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Optimal robust reinsurance-investment strategies for insurers with mean reversion and mispricing. (English) Zbl 1402.91196

Summary: This paper considers how to optimize reinsurance and investment decisions for an insurer who has aversion to model ambiguity, who wants to take into consideration time-varying investment conditions via mean reverting models, and who wants to take advantage of statistical arbitrage opportunities afforded by mispricing of stocks. We work under a complex realistic environment: The surplus process is described by a jump-diffusion model and the financial market contains a market index, a risk-free asset, and a pair of mispriced stocks, where the expected return rate of the stocks and the mispricing follow mean reverting stochastic processes which take into account liquidity constraints. The insurer is allowed to purchase reinsurance and to invest in the financial market. We formulate an optimal robust reinsurance-investment problem under the assumption that the insurer is ambiguity-averse to the uncertainty from the financial market and to the uncertainty of the insured’s claims. Ambiguity aversion is an aversion to the uncertainty taken by making investment decisions based on a misspecified model. By employing the dynamic programming approach, we derive explicit formulae for the optimal robust reinsurance-investment strategy and the optimal value function. Numerical examples are presented to illustrate the impact of some parameters on the optimal strategy and on utility loss functions. Among our various practical findings and recommendations, we find that strengthened market liquidity significantly increases the demand for hedging from the mispriced market, to take advantage of the statistical arbitrage it affords.

MSC:

91B30 Risk theory, insurance (MSC2010)
90C39 Dynamic programming
91G10 Portfolio theory
93E20 Optimal stochastic control
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[1] Anderson, E., Hansen, L.P., Sargent, T., 2000. Robustness, detection and the price of risk. Manuscript, Stanford, 19.; Anderson, E., Hansen, L.P., Sargent, T., 2000. Robustness, detection and the price of risk. Manuscript, Stanford, 19.
[2] Anderson, E.; Hansen, L.; Sargent, T., A semigroups for model specification, robustness, price of risk, and model detection, J. Eur. Econom. Assoc., 1, 68-123, (2003)
[3] Baev, A. V.; Bondarev, B. V., On the ruin probability of an insuance company dealing in a BS-market, Theory Probab. Math. Statist., 74, 11-23, (2007) · Zbl 1150.91421
[4] Bai, L. H.; Guo, J. Y., Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42, 968-975, (2008) · Zbl 1147.93046
[5] Bäuerle, N., Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res, 62, 1, 159-165, (2005) · Zbl 1101.93081
[6] Cochrane, J., Asset pricing, (2001), Princeton University Press Princeton · Zbl 1140.91041
[7] Fouque, J. P.; Papanicolaou, G.; Sircar, R., Derivatives in financial markets with stochastic volatility, (2000), Cambridge University Press Cambridge, UK · Zbl 0954.91025
[8] Gu, A. L.; Guo, X. P.; Li, Z. F.; Zeng, Y., Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom., 51, 674-684, (2012) · Zbl 1285.91057
[9] Gu, A. L.; Li, Z. F.; Zeng, Y., Optimal investment strategy under Ornstein-Uhlenbeck model for a DC pension plan, Acta Math. Appl. Sin., 36, 4, 715-726, (2013) · Zbl 1299.91119
[10] Gu, A. L.; Viens, F. G.; Yi, B., Optimal reinsurance and investment strategies for insurers with mispricing and model ambiguity, Insurance Math. Econom., 72, 235-249, (2017) · Zbl 1394.91216
[11] Honda, T.; Kamimura, S., On the verification theorem of dynamic portfolio-consumption problems with stochastic market price of risk, Asia-Pac. Financ. Mark., 18, 151-166, (2011) · Zbl 1278.91140
[12] Jurek, J.W., Yang, H., 2006. Dynamic Portfolio Selection in Arbitrage. Working Paper, Harvard University.; Jurek, J.W., Yang, H., 2006. Dynamic Portfolio Selection in Arbitrage. Working Paper, Harvard University.
[13] Korn, R.; Menkens, O.; Steffensen, M., Worst-case-optimal dynamic reinsurance for large claims, Eur. Actuar. J., 2, 1, 21-48, (2012) · Zbl 1269.91044
[14] Li, D.; Zeng, Y.; Yang, H., Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scand. Actuar. J., (2017)
[15] Liang, Z. B.; Yuen, K. C.; Guo, J. Y., Optimal reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Math. Econ., 49, 207-215, (2011) · Zbl 1218.91084
[16] Liu, J.; Longstaff, F. A., Losing money on arbitrage: optimal dynamic portfolio choice in markets with arbitrage opportunities, Rev. Financ. Stud., 17, 611-641, (2004)
[17] Liu, J.; Timmermann, A., Optimal convergence trade strategies, Rev. Financ. Stud., 26, 1048-1086, (2013)
[18] Maenhout, P. J., Robust portfolio rules and asset pricing, Rev. Financ. Stud., 17, 951-983, (2004)
[19] Maenhout, P. J., Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium, J. Econom. Theory, 128, 136-163, (2006) · Zbl 1152.91535
[20] Mataramvura, S.; Øksendal, B., Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics, 80, 4, 317-337, (2008) · Zbl 1145.93054
[21] Merton, R. C., On estimating the expected return on the market: an exploratory investigation, J. Financ. Econ., 8, 4, 323-361, (1980)
[22] Mukherji, S., Are stock returns still mean-reverting?, Rev. Financ. Econ., 20, 22-27, (2011)
[23] Poterba, J. P.; Summers, L. H., Mean reversion in stock prices: evidence and implications, J. Financ. Econ., 22, 27-59, (1988)
[24] Pun, C. S.; Wong, H. Y., Robust investment-reinsurance optimization with multiscale stochastic volatility, Insurance Math. Econom., 62, 245-256, (2015) · Zbl 1318.91122
[25] Rapach, D.; Zhou, G., Forecasting stock return, (Handbook of Economic Forecasting, Vol. 2, (2013)), 328-383, (Chapter 6)
[26] Schmidli, H., On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12, 890-907, (2002) · Zbl 1021.60061
[27] Uppal, R.; Wang, T., Model misspecification and underdiversification, J. Finance, 58, 2465-2486, (2003)
[28] Yi, B.; Li, Z. F.; Viens, F.; Zeng, Y., Robust optimal control for an insurer with reinsurance and investment under heston’s stochastic volatility model, Insurance Math. Econom., 53, 601-614, (2013) · Zbl 1290.91103
[29] Yi, B.; Viens, F.; Law, B.; Li, Z. F., Dynamic portfolio selection with mispricing and model ambiguity, Ann. Finance, 11, 1, 37-75, (2015) · Zbl 1311.91176
[30] Yi, B.; Viens, F.; Li, Z. F.; Zeng, Y., Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scand. Actuar. J., 8, 725-751, (2015) · Zbl 1401.91208
[31] Zeng, Y.; Li, D. P.; Gu, A. L., Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance Math. Econom., 66, 138-152, (2016) · Zbl 1348.91192
[32] Zeng, Y.; Li, Z. F., Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49, 145-154, (2011) · Zbl 1218.91167
[33] Zhang, X.; Siu, T. K., Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Math. Econom., 45, 81-88, (2009) · Zbl 1231.91257
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