# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2019114

## Optimal investment for an insurer under liquid reserves

 School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, China

* Corresponding author: Haili Yuan

Received  January 2019 Revised  March 2019 Published  September 2019

Fund Project: Supported by National Natural Science Foundation of China (Nos. 11201352, 11771343).

In this paper, we study the optimal investment problem for an insurer, who is allowed to invest in a financial market which consists of $N$ risky securities modeled by an $N$-dimensional Itô process. The surplus of the insurer is modeled by a general risk model. For the insurer's wealth, some money (called liquid reserves) can only be used to cope with risk, and can not be invested in the financial market. We suggest that the liquid reserve is a proportion of the total claim amount. By the martingale approach, we derive the optimal strategies for the CARA and the quadratic utilities, respectively.

Citation: Haili Yuan, Yijun Hu. Optimal investment for an insurer under liquid reserves. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019114
##### References:
 [1] H. Albrecher, C. Constantinescu, Z. Palmowski, G. Regensburger and M. Rosenkranz, Exact and asymptotic results for insurance risk models with surplus-dependent premiums, SIAM Journal on Applied Mathematics, 73 (2013), 47-66.  doi: 10.1137/110852000.  Google Scholar [2] S. Asmussen and H. Albrecher, Ruin Probabilities, Second edition, Advanced Series on Statistical Science & Applied Probability, 14. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/9789814282536.  Google Scholar [3] T. Belkina, C. Hipp, S. Z. Luo and M. Taksar, Optimal constrained investment in the Cramer-Lundberg model, Scandinavian Actuarial Journal, 5 (2014), 383-404.  doi: 10.1080/03461238.2012.699001.  Google Scholar [4] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar [5] J. Cai, R. H. Feng and G. E. Willmot, The compound Poisson surplus model with interest and liquid reserves: Analysis of the Gerber-Shiu discounted penalty function, Methodology and Computing in Applied Probability, 11 (2009), 401-423.  doi: 10.1007/s11009-007-9050-6.  Google Scholar [6] J. Cai, R. H. Feng and G. E. Willmot, Analysis of the compound Poisson surplus model with liquid reserves, interest and dividends, Astin Bulletin, 39 (2009), 225-247.  doi: 10.2143/AST.39.1.2038063.  Google Scholar [7] E. C. K. Cheung and D. Landriault, On a risk model with surplus-dependent premium and tax rates, Methodology and Computing in Applied Probability, 14 (2012), 233-251.  doi: 10.1007/s11009-010-9197-4.  Google Scholar [8] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar [9] P. Embrechts and H. Schmidli, Ruin estimation for a general insurance risk model, Advances in Applied Probability, 26 (1994), 404-422.  doi: 10.2307/1427443.  Google Scholar [10] H. U. Gerber and E. S. W. Shiu, The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: Mathematics and Economics, 21 (1997), 129-137.  doi: 10.1016/S0167-6687(97)00027-9.  Google Scholar [11] H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78.  doi: 10.1080/10920277.1998.10595671.  Google Scholar [12] W. J. Guo, Optimal portfolio choice for an insurer with loss aversion, Insurance: Mathematics and Economics, 58 (2014), 217-222.  doi: 10.1016/j.insmatheco.2014.07.004.  Google Scholar [13] C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar [14] I. Karatzas, J. P. Lehoczky, S. E. Shreve and G.-L. Xu, Martingale and duality methods for utility maximization in incomplete markets, SIAM Journal on Control and Optimization, 29 (1991), 702-730.  doi: 10.1137/0329039.  Google Scholar [15] X. S. Lin and G. E. Willmot, The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: Mathematics and Economics, 27 (2000), 19-44.  doi: 10.1016/S0167-6687(00)00038-X.  Google Scholar [16] C. S. Liu and H. L. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.  Google Scholar [17] S. Z. Luo, Ruin minimization for insurers with borrowing constraints, North American Actuarial Journal, 12 (2008), 143-174.  doi: 10.1080/10920277.2008.10597508.  Google Scholar [18] H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar [19] R. S. Perera, Optimal consumption, investment and insurance with insurable risk for an investor in a Lévy market, Insurance: Mathematics and Economics, 46 (2010), 479-484.  doi: 10.1016/j.insmatheco.2010.01.005.  Google Scholar [20] M. Schweizer, Approximating random variables by stochastic integrals, The Annals of Probability, 22 (1994), 1536-1575.  doi: 10.1214/aop/1176988611.  Google Scholar [21] Z. W. Wang, J. M. Xia and L. H. Zhang, Optimal investment for an insurer: The martingale approach, Insurance: Mathematics and Economics, 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003.  Google Scholar [22] S. X. Xie, Continuous-time mean-variance portfolio selection with liability and regime switching, Insurance: Mathematics and Economics, 45 (2009), 148-155.  doi: 10.1016/j.insmatheco.2009.05.005.  Google Scholar [23] H. L. Yuan and Y. J. Hu, The compound Poisson risk model with interest and a threshold strategy, Stochastic Models, 25 (2009), 197-220.  doi: 10.1080/15326340902869846.  Google Scholar [24] X.-L. Zhang, K.-C. Zhang and X.-J. Yu, Optimal proportional reinsurance and investment with transaction costs, Ⅰ: Maximizing the terminal wealth, Insurance: Mathematics and Economics, 44 (2009), 473-478.  doi: 10.1016/j.insmatheco.2009.01.004.  Google Scholar [25] J. M. Zhou, X. Q. Yang and J. Y. Guo, Portfolio selection and risk control for an insurer in the Lévy market under mean-variance criterion, Statistics and Probability letters, 126 (2017), 139-149.  doi: 10.1016/j.spl.2017.03.008.  Google Scholar [26] Q. Zhou, Optimal investment for an insurer in the Lévy market: The martingale approach, Statistics and Probability Letters, 79 (2009), 1602-1607.  doi: 10.1016/j.spl.2009.03.027.  Google Scholar [27] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance: Mathematics and Economics, 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006.  Google Scholar

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##### References:
 [1] H. Albrecher, C. Constantinescu, Z. Palmowski, G. Regensburger and M. Rosenkranz, Exact and asymptotic results for insurance risk models with surplus-dependent premiums, SIAM Journal on Applied Mathematics, 73 (2013), 47-66.  doi: 10.1137/110852000.  Google Scholar [2] S. Asmussen and H. Albrecher, Ruin Probabilities, Second edition, Advanced Series on Statistical Science & Applied Probability, 14. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/9789814282536.  Google Scholar [3] T. Belkina, C. Hipp, S. Z. Luo and M. Taksar, Optimal constrained investment in the Cramer-Lundberg model, Scandinavian Actuarial Journal, 5 (2014), 383-404.  doi: 10.1080/03461238.2012.699001.  Google Scholar [4] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar [5] J. Cai, R. H. Feng and G. E. Willmot, The compound Poisson surplus model with interest and liquid reserves: Analysis of the Gerber-Shiu discounted penalty function, Methodology and Computing in Applied Probability, 11 (2009), 401-423.  doi: 10.1007/s11009-007-9050-6.  Google Scholar [6] J. Cai, R. H. Feng and G. E. Willmot, Analysis of the compound Poisson surplus model with liquid reserves, interest and dividends, Astin Bulletin, 39 (2009), 225-247.  doi: 10.2143/AST.39.1.2038063.  Google Scholar [7] E. C. K. Cheung and D. Landriault, On a risk model with surplus-dependent premium and tax rates, Methodology and Computing in Applied Probability, 14 (2012), 233-251.  doi: 10.1007/s11009-010-9197-4.  Google Scholar [8] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar [9] P. Embrechts and H. Schmidli, Ruin estimation for a general insurance risk model, Advances in Applied Probability, 26 (1994), 404-422.  doi: 10.2307/1427443.  Google Scholar [10] H. U. Gerber and E. S. W. Shiu, The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: Mathematics and Economics, 21 (1997), 129-137.  doi: 10.1016/S0167-6687(97)00027-9.  Google Scholar [11] H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78.  doi: 10.1080/10920277.1998.10595671.  Google Scholar [12] W. J. Guo, Optimal portfolio choice for an insurer with loss aversion, Insurance: Mathematics and Economics, 58 (2014), 217-222.  doi: 10.1016/j.insmatheco.2014.07.004.  Google Scholar [13] C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar [14] I. Karatzas, J. P. Lehoczky, S. E. Shreve and G.-L. Xu, Martingale and duality methods for utility maximization in incomplete markets, SIAM Journal on Control and Optimization, 29 (1991), 702-730.  doi: 10.1137/0329039.  Google Scholar [15] X. S. Lin and G. E. Willmot, The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: Mathematics and Economics, 27 (2000), 19-44.  doi: 10.1016/S0167-6687(00)00038-X.  Google Scholar [16] C. S. Liu and H. L. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.  Google Scholar [17] S. Z. Luo, Ruin minimization for insurers with borrowing constraints, North American Actuarial Journal, 12 (2008), 143-174.  doi: 10.1080/10920277.2008.10597508.  Google Scholar [18] H. M. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91.   Google Scholar [19] R. S. Perera, Optimal consumption, investment and insurance with insurable risk for an investor in a Lévy market, Insurance: Mathematics and Economics, 46 (2010), 479-484.  doi: 10.1016/j.insmatheco.2010.01.005.  Google Scholar [20] M. Schweizer, Approximating random variables by stochastic integrals, The Annals of Probability, 22 (1994), 1536-1575.  doi: 10.1214/aop/1176988611.  Google Scholar [21] Z. W. Wang, J. M. Xia and L. H. Zhang, Optimal investment for an insurer: The martingale approach, Insurance: Mathematics and Economics, 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003.  Google Scholar [22] S. X. Xie, Continuous-time mean-variance portfolio selection with liability and regime switching, Insurance: Mathematics and Economics, 45 (2009), 148-155.  doi: 10.1016/j.insmatheco.2009.05.005.  Google Scholar [23] H. L. Yuan and Y. J. Hu, The compound Poisson risk model with interest and a threshold strategy, Stochastic Models, 25 (2009), 197-220.  doi: 10.1080/15326340902869846.  Google Scholar [24] X.-L. Zhang, K.-C. Zhang and X.-J. Yu, Optimal proportional reinsurance and investment with transaction costs, Ⅰ: Maximizing the terminal wealth, Insurance: Mathematics and Economics, 44 (2009), 473-478.  doi: 10.1016/j.insmatheco.2009.01.004.  Google Scholar [25] J. M. Zhou, X. Q. Yang and J. Y. Guo, Portfolio selection and risk control for an insurer in the Lévy market under mean-variance criterion, Statistics and Probability letters, 126 (2017), 139-149.  doi: 10.1016/j.spl.2017.03.008.  Google Scholar [26] Q. Zhou, Optimal investment for an insurer in the Lévy market: The martingale approach, Statistics and Probability Letters, 79 (2009), 1602-1607.  doi: 10.1016/j.spl.2009.03.027.  Google Scholar [27] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance: Mathematics and Economics, 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006.  Google Scholar
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