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Optimal Reinsurance-Investment Strategies for Insurers Under Mean-Car Criteria JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2012.8.673 MANAGEMENT OPTIMIZATION Volume 8, Number 3, August 2012 pp. 673–690 OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS UNDER MEAN-CAR CRITERIA Yan Zeng Lingnan (University) College Sun Yat-sen University, Guangzhou 510275, China Zhongfei Li1 Lingnan (University) College/Sun Yat-sen Business School Sun Yat-sen University, Guangzhou 510275, China (Communicated by Vladimir Veliov) Abstract. This paper considers an optimal reinsurance-investment problem for an insurer, who aims to minimize the risk measured by Capital-at-Risk (CaR) with the constraint that the expected terminal wealth is not less than a predefined level. The surplus of the insurer is described by a Brownian motion with drift. The insurer can control her/his risk by purchasing pro- portional reinsurance, acquiring new business, and investing her/his surplus in a financial market consisting of one risk-free asset and multiple risky assets. Three mean-CaR models are constructed. By transforming these models into bilevel optimization problems, we derive the explicit expressions of the optimal deterministic rebalance reinsurance-investment strategies and the mean-CaR efficient frontiers. Sensitivity analysis of the results and a numerical example are provided. 1. Introduction. Since reinsurance is an effective way to spread risk and invest- ment is an increasing important element in the insurance business, many optimiza- tion problems have arisen in the last decades. These problems are subject to the control of reinsurance and/or investment, with various objectives in insurance risk management. This topic has attracted a great deal of interest, and has been ex- tensively investigated in literature. For example, Browne [7], Cao and Wan [8], Irgens and Paulsen [16], Liang [20], Xu et al. [34], and Yang and Zhang [35] focus on studying the optimal reinsurance and/or investment strategies to maximize the expected utility from the terminal wealth of insurers. Azcue and Muler [2], Bai and Guo [3], Hipp and Plum [14], Liang and Guo ([21], [22]), Liu and Yang [23], Luo [24], Luo et al. [25], Promislow and Young [29], Schmidli ([30], [31]), and Taksar and 2000 Mathematics Subject Classification. Primary: 90C26; Secondary: 91B28, 49N15. Key words and phrases. Optimal proportional reinsurance-investment strategy, insurers, Capital-at-Risk, Hamilton-Jacobi-Bellman equation. This research was partially supported by the National Science Foundation for Distinguished Young Scholars (No. 70825002), China Postdoctoral Science Foundation funded project (No. 2011M501351), Humanity and Social Science Foundation of Ministry of Education of China (No. 12YJCZH267), the High-level Talent Project of Guangdong“Research on Models and Strategies for Optimal Reinsurance, Investment and Dividend”, Philosophy and Social Science Program- ming Foundation of Guangdong Province (No. GD11YYJ07), and “985 Project” of Sun Yat-sen University. 1Corresponding author. 673 674 YANZENGANDZHONGFEILI Markussen [32] investigate the optimal reinsurance and/or investment strategies in the sense of minimizing the ruin probability of insurers. Bai and Zhang [4], B¨auerle [5], Delong and Gerrard [9], Li et al. [19], Wang et al. [33], Zeng and Li [36], and Zeng et al. [37] concentrate on seeking the optimal reinsurance and/or investment strategies under mean-variance criterion. In the above mentioned papers, three criteria: (1) maximizing the expected utility from the terminal wealth, (2) minimizing the ruin probability, and (3) mean- variance criterion, are used to determine the optimal strategies. However, there are some drawbacks of adopting these three criteria to determine the optimal strategies for insurers. Maximizing the expected utility from the terminal wealth criterion can not illustrate the magnitude of risk. Even for some special utility functions, such as the exponential utility function, the optimal investment strategy is independent of the parameters of the insurance market; thus, the strategy is unrealistic. Min- imizing the ruin probability criterion emphasizes only risk rather than profit, and this focus only on risk does not coincide with the investment intentions of insur- ers. Mean-variance criterion can reveal profit and risk directly, but the variance as a measure of risk has the drawback that it penalizes both upside and downside moments in the portfolio value, which is inconsistent with what insurers expect. To complement the deficiencies, scholars have proposed some alternative risk mea- sures, such as Value-at-Risk (VaR), semivariance, shortfall probability, the limited expected loss, and so on. Among them, VaR is the most notable risk measure, and has been widely accepted by the banks and regulators as the benchmark for control- ling market risk. Recently, Emmer et al. ([10], [11]) define a VaR-based related risk measure called Capital-at-Risk (CaR), and incorporate it in the portfolio optimiza- tion problems. They derive explicitly the optimal investment strategies when the trading strategies are restricted to constant rebalance portfolio (CRP) strategies. Li et al. [18] formulate three mean-CaR models to investigate the continuous-time optimal portfolio selection problems, and derive the corresponding explicit optimal CRP strategies. The CRP strategies mentioned above are the investment strategies which suppose that the proportions of the total wealth invested in each available assets are the same at any time point, regardless of the level of wealth. It is worth pointing out that the CRP strategies are dynamic investment strategies in that they require trading over time, and are also known as the constant mix strategies. These strategies are widely studied in the existing literature. Merton ([26], [27]) indicates that this form of strategies is optimal to the portfolio selection problems of max- imizing expected utility with constant relative risk-aversion. Black and Perold [6] and Perold and Shapre [28] use those strategies in asset allocation practice. Irgens and Paulson [16] investigate the optimal strategies of risk exposure, reinsurance and investments for insurers with the restriction of adopting the constant mix strategies to maximize the expected utility from the terminal wealth. In this paper, we consider an optimal proportional reinsurance-investment prob- lem for an insurer by adopting CaR to measure the risk. Specifically, the surplus of the insurer is described by a diffusion approximation model which has been widely adopted in the existing papers, for example, in Browne [7], Luo [24], Promislow and Young [29], Taksar and Markussen [32], and so on, to investigate optimization problems related to insurers. The insurer in this paper is allowed to purchase pro- portional reinsurance, acquire new business (as a reinsurer of other insurers), and invest her/his surplus in a financial market, which consists of one risk-free asset OPTIMAL REINSURANCE-INVESTMENT STRATEGIES FOR INSURERS 675 and multiple risky assets. We construct three mean-CaR models in order to min- imize the risk measured by CaR under the constraint that the expected terminal wealth is not less than a predetermined level. To ensure the tractability of the optimization problems in this paper, we restrict ourselves to adopting deterministic rebalance (DR) strategies. In this paper, a DR strategy is a reinsurance-investment strategy which requires that the risk exposure (the retention level of proportional reinsurance/new business) is proportionate to the total wealth with a proportion- ality factor, which is a deterministic function with respect to (w.r.t.) time t, and the proportions of the total wealth invested in each of available assets are also de- terministic functions w.r.t time t. The DR strategy is an extension of the CRP strategy, and results in much broader trading strategies. Such DR strategy also has the advantages of the CRP strategy, which are discussed in Sections 3.3 and 3.4 of Emmer et al. [11] and Korn [17]. By transforming the original optimization prob- lems into bilevel optimization problems, we derive the explicit expressions of the optimal proportional reinsurance-investment strategies and the mean-CaR efficient frontiers of the three mean-CaR models. We generalize the assumptions of Emmer et al. [11], Irgens and Paulsen [16] and Li et al. [18] in the sense that the market coefficients as well as the trading strategies are time dependent, and generalize the models of Emmer et al. [11] and Li et al. [18] in the sense that the control of risk exposure is introduced. In addition, our results can reduce to the ones of Emmer et al. [11] and Li et al. [18]. In the next section we state the basic assumptions of this paper. Section 3 provides the definition and some properties of CaR. Section 4 solves three mean- CaR models. Section 5 provides sensitivity analysis of our results, and presents a numerical example to illustrate our results. Section 6 concludes this paper. 2. Basic assumptions. To provide a rigorous mathematical formulation of the optimization problems, we start with a filtered probability space (Ω, , F := Ft : 0 t T , P ), where T is a finite and positive constant, representingF the{ time ≤ ≤ } horizon; the filtration Ft represents the information available at time t, and any decision is made based upon this information. All stochastic processes introduced below are assumed to be the adapted processes in this space. 2.1. The surplus process. In this paper, we consider an insurer whose surplus process is described by a diffusion approximation model. For the purpose of moti- vation, we start from the classical Cram´er-Lundberg (C-L) model. In this model, without reinsurance and investment, the surplus process of the insurer is given by Nt R(t)= x + pt Z , (1) − i i=1 X where x is the initial surplus of the insurer; p is the premium rate; Nt, 0 t T , representing the number of claims occurring in [0,t], is a homogeneous{ ≤ Poisson≤ } process with intensity λ; the claim sizes Zi, i = 1, 2, are assumed to be independent and identically distributed positive{ random variables···} with finite first 2 and second-order moments given by µ∞ and σ∞, respectively.
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