Solution of Hamilton-Jacobi-Bellman Equation in Optimal Reinsurance Strategy under Dynamic VaR Constraint

Abstract

This paper analyzes the optimal reinsurance strategy for insurers with a generalized mean-variance premium principle. The surplus process of the insurer is described by the diffusion model which is an approximation of the classical Cramér-Lunderberg model. We assume the dynamic VaR constraints for proportional reinsurance. We obtain the closed form expression of the optimal reinsurance strategy and corresponding survival probability under proportional reinsurance.

1. Introduction

In practice, reinsurance is an important way for an insurer to control its risk exposure. In the actuarial literature, the optimal reinsurance problem of minimising ruin probability or equivalently maximising survival probability has been studied extensively in the past two decades. As one type of typical reinsurance strategy, proportional reinsurance has received great attention from both the academics and practitioners. Among others, Choulli et al. (2003), Højgaard and Taksar [1, 2], Schmidli [3, 4], Taksar [5], and Zhang et al. [6] work on the proportional reinsurance.

In the existing literature, the expected value principle is commonly used as the reinsurance premium principle due to its simplicity and popularity in practice. For details, the readers are referred to Bäuerle [7], Bai and Zhang [8], and Liang and Bayraktar [9]. Generally speaking, expected value principle is commonly used in life insurance whose claim frequency and claim sizes are stable and smooth, while the variance premium principle is extensively used in property insurance; see Zhou and Yuen [10] and Sun et al. [11]. Similarly to Zhang et al. [6], in this paper, we focus on a generalized mean-variance premium principle, which includes the expected value principle and the variance principle as special cases.

More recently, the problem of optimal reinsurance design has been studied by using risk measures such as the Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and conditional tail expectation (CTE) (to name a few, Cai and Tan [12], Cheung et al. [13], and Cai et al. [14, 15]). Latterly static risk measures have been extended to the dynamic version; see Yiu [16], Alexander and Baptista [17], Cuoco et al. [18], Chen et al. [19], and Zhang et al. [6], all of which investigate the optimal reinsurance problem under dynamic VaR constraint.

In this paper, we investigate an optimal proportional reinsurance problem under dynamic VaR constraint. Assume that an insurer aims to maximize the survival probability. With this assumption, we obtain the closed form expressions. The rest of the paper is organized as follows. In Section 2, we provide a general formulation of the optimal reinsurance problem. Then we investigate the insurance company’s maximum survival probability under dynamic VaR constraints, and the corresponding optimal reinsurance strategy is given in proportional reinsurance settings in Section 3.

2. Formulation

Let be a probability space with a filtration . Consider a Cramér-Lundberg model with the surplus process of an insurance company being given by where is the initial surplus, the claim arrival process is a Poisson process with constant intensity , and the random variables , , are i.i.d claim sizes independent of . We let denote the -th claim occurrence time and denote the claim size distribution with finite first and second moments , . The premium rate is assumed to be calculated via the expected value principle; that is, where is the relative loading factor.

In this paper, the insurer can purchase proportional reinsurance to adjust the exposure to insurance risk. The proportional reinsurance level is associated with the risk exposure at time . We assume for all , and it means the insurer purchases proportional reinsurance. In this case, for each claim, the insurer only pays its , while the reinsurer pays the rest for each claim.

For a chosen reinsurance policy , let denote the associated surplus process; that is, is the surplus of insurer at time t. This process then evolves as where is the net reinsurance rate which the reinsurer receives from the insurer. We assume that the reinsurance premium is calculated by the following generalized mean-variance principle , where , and and denote the expectation and variance, respectively. Thus we haveand the premium rate for the insurer is

According to Grandell (1991), the surplus process after reinsurance can be approximated by the following diffusion process:where is a standard Brownian motion.

We define the ruin time where the superscript emphasises that the surplus process and the ruin time are controlled by an admissible policy . Denote the survival probability given the initial surplus by and the maximum survival probability by

Our objective is to find the value function and the optimal policy such that

3. Maximizing Survival Probability

Under the proportional reinsurance, the insurer could transfer a fraction of the incoming claims to a reinsurer, where is -measurable and satisfies for all . The diffusion approximation of insurance company’s claim process becomes where is a standard Brownian motion. The insurer’s surplus process satisfies the stochastic differential equation

Taking is small enough, we assume that risk exposure does not change over the short time period . This means that the risk exposure remains roughly constant in the given time period; that is, , , . This setting is reasonable because the insurer can only adjust its reinsurance business at discrete time; and the decision made is based on the holding at time . Thus, we rewrite the claim dynamics as

3.1. Dynamic VaR, CVaR, and Worst-Case CVaR

For a given confidence level and a given horizon , the VaR at time of a proportional reinsurance policy , denoted by , is defined as The dynamic Conditional Value-at-Risk is given by The dynamic worst-case CVaR is defined as where

Proposition 1 (Zhang et al. [6]). where and denote the probability density function and the cumulative distribution function of a standard normal random variable, respectively. is the inverse function of .

3.2. HJB Equation

Using the dynamic programming technique, we obtain that the value function satisfies the following Hamilton-Jacobi-Bellman (HJB) equation:where () is a constant.

Next we try to construct a solution of the HJB equation (19) with the boundary condition (20). Suppose that , with , satisfies (19) and (20).

Theorem 2. (a) If , the functionis a smooth () solution to the HJB equation, where, , . The maximum of the left side of HJB equation is attained at(b) If , the functionis a smooth () solution to the HJB equation, where, . The maximum of the left side of HJB equation is attained at

Proof. We solve the HJB equation analytically. First we need to determine the optimal strategy . Differentiating the terms inside the maximum in (19) with respect to and setting to 0 yieldThe dynamic VaR constraint implies , when is defined by (27). Normally, we take ; hence, is always positive.
(1) For , we have . Then, from obtained from the dynamic VaR constraint and the requirement that the retained proportion of claims is always within , we have .
(a) If , we let , and then the HJB equation becomeswhich impliesInserting it into (28), we obtain(i)If , we have ; consequently , and then the HJB equation becomes (30).(ii)If , we have , where conflict exits.(b) If , we have , and then the HJB equation becomeswhich implies(i)If , we have , where conflict exits.(ii)If , we have ; consequently , and then the HJB equation becomes (32).(2) When , we have ; thus .
(a) If , we have , and then the HJB equation becomes We havewhich impliesWe have when , and we have when . We have and when .
For , we have the following. (i)If , we have ; consequently , and then the HJB equation becomes (34).(ii)If , we have , where conflict exits.For , we have ; therefore is convex for small . Through the analysis of the HJB equation (19), for , the maximum of the left side of the HJB is attained at and the HJB equation becomes (34).
(b) When , it is reasonable to let . Similar to (a), we have the following conclusions.
For , the optimal strategy is obtained at .
For , we have the following. (i)If , we have , where conflict exits.(ii)If , we have ; consequently, , and then the HJB equation becomes (32). From the previous analysis, we have the following conclusions. (i)If , the maximum of the left side of HJB equation is attained at(ii)If , the maximum of the left side of HJB equation is attained atIn the following, we will solve the HJB equation in each situation.
For and , the HJB equation is (34), which is equivalent to (35). Taking integral from to , we obtainwhere , , and . Applying the boundary condition we obtainFor and , the corresponding HJB is (29), which is equivalent to (30). Taking integral form to , we obtainApplying the boundary condition we obtainConsidering that is twice continuously differentiable, it should satisfy ; that is,which leads toThus, if , we have the functionIf and , the HJB equation is (34), and the HJB equation is (32) for and . From the procedure that is similar to the previous analysis we can get the following function is a solution to HJB; that is,whereThis ends the proof.

When the value function is twice continuously differentiable, then it is the unique solution of the HJB equation (see, e.g., [20]), and we have the following result.

Proposition 3. The value function coincides with the smooth function defined in Theorem 2 and the optimal control, which represents the optimal proportional reinsurance strategy, is described by the in Theorem 2, where is the corresponding surplus process.

Remark 4. When , Theorem 2 coincides with theorem 3.1 in Zhang et al. [6].

Corollary 5. When , the generalized mean-variance premium principle is mean-variance premium principle, and we have the following:is a smooth () solution to the HJB equation, where. The maximum of the left side of HJB equation is attained at

Corollary 6. When there is no dynamic VaR, CVaR, or wcCVaR constraints, that is, , and the model becomes the unconstrained reinsurance problem, we have the following.
(a) If , the optimal reinsurance strategy is , and the optimal survival probability is(b) If , the optimal reinsurance strategy is , and the optimal survival probability is

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the manuscript. All authors read and approved the final manuscript.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (No. 11501319, No. 11571198), the Education Department of Shandong Province Science and Technology Plan Project (No. J15L105), the Natural Science Foundation of Shandong Province (No. ZR2015AL013), and the China Postdoctoral Science Foundation (No. 2015M582064).