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July
2021, 17(4): 1887-1912.
doi: 10.3934/jimo.2020051
Optimal mean-variance reinsurance in a financial market with stochastic rate of return
| 1. | College of Science, Civil Aviation University of China, Tianjin 300300, China |
| 2. | School of Mathematical Sciences, Nankai University, Tianjin 300071, China |
| 3. | School of Statistics, Qufu Normal University, Qufu, Shandong 273165, China |
In this paper, we investigate the optimal investment and reinsurance strategies for a mean-variance insurer when the surplus process is represented by a Cramér-Lundberg model. It is assumed that the instantaneous rate of investment return is stochastic and follows an Ornstein-Uhlenbeck (OU) process, which could describe the features of bull and bear markets. To solve the mean-variance optimization problem, we adopt a backward stochastic differential equation (BSDE) approach and derive explicit expressions for both the efficient strategy and efficient frontier. Finally, numerical examples are presented to illustrate our results.
Keywords:
Investment-reinsurance strategy,
Rate of investment return,
Ornstein-Uhlenbeck process,
Backward stochastic differential equation,
Efficient frontier.
Mathematics Subject Classification:
Primary: 93E20; Secondary: 91B30, 91G80.
Citation:
Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial and Management Optimization,
2021, 17
(4)
: 1887-1912.
doi: 10.3934/jimo.2020051
![]()
References:
| [1] |
P. Azcue and N. Muler,
Optimal reinsurance and dividend distribution policies in the Cramér Lundberg model, Mathematical Finance, 15 (2005), 261-308.
doi: 10.1111/j.0960-1627.2005.00220.x. |
| [2] |
S. Asmussen, B. Højgaard and M. Taksar,
Optimal risk control and dividend distribution policies: Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.
doi: 10.1007/s007800050075. |
| [3] |
N. Bäuerle,
Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.
doi: 10.1007/s00186-005-0446-1. |
| [4] |
L. H. Bai and H. Y. Zhang,
Dynamic mean-variance problem with constraint risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205.
doi: 10.1007/s00186-007-0195-4. |
| [5] |
C. Bender and M. Kohlmann, BSDEs with Stochastic Lipschitz Condition, Universität Konstanz, Fakultät für Mathematik and Informatik, 2000. |
| [6] |
J. N. Bi and J. Y. Guo,
Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, Journal of Optimization Theory and Applications, 157 (2013), 252-275.
doi: 10.1007/s10957-012-0138-y. |
| [7] |
J. N. Bi, Z. B. Liang and F. J. Xu,
Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insurance: Mathematics and Economics, 70 (2016), 245-258.
doi: 10.1016/j.insmatheco.2016.06.012. |
| [8] |
T. R. Bielecki, H. Q. Jin, S. R. Pliskaz and X. Y. Zhou,
Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.
doi: 10.1111/j.0960-1627.2005.00218.x. |
| [9] |
S. Browne,
Optimal investment policies for a firm with a random risk process: Exponentional utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.
doi: 10.1287/moor.20.4.937. |
| [10] |
F.J. Fabozzi and J. C. Francis,
Mutual fund systematic risk for bull and bear markets: An empirical examination,, Journal of Finance, 34 (1979), 1243-1250.
doi: 10.1111/j.1540-6261.1979.tb00069.x. |
| [11] |
W. H. Fleming and H. M. Soner, Controled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. |
| [12] |
M. Kobylanski,
Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 28 (2000), 558-602.
doi: 10.1214/aop/1019160253. |
| [13] |
X. Li, X. Y. Zhou and A. E. B. Lim,
Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.
doi: 10.1137/S0363012900378504. |
| [14] |
Z. B. Liang, K. C. Yuen and J. Y. Guo,
Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Mathematics and Economics, 49 (2011), 207-215.
doi: 10.1016/j.insmatheco.2011.04.005. |
| [15] |
A. E. B. Lim,
Mean-variance hedging when there are jumps, SIAM Journal on Control and Optimization, 44 (2005), 1893-1922.
doi: 10.1137/040610933. |
| [16] |
A. E. B. Lim,
Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161.
doi: 10.1287/moor.1030.0065. |
| [17] |
H. M. Markowitz,
Portfolio selection, Journal of Finance, 7 (1952), 77-91.
|
| [18] |
R. C. Merton,
Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183.
doi: 10.2307/3003143. |
| [19] |
E. Pardoux and A. Rǎşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied probability, 69. Springer, Switzerland, 2014.
doi: 10.1007/978-3-319-05714-9. |
| [20] |
R. Rishel, Optimal portfolio management with partial observation and power utility function, Stochastic Analysis, Control, Optimization and Applications, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, (1999), 605–619. |
| [21] |
H. Schmidli,
On minimizing the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907.
doi: 10.1214/aoap/1031863173. |
| [22] |
Y. Shen,
Mean-variance portfolio selection in a complete market with unbounded random coefficients, Automatica J. IFAC, 55 (2015), 165-175.
doi: 10.1016/j.automatica.2015.03.009. |
| [23] |
Y. Shen and Y. Zeng,
Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process, Insurance: Mathematics and Economics, 62 (2015), 118-137.
doi: 10.1016/j.insmatheco.2015.03.009. |
| [24] |
Y. Shen, X. Zhang and T. K. Siu,
Mean-variance portfolio selection under a constant elasticity of variance model, Operations Research Letters, 42 (2014), 337-342.
doi: 10.1016/j.orl.2014.05.008. |
| [25] |
Z. Y. Sun,
Upper bounds for ruin probabilities under model uncertainty, Communications in Statistics-Theory and Methods, 48 (2019), 4511-4527.
doi: 10.1080/03610926.2018.1491991. |
| [26] |
Z. Y. Sun and J. Y. Guo,
Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility, Mathematical Methods of Operations Research, 88 (2018), 59-79.
doi: 10.1007/s00186-017-0628-7. |
| [27] |
Z. Y. Sun and X. P. Guo,
Equilibrium for a time-inconsistent stochastic linear-quadratic control system with jumps and its application to the mean-variance problem, Journal of Optimization Theory and Applications, 181 (2019), 383-410.
doi: 10.1007/s10957-018-01471-x. |
| [28] |
Z. Y. Sun, K. C. Yuen and J. Y. Guo, A BSDE approach to a class of dependent risk model of mean-variance insurers with stochastic volatility and no-short selling, Journal of Computational and Applied Mathematics, 366 (2019), 112413, 21 pp.
doi: 10.1016/j.cam.2019.112413. |
| [29] |
Z. Y. Sun, X. Zhang and K. C. Yuen, Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option, Scandinavian Actuarial Journal, (2019), DOI: https://doi.org/10.1080/03461238.2019.1658619.
doi: 10.1080/03461238.2019.1658619. |
| [30] |
Z. Y. Sun, X. X. Zheng and X. Zhang,
Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686.
doi: 10.1016/j.jmaa.2016.09.053. |
| [31] |
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. |
| [32] |
H. L. Yang and L. H. Zhang,
Optimal investment for an insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.
doi: 10.1016/j.insmatheco.2005.06.009. |
| [33] |
J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamilton Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
| [34] |
M. Zhang and P. Chen,
Mean-variance asset-liability management under constant elasticity of variance process, Insurance: Mathematics and Economics, 70 (2016), 11-18.
doi: 10.1016/j.insmatheco.2016.05.019. |
| [35] |
X. Y. Zhou and D. Li,
Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.
doi: 10.1007/s002450010003. |
| [36] |
X. Y. Zhou and G. Yin,
Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.
doi: 10.1137/S0363012902405583. |
show all references
References:
| [1] |
P. Azcue and N. Muler,
Optimal reinsurance and dividend distribution policies in the Cramér Lundberg model, Mathematical Finance, 15 (2005), 261-308.
doi: 10.1111/j.0960-1627.2005.00220.x. |
| [2] |
S. Asmussen, B. Højgaard and M. Taksar,
Optimal risk control and dividend distribution policies: Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.
doi: 10.1007/s007800050075. |
| [3] |
N. Bäuerle,
Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.
doi: 10.1007/s00186-005-0446-1. |
| [4] |
L. H. Bai and H. Y. Zhang,
Dynamic mean-variance problem with constraint risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205.
doi: 10.1007/s00186-007-0195-4. |
| [5] |
C. Bender and M. Kohlmann, BSDEs with Stochastic Lipschitz Condition, Universität Konstanz, Fakultät für Mathematik and Informatik, 2000. |
| [6] |
J. N. Bi and J. Y. Guo,
Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, Journal of Optimization Theory and Applications, 157 (2013), 252-275.
doi: 10.1007/s10957-012-0138-y. |
| [7] |
J. N. Bi, Z. B. Liang and F. J. Xu,
Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insurance: Mathematics and Economics, 70 (2016), 245-258.
doi: 10.1016/j.insmatheco.2016.06.012. |
| [8] |
T. R. Bielecki, H. Q. Jin, S. R. Pliskaz and X. Y. Zhou,
Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15 (2005), 213-244.
doi: 10.1111/j.0960-1627.2005.00218.x. |
| [9] |
S. Browne,
Optimal investment policies for a firm with a random risk process: Exponentional utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.
doi: 10.1287/moor.20.4.937. |
| [10] |
F.J. Fabozzi and J. C. Francis,
Mutual fund systematic risk for bull and bear markets: An empirical examination,, Journal of Finance, 34 (1979), 1243-1250.
doi: 10.1111/j.1540-6261.1979.tb00069.x. |
| [11] |
W. H. Fleming and H. M. Soner, Controled Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. |
| [12] |
M. Kobylanski,
Backward stochastic differential equations and partial differential equations with quadratic growth, The Annals of Probability, 28 (2000), 558-602.
doi: 10.1214/aop/1019160253. |
| [13] |
X. Li, X. Y. Zhou and A. E. B. Lim,
Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555.
doi: 10.1137/S0363012900378504. |
| [14] |
Z. B. Liang, K. C. Yuen and J. Y. Guo,
Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Mathematics and Economics, 49 (2011), 207-215.
doi: 10.1016/j.insmatheco.2011.04.005. |
| [15] |
A. E. B. Lim,
Mean-variance hedging when there are jumps, SIAM Journal on Control and Optimization, 44 (2005), 1893-1922.
doi: 10.1137/040610933. |
| [16] |
A. E. B. Lim,
Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29 (2004), 132-161.
doi: 10.1287/moor.1030.0065. |
| [17] |
H. M. Markowitz,
Portfolio selection, Journal of Finance, 7 (1952), 77-91.
|
| [18] |
R. C. Merton,
Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183.
doi: 10.2307/3003143. |
| [19] |
E. Pardoux and A. Rǎşcanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied probability, 69. Springer, Switzerland, 2014.
doi: 10.1007/978-3-319-05714-9. |
| [20] |
R. Rishel, Optimal portfolio management with partial observation and power utility function, Stochastic Analysis, Control, Optimization and Applications, Systems Control Found. Appl., Birkhäuser Boston, Boston, MA, (1999), 605–619. |
| [21] |
H. Schmidli,
On minimizing the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907.
doi: 10.1214/aoap/1031863173. |
| [22] |
Y. Shen,
Mean-variance portfolio selection in a complete market with unbounded random coefficients, Automatica J. IFAC, 55 (2015), 165-175.
doi: 10.1016/j.automatica.2015.03.009. |
| [23] |
Y. Shen and Y. Zeng,
Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process, Insurance: Mathematics and Economics, 62 (2015), 118-137.
doi: 10.1016/j.insmatheco.2015.03.009. |
| [24] |
Y. Shen, X. Zhang and T. K. Siu,
Mean-variance portfolio selection under a constant elasticity of variance model, Operations Research Letters, 42 (2014), 337-342.
doi: 10.1016/j.orl.2014.05.008. |
| [25] |
Z. Y. Sun,
Upper bounds for ruin probabilities under model uncertainty, Communications in Statistics-Theory and Methods, 48 (2019), 4511-4527.
doi: 10.1080/03610926.2018.1491991. |
| [26] |
Z. Y. Sun and J. Y. Guo,
Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility, Mathematical Methods of Operations Research, 88 (2018), 59-79.
doi: 10.1007/s00186-017-0628-7. |
| [27] |
Z. Y. Sun and X. P. Guo,
Equilibrium for a time-inconsistent stochastic linear-quadratic control system with jumps and its application to the mean-variance problem, Journal of Optimization Theory and Applications, 181 (2019), 383-410.
doi: 10.1007/s10957-018-01471-x. |
| [28] |
Z. Y. Sun, K. C. Yuen and J. Y. Guo, A BSDE approach to a class of dependent risk model of mean-variance insurers with stochastic volatility and no-short selling, Journal of Computational and Applied Mathematics, 366 (2019), 112413, 21 pp.
doi: 10.1016/j.cam.2019.112413. |
| [29] |
Z. Y. Sun, X. Zhang and K. C. Yuen, Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option, Scandinavian Actuarial Journal, (2019), DOI: https://doi.org/10.1080/03461238.2019.1658619.
doi: 10.1080/03461238.2019.1658619. |
| [30] |
Z. Y. Sun, X. X. Zheng and X. Zhang,
Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686.
doi: 10.1016/j.jmaa.2016.09.053. |
| [31] |
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. |
| [32] |
H. L. Yang and L. H. Zhang,
Optimal investment for an insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.
doi: 10.1016/j.insmatheco.2005.06.009. |
| [33] |
J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamilton Systems and HJB Equations, Applications of Mathematics (New York), 43. Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
| [34] |
M. Zhang and P. Chen,
Mean-variance asset-liability management under constant elasticity of variance process, Insurance: Mathematics and Economics, 70 (2016), 11-18.
doi: 10.1016/j.insmatheco.2016.05.019. |
| [35] |
X. Y. Zhou and D. Li,
Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.
doi: 10.1007/s002450010003. |
| [36] |
X. Y. Zhou and G. Yin,
Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.
doi: 10.1137/S0363012902405583. |
Figure 2.
the optimal investment strategy with $ \alpha = -0.04, \beta = 0.03 $ and $ \alpha = \beta = 0 $
Figure 3.
the optimal reinsurance strategy with $ \alpha = -0.04, \beta = 0.03 $ and $ \alpha = \beta = 0 $
Figure 4.
the effect of $ \alpha $ on the efficient frontier
Figure 5.
the effect of $ \lambda $ on the efficient frontier
Figure 6.
the effect of $ \eta $ on the efficient frontier
Figure 7.
the effect of $ r $ on the efficient frontier
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