Optimal Asset Allocation for a Mean-Variance-CVaR Insurer under Regulatory Constraints

In this paper, we introduce the mean-variance-CVaR criteria into the study of asset allocation for insurers. Considering that the financial market consists of one risk-free asset and multiple risky assets with regulatory constraints, an optimization problem is established for an insurer with underwriting business. Based on practical financial and insurance data, an empirical study is carried out. The results show that the mean-variance-CVaR model is able to provide more potential investment strategies for an insurer. The regulatory policy released by China Insurance Regulatory Commission plays a key role in controlling investment risk for Chinese insurers.


Introduction
There has been much attention to the optimal asset allocation problem for insurers recently in the field of risk management and insurance. Early studies about the asset allocation problem for insurers can be seen in Kahane and Nye (1975) [1], Krous (1970) [2], Lambert and Hofflander (1966) [3]. Briys (1985) [4] studies the investment behavior of insurers by maximizing the expected utility based on deterministic underwriting framework. Later, there have been some extended works focusing on the asset allocation problem for insurers based on different optimization objectives (see, e.g., Rong et al. (2001) [5], Rong and Li (2004) [6], Chen et al. (2006) [7], Zhao et al. (2011) [8], Zhao et al. (2018) [9]). utility of return or minimize the risk of portfolios. Specifically, for the latter, variance, Value-at-Risk (VaR) or VaR-based risk measure are generally used to characterize risk. As we know, variance, which is generally cared about by traditional fund managers, only focuses on characterizing the fluctuation around the expected return, but ignores the risk in worst-case scenarios. While considering the risk of extreme loss, VaR and Conditional Value-at-Risk (CVaR), etc. are widely used, see, Guo and Li (2009) [10], Xu et al. (2016) [11], Banihashemi (2017) [12], Yu et al. (2011) [13] and the references therein. In particular, CVaR has attracted more attention since it has good theoretical properties, which is consistent with financial practice, see Artzner et al. (1999) [14].
Since different risk measures describe different risk characteristics, it is noteworthy to incorporate two risk measures in an optimization objective. Roman et al. (2007) [15] employ variance and CVaR in the optimal investment problem to get a balanced policy for addressing the requirements of both traditional fund managers and regulators. They find that the mean-variance (MV) model pioneered by Markowitz (1952) [16] and the mean-CVaR model might lead to very different conclusion. The portfolio derived from MV model may have an excessively large CVaR while that from mean-CVaR model may have an unacceptable variance. However, the proposed mean-variance-CVaR model can generate a series of portfolios which are generally disregarded by MV and mean-CVaR model. The mean-variance-CVaR model is used for optimizing investment strategy of China sovereign wealth funds by Yu and Ma (2014) [17]. Using a linear weighted sum method, Younes et al. (2014) [18] devote to simplify the optimization of mean-variance-CVaR model. Chen (2016) [19] analyzes the optimal bond portfolio for commercial bank based on mean-variance-CVaR criterion. And Gao et al. (2016) [20] expand the above static mean-variance-CVaR model to dynamic portfolio selection and derive the analytical forms of the portfolio policy for mean-variance-CVaR optimization models.
So far, few papers have incorporated mean-variance-CVaR criterion into the study of asset allocation strategies for insurance companies. Meanwhile, when Chinese insurers invest in the risk market, they have to consider the constraints on the proportion of investment channels imposed by China Insurance Regulatory Commission (CIRC). Therefore, the main contribution of this paper is that new optimization criterion and regulatory constraints are incorporated into the study of optimal asset allocation problem in insurance risk management. We construct an optimization model based on mean-variance-CVaR criteria for an insurer under the regulatory policies imposed by CIRC. The insurer can invest in a financial market with one risk-free asset and multiple risky assets and underwriting business is also involved. Based on the historical data of the insurance industry and financial market from 2013 to 2017, an empirical study is carried out and the results are analyzed.
The rest of this paper is constructed as follows. In Section 2, we introduce the related risk measures, the return of an insurer and the optimization model with is the covariance matrix of r and ij σ is the covariance of i r and j r . Given a certain level of confidence 1 α − , VaR measures the greatest potential loss for a portfolio over specific holding period. If the cumulative distribution function for return Y is F, VaR can be defined as, Since VaR does not satisfy the subadditivity (see, Artzner (1999) [14]), a coherent risk measure CVaR is proposed and defined as follows, Considering the discrete case, Rockafeller andUryasev (2000, 2002) [20] [21] propose a linear programming model for calculating CVaR, which has been widely used. Suppose that there are T scenarios, every scenario ( ) Then, CVaR can be formulated as,

Return of an Insurer
We assume that an insurer is allowed to invest in n kinds of assets, which consist of one risk-free asset and 1 n − risky assets. Taking account of underwriting business, the return of the insurer can be expressed as, 1 1 where b r is the underwriting return of the insurer; g is the utilization rate of investment capital; f r and , (5) can be rewritten as,

Mean-Variance-CVaR Model
Given that the confidence level is 1 α − and the target return d, the optimization problem is established in T scenarios. Based on the idea in Rockafeller and Uryasev (2000) [22], we let every scenario have the same probability, that is where ( ) cons ⋅ denotes the relevant regulatory constraints on investment, d is the target return and c is a given CVaR limit. In addition, since which can be rewritten as Equation (10) [15]).

Data Selection
According to the actual investment situation of Chinese insurance companies, we consider six kinds of risky assets which consist of government bond (Bond), financial bond (F_bond), corporate bond (C_bond), fund, stock and overseas investment (Overseas) and one risk-free asset (bank deposit).

Rate of Underwriting Profit
Related data is shown in Table 1.

Utilization Rate of Capital
According to CIRC's annual statistics report, we organize and calculate the utilization rate of capital in Chinese insurance industry, as shown in Table 2.

Regulatory Policies
In terms of the latest regulatory policy 1 released by the CIRC and in order to facilitate the calculation, we streamline and summarize the limits of investment ratio for insurers as shown in Table 4.
Since , 1, , 7 k k w gx k = =  , considering the above limits combined with the assumption that short selling is not allowed, Equation (11) can be elaborated as,

Descriptive Statistical Analysis
First of all, we conduct a descriptive statistical analysis of the collected quarterly data of assets and obtain (Table 5). We can conclude that corporate bond has the lowest fluctuation and a relatively high mean value; government bond and financial bond have low fluctuation while low mean value; fund, stock and oversea investment have high return while high risk.

Conclusions
Motivated by Roman et al. (2007) [15], this paper has studied the optimal asset allocation problem under mean-variance-CVaR criteria for an insurer with investment business and underwriting business. Furthermore, the policy Y. Shi et al.
constraints imposed by CIRC for investment business are considered.
Based on the practical data from Chinese insurance industry and financial market in the period of 2013-2017, an empirical study has been conducted. The empirical results show that there are fewer kinds of asset in the optimal portfolio derived from mean-CVaR than that from mean-variance model and mean-CVaR model does a better job in controlling the tail-risk but ignoring the variance while mean-variance model is opposite. This is consistent with the results in Roman et al. (2007) [15]. Furthermore, we find that, with the target return increasing gradually, as expected, less and less proportion of risk-free asset will be invested. Specifically, more consideration of variance leads to less risk-free asset investing if the target return is relatively low, while leads to more risk-free asset investing if the target return is relatively high. By adjusting the parameters of the constraint for CVaR, mean-variance-CVaR model can take into account the preference between regulators' requirements for short tails and classical fund managers' requirements for small variance, which is more reasonable for insurers under regulatory constraints. In addition, we can find that the constraints imposed by CIRC do work well to limit high-risk investment, especially when insurers intend to go for a high target return. However, current investment channels still can not satisfy the requirement of insurers for making up underwriting losses. Therefore, it's considerable to further expand the investment channels for Chinese insurers to gain more investment profit.
The main contribution of this paper is introducing mean-variance-CVaR model and policy constraints into the empirical study of insurance risk management and some meaningful results are obtained. The study could be extended to a continuous-time setting and the CVaR item in the model could also be incorporated into the objective function to simplify the optimization. We will explore these topics in the following study.