Optimal per-loss reinsurance and investment to minimize the probability of drawdown

Xia Han; Zhibin Liang; Yu Yuan; Caibin Zhang

doi:10.3934/jimo.2021145

Article Contents

2022, Volume 18, Issue 6: 4011-4041. Doi: 10.3934/jimo.2021145

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Optimal per-loss reinsurance and investment to minimize the probability of drawdown

1.
Department of Statistics and Actuarial Science, University of Waterloo, Ontario, N2L 3G1, Canada
2.
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu, 210023 China
3.
School of Management Science and Engineering, Nanjing University of Information Science and Technology, Jiangsu, 210044, China
4.
School of Finance, Nanjing University of Finance and Economics, Jiangsu, 210023, China

^*Corresponding author: Zhibin Liang
^*Corresponding author: Zhibin Liang

Received: December 31, 2020

Revised: April 30, 2021

Early access: September 2021

Published: September 2022

This research was supported by the National Natural Science Foundation of China (Grant No.12071224)

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Abstract

In this paper, we study an optimal reinsurance-investment problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. We assume that the insurer can purchase per-loss reinsurance for each line of business and invest its surplus in a financial market consisting of a risk-free asset and a risky asset. Under the criterion of minimizing the probability of drawdown, the closed-form expressions for the optimal reinsurance-investment strategy and the corresponding value function are obtained. We show that the optimal reinsurance strategy is in the form of pure excess-of-loss reinsurance strategy under the expected value principle, and under the variance premium principle, the optimal reinsurance strategy is in the form of pure quota-share reinsurance. Furthermore, we extend our model to the case where the insurance company involves $ n $ $ (n\geq3) $ dependent classes of insurance business and the optimal results are derived explicitly as well.

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Mathematics Subject Classification: Primary: 93E20; Secondary: 91G05, 91G10.

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Figure 1. The function $ G(7,d) $

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Figure 2. The effect of $ u $ on optimal strategy

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Figure 3. The effect of η₁ on optimal strategy (η₂ = 0.25)

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Figure 4. The effect of $ \eta_2 $ on optimal strategy ($ \eta_1 = 0.4 $)

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Figure 5. The effect of $ \lambda $ on optimal strategy

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Figure 6. The effect of $ \lambda $ on correlation coefficient

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Figure 7. Comparison between the optimal strategies

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Figure 8. Comparison between the value functions

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