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.

[25]

D.-L. Sheng, Explicit solution of reinsurance-investment problem for an insurer with dynamic income under Vasicek model, Adv. Math. Phys., 2016 (2016), Art. ID 1967872, 13 pp. doi: 10.1155/2016/1967872.

[26]

Z. SunX. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, J Math. Anal. Appl., 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053.

[27]

Y. WangX. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 328 (2018), 414-431.  doi: 10.1016/j.cam.2017.08.001.

[28]

J. XiaoZ. Hong and C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302-310.  doi: 10.1016/j.insmatheco.2006.04.007.

[29]

B. YiZ. LiF. G. Viens and Y. Zeng, Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model, Insurance Math. Econom., 53 (2013), 601-614.  doi: 10.1016/j.insmatheco.2013.08.011.

[30]

H. ZhaoX. 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 Math. Econom., 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004.

[31]

X. ZhengJ. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a CEV model, Insurance Math. Econom., 67 (2016), 77-87.  doi: 10.1016/j.insmatheco.2015.12.008.

[32]

B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance Math. Econom., 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006.

[33]

Y. Zhang and P. Zhao, Optimal reinsurance-investment problem with dependent risks based on Legendre transform, Journal of Industrial & Management Optimization, (2019). doi: 10.3934/jimo.2019011.

[34]

H. ZhaoC. WengY. Shen and Y. Zeng, Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models, Sci. China Math., 60 (2017), 317-344.  doi: 10.1007/s11425-015-0542-7.

Figure 1.  Effect of $ x $ on $ q^*_{HARA} $
Figure 2.  Effects of $ y $ on pH ARA*
Figure 3.  Effect of $ r $ on $ q^*_{\exp} $ and $ p^*_{\exp} $
Figure 4.  Effect of v on qexp* and pexp*
Figure 5.  Effect of r on qexp* and pexp*
Figure 6.  Effect of x on π1 HARA*
Figure 7.  Effect of y on π2HARA*
Figure 8.  Effect of v on π1HARA* and π2HARA*
Figure 9.  Effect of β on π1 exp*
Figure 10.  Effect of α on π1 exp*
Figure 11.  Effect of σ on π1 exp*
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