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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

Yan Zhang 1,2, , Peibiao Zhao 1,, , Xinghu Teng 2, and  Lei Mao 2,

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  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.

Keywords: Investment, Reinsurance, HARA utility, Legendre transform, Heston's SV model.
Mathematics Subject Classification: Primary: 91B30, 93E20; Secondary: 62P05.
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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

M. Gu, Y. Yang, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. Huang, X. 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.  Google Scholar

[15]

Y. Huang, Y. Ouyang, L. 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.  Google Scholar

[16]

D. Li, X. 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.  Google Scholar

[17]

D. Li, X. 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.  Google Scholar

[18]

D. Li, X. 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.  Google Scholar

[19]

Z. Li, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

Z. Sun, X. 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.  Google Scholar

[27]

Y. Wang, X. 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.  Google Scholar

[28]

J. Xiao, Z. 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.  Google Scholar

[29]

B. Yi, Z. Li, F. 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.  Google Scholar

[30]

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

[31]

X. Zheng, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Zhao, C. Weng, Y. 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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

M. Gu, Y. Yang, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

Y. Huang, X. 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.  Google Scholar

[15]

Y. Huang, Y. Ouyang, L. 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.  Google Scholar

[16]

D. Li, X. 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.  Google Scholar

[17]

D. Li, X. 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.  Google Scholar

[18]

D. Li, X. 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.  Google Scholar

[19]

Z. Li, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

Z. Sun, X. 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.  Google Scholar

[27]

Y. Wang, X. 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.  Google Scholar

[28]

J. Xiao, Z. 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.  Google Scholar

[29]

B. Yi, Z. Li, F. 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.  Google Scholar

[30]

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

[31]

X. Zheng, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Zhao, C. Weng, Y. 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.  Google Scholar

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