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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.

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) [

In the above literature, an insurer intends to either maximize the expected 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) [

Since different risk measures describe different risk characteristics, it is noteworthy to incorporate two risk measures in an optimization objective. Roman et al. (2007) [

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 regulatory constraints. Section 3 presents the data selection, regulatory policies from CIRC and conducts the empirical study. Section 4 concludes the paper.

Suppose that r = ( r 1 , r 2 , ⋯ , r n ) ′ represents return vector, in which n is the number of assets and the corresponding weight vector is denoted by x = ( x 1 , x 2 , ⋯ , x n ) ′ , so the total return is Y = r ′ x . Variance measures the fluctuation of a random variable around its expected value. Since Y = r ′ x , variance of Y can be expressed as,

V ( Y ) = V ( r ′ x ) = ∑ i = 1 n ∑ j = 1 n x i x j σ i j = x ′ Σ x , (1)

where Σ = ( σ i j ) n × n is the covariance matrix of r and σ i j is the covariance of r i and r j .

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,

V a R 1 − α = − F − 1 ( α ) . (2)

Since VaR does not satisfy the subadditivity (see, Artzner (1999) [

C V a R 1 − α = E [ − Y | − Y ≥ V a R 1 − α ] . (3)

Considering the discrete case, Rockafeller and Uryasev (2000, 2002) [

C V a R 1 − α = min x , ξ ξ + 1 1 − α ∑ i = 1 T p i [ − ∑ j = 1 n x j r j i − ξ ] + (4)

where ξ is V a R 1 − α .

We assume that an insurer is allowed to invest in n kinds of assets, which consist of one risk-free asset and n − 1 risky assets. Taking account of underwriting business, the return of the insurer can be expressed as,

r p = r b + g ( 1 − ∑ k = 1 n − 1 x i ) r f + g ∑ k = 1 n − 1 x k r k , k = 1 , 2 , ⋯ , n − 1 , (5)

where r b is the underwriting return of the insurer; g is the utilization rate of investment capital; r f and 1 − ∑ i = 1 n − 1 x k are the return and investment weight of risk-free asset respectively.

Letting w = ( w 0 , w 1 , ⋯ , w n − 1 ) ′ = ( 1, g x 1 , g x 2 , ⋯ , g x n − 1 ) ′ and R = ( R 0 , R 1 , ⋯ , R n − 1 ) ′ = ( r b + g r f , r 1 − r f , ⋯ , r n − 1 − r f ) ′ , Equation (5) can be rewritten as,

r p = R ′ w . (6)

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) [

probability, that is p i = 1 T . And then motivated by Roman et al. (2007) [

mean-variance-CVaR model with regulatory constraints is constructed as follows,

min V ( R ′ w ) , (7)

s . t . C V a R ( r p ) ≤ c , (8)

E [ R ′ w ] = d , (9)

0 ≤ ∑ k = 1 n − 1 w k ≤ g , (10)

c o n s ( w k ) , k = 1,2, ⋯ , n − 1, (11)

where c o n s ( ⋅ ) denotes the relevant regulatory constraints on investment, d is the target return and c is a given CVaR limit. In addition, since ∑ k = 1 n − 1 x k is the investment proportion without considering risk-free asset and short selling is not allowed in real Chinese stock, fund and bond market, 0 ≤ ∑ k = 1 n − 1 x k ≤ 1 should be satisfied. Multiplied by g on both sides, we get 0 ≤ g ∑ k = 1 n − 1 x k ≤ g , which can be rewritten as Equation (10) due to the fact that w k = g x k , k = 1 , 2 , ⋯ , n − 1 .

Remark 1 Different from Roman et al. (2007) [

According to Roman et al. (2007) [

max E ( R ′ w ) , (12)

s . t . 0 ≤ ∑ k = 1 n − 1 w k ≤ g . (13)

Furthermore, for a specific d * ∈ [ d min , d max ] , the level c of C V a R α must lie in the interval [ c d * , min , c d * , max ] , where c d * , min is the minimum value of C V a R 1 − α for the expected return d * and c d * , max is the C V a R 1 − α of the portfolio generated by MV model with d * (more details, see Roman et al. [

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).

We collect insurance industry data from China Insurance Regulatory Commission and the statistical data of the national insurance industry in China Insurance Yearbook 2013 to 2017. We are able to get Chinese insurance industry data of premium income, total profit and investment profit (unit: hundred million RMB), which can be denoted as I, R t and R i respectively. Then underwriting profit R u can be calculated by R u = R t − R i , and hence the rate of underwriting profit is r b = ( R t − R i ) / I = R u / I . Related data is shown in

According to CIRC’s annual statistics report, we organize and calculate the utilization rate of capital in Chinese insurance industry, as shown in

Year | Underwriting Profit (R_{u}) | Premium Income (I) | Rate of Underwriting Profit (r_{b}) |
---|---|---|---|

2013 | −2666.90 | 17,222 | −15.49% |

2014 | −3312.20 | 20,235 | −16.37% |

2015 | −4980.00 | 24,283 | −20.51% |

2016 | −5091.88 | 30,959 | −16.64% |

2017 | −5784.91 | 36,581 | −15.81% |

mean | −4367.20 | 25,856 | −16.89% |

Year | The utilization rate of capital (g) |
---|---|

2013 | 92.74% |

2014 | 91.85% |

2015 | 90.45% |

2016 | 88.58% |

2017 | 89.07% |

mean | 90.54% |

We collect data in Chinese financial market. Here three-year bank deposit interest rate 2.75% is regarded as a return of risk-free asset (r_{f}), which can be found from the website of the People’s Bank of China.

The data of government bond (r_{1}), financial bond (r_{2}) and corporate bond (r_{3}) are collected from Bond Index, Financial Bond Index and Corporate Bond Index in the website of China Securities Index co.Ltd respectively. And for data of fund (r_{4}), stock (r_{5}) and overseas investment (r_{6}), we choose SSE Fund Index, the SSE Composite Index and Hang Seng Composite index respectively (main channel for overseas investment is HongKong securities market because of the policy constraints). Such data is obtained from the website of Shanghai Stock Exchange and HongKong Stock Exchange. We get the daily data of all these indexes from January 1, 2013 to December 31, 2017, and calculate daily logarithm rate of returns. Then the daily logarithm rate of returns of all indexes is transformed to the quarterly rate of returns for each risk asset, which is shown in

1In February 19, 2014, CIRC released policy “Notice on Strengthening and Improving the Regulation of Utilization Ratio of Insurance Capital” to regulate the investment business of Chinese insurers.

In terms of the latest regulatory policy^{1} released by the CIRC and in order to

Time | Bond (r_{1}) | F_bond (r_{2}) | C_bond (r_{3}) | Fund (r_{4}) | Stock (r_{5}) | Overseas (r_{6}) |
---|---|---|---|---|---|---|

2013.1 | 0.0106 | 0.0167 | 0.0238 | −0.0031 | −0.0193 | −0.0474 |

2013.2 | 0.0099 | 0.0076 | 0.0147 | −0.0692 | −0.1133 | −0.0776 |

2013.3 | −0.0231 | −0.0136 | −0.0033 | 0.1015 | 0.0971 | 0.1208 |

2013.4 | −0.0263 | −0.0267 | −0.0178 | −0.0304 | −0.0582 | 0.0177 |

2014.1 | 0.0246 | 0.0242 | 0.0296 | −0.0491 | −0.0294 | −0.0430 |

2014.2 | 0.0334 | 0.0356 | 0.0376 | 0.0170 | 0.0015 | 0.0318 |

2014.3 | 0.0126 | 0.0226 | 0.0252 | 0.1055 | 0.1501 | −0.0073 |

2014.4 | 0.0348 | 0.0307 | 0.0190 | 0.2846 | 0.3418 | 0.0285 |

2015.1 | 0.0090 | 0.0004 | 0.0150 | 0.1048 | 0.1288 | 0.0540 |

2015.2 | 0.0189 | 0.0212 | 0.0290 | 0.0779 | 0.0611 | 0.0372 |

2015.3 | 0.0220 | 0.0230 | 0.0298 | −0.2097 | −0.2543 | −0.1613 |

2015.4 | 0.0269 | 0.0287 | 0.0296 | 0.0889 | 0.0468 | −0.0515 |

2016.1 | 0.0198 | 0.0065 | 0.0171 | −0.0432 | −0.0911 | −0.0410 |

2016.2 | −0.0015 | 0.0082 | 0.0052 | −0.0020 | −0.0257 | 0.0065 |

2016.3 | 0.0220 | 0.0167 | 0.0231 | 0.0071 | 0.0387 | 0.1150 |

2016.4 | −0.0198 | −0.0241 | −0.0196 | 0.0123 | 0.0286 | −0.0600 |

2017.1 | −0.0034 | −0.0038 | 0.0028 | 0.0176 | 0.0421 | 0.1081 |

2017.2 | −0.0110 | 0.0053 | 0.0068 | 0.0165 | −0.0229 | 0.0587 |

2017.3 | 0.0010 | 0.0059 | 0.0129 | 0.0357 | 0.0542 | 0.1122 |

2017.4 | −0.0055 | −0.0143 | −0.0050 | 0.0101 | 0.0212 | 0.0071 |

facilitate the calculation, we streamline and summarize the limits of investment ratio for insurers as shown in

Since w k = g x k , k = 1 , ⋯ , 7 , considering the above limits combined with the assumption that short selling is not allowed, Equation (11) can be elaborated as,

0 ≤ w 1 , w 2 ≤ g , (14)

0 ≤ w 3 ≤ 0.5 ∗ g , (15)

0 ≤ w 4 ≤ 0.2 ∗ g , (16)

0 ≤ w 5 ≤ 0.2 ∗ g , (17)

0 ≤ w 4 + w 5 ≤ 0.3 ∗ g , (18)

0 ≤ w 6 ≤ 0.15 ∗ g , (19)

g − ∑ k = 1 n − 1 w k ≥ 0.05 ∗ g , k = 1 , ⋯ , 7. (20)

First of all, we conduct a descriptive statistical analysis of the collected quarterly data of assets and obtain (

Type of assets | Limits of investment ratio | |
---|---|---|

Bond | ≤100% | |

F_bond | ≤100% | |

C_bond | ≤50% | |

Fund | ≤20% | Total ratio |

Stock | ≤20% | ≤30% |

Overseas | ≤15% | |

Risk free asset | ≥5% |

Mean | Std | Max | Min | |
---|---|---|---|---|

Deposit (r_{f}) | 0.0069 | - | - | - |

E(r_{f}) = Bond (r_{1}) | 0.0077 | 0.0184 | 0.0348 | −0.0263 |

F_bond (r_{2}) | 0.0085 | 0.0180 | 0.0356 | −0.0267 |

C_bond (r_{3}) | 0.0138 | 0.0161 | 0.0376 | −0.0196 |

Fund (r_{4}) | 0.0236 | 0.0953 | 0.2846 | −0.2097 |

Stock (r_{5}) | 0.0199 | 0.1175 | 0.3418 | −0.2543 |

Overseas (r_{6}) | 0.0104 | 0.0740 | 0.1208 | −0.1613 |

For mean-variance-CVaR model we consider five levels of expected return, which divide the interval [ d min , d max ] into 4 equal parts: d 1 = d min = − 0.1421 , d 2 = − 0.1351 , d 3 = − 0.1281 , d 4 = − 0.1211 , d 5 = d max = − 0.1143 . For each level of expected return d * ∈ { d 1 , d 2 , d 3 , d 4 , d 5 } , we determine c d * , min and c d * , max (details can be seen in Section 2.3). In addition, the interval [ c d * , min , c d * , max ] for CVaR is equally divided into 4 equal parts, so five CVaR levels c * ∈ [ c d * , min , c d * , max ] . We solve the mean-variance-CVaR model with policy constraints for every combination of d * and c * . The results are shown in

With the increasing of the target return, the proportion of bank deposit declines remarkably and the proportion of corporate bond, fund, stock and overseas investment climb up steadily. Corporate bond is the first increasing asset in a portfolio to meet the requirement of higher target return, and reaches the ceiling quickly. Afterwards, in order to further increase the expected return of the portfolio, the proportion of financial bond, fund, overseas investment and stock increase gradually and reach their ceilings finally. It is obvious that the regulatory constraints are active to prevent an insurer investing intensively in single high-return risky asset and to guarantee a certain level of retention of bank deposit.

For every specific level of target return, a larger c * means that the model considers more variance-controlling than CVaR-controlling. In particular, when c = c d * , min , the portfolio is just the mean-CVaR portfolio, and when c = c d * , max , the portfolio is the mean-variance portfolio. We can see that a larger c * leads to a more diversified portfolio, which has a smaller variance but larger CVaR ( c * ). These findings are in line with that in Roman et al. (2007) [

d * = − 0.1421 | |||||
---|---|---|---|---|---|

c * | −0.0318 | −0.0315 | −0.0313 | −0.0310 | −0.0307 |

Deposit | 0.8969 | 0.8778 | 0.8703 | 0.8774 | 0.8672 |

Bond | 0.0033 | 0.0242 | 0.0342 | 0.0243 | 0.0313 |

F_bond | 0 | 0 | 0 | 0.0023 | 0.0007 |

C_bond | 0.0831 | 0.0848 | 0.0863 | 0.0887 | 0.0993 |

Fund | 0.0070 | 0.0058 | 0.0055 | 0.0053 | 0.0011 |

Stock | 0 | 0 | 0 | 0.0004 | 0.0001 |

Overseas | 0.0097 | 0.0073 | 0.0037 | 0.0017 | 0.0004 |

var (10^{−4}) | 0.211 | 0.197 | 0.188 | 0.185 | 0.178 |

d * = − 0.1351 | |||||

c * | −0.0253 | −0.0252 | −0.0251 | −0.0249 | −0.0248 |

Deposit | 0.6466 | 0.6433 | 0.6398 | 0.6362 | 0.6327 |

Bond | 0 | 0 | 0.0001 | 0.0002 | 0.0002 |

F_bonds | 0 | 0.0001 | 0.0001 | 0.0002 | 0.0002 |

C_bond | 0.3304 | 0.3356 | 0.3415 | 0.3471 | 0.3527 |

Fund | 0.0229 | 0.0207 | 0.0181 | 0.0157 | 0.0135 |

Stock | 0 | 0.0001 | 0 | 0.0001 | 0.0002 |

Overseas | 0 | 0.0002 | 0.0004 | 0.0005 | 0.0006 |

var (10^{−4}) | 0.297 | 0.295 | 0.293 | 0.292 | 0.291 |

d * = − 0.1281 | |||||

c * | −0.0161 | −0.0151 | −0.0141 | −0.0130 | −0.0120 |

Deposit | 0.2818 | 0.3039 | 0.3716 | 0.4072 | 0.4279 |

Bond | 0 | 0 | 0 | 0 | 0.0001 |

F_bonds | 0.0647 | 0.0673 | 0.0140 | 0.0007 | 0.0001 |

C_bond | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.4999 |

Fund | 0.0376 | 0.0438 | 0.0544 | 0.0620 | 0.0675 |

Stock | 0 | 0 | 0 | 0 | 0 |

Overseas | 0.1159 | 0.0849 | 0.0600 | 0.0302 | 0.0045 |

var (10^{−4}) | 1.5 | 1.25 | 1.05 | 0.895 | 0.819 |

d * = − 0.1211 | |||||

c * | 0.0114 | 0.0128 | 0.0141 | 0.0155 | 0.0168 |

Deposits | 0.0500 | 0.0535 | 0.0562 | 0.1324 | 0.2186 |

Bond | 0 | 0.0002 | 0 | 0.0001 | 0.0003 |

F_bond | 0.1642 | 0.1973 | 0.2329 | 0.1772 | 0.1081 |

C_bond | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 |

Fund | 0.1358 | 0.1415 | 0.1473 | 0.1598 | 0.1730 |
---|---|---|---|---|---|

Stock | 0 | 0 | 0 | 0 | 0 |

Overseas | 0.1500 | 0.1075 | 0.0636 | 0.0304 | 0 |

var (10^{−4}) | 4.37 | 3.82 | 3.42 | 3.17 | 2.99 |

d * = − 0.1143 | |||||

c * | 0.0538 | 0.0540 | 0.0541 | 0.0543 | 0.0544 |

Deposits | 0.0500 | 0.0500 | 0.0500 | 0.0500 | 0.0500 |

Bond | 0 | 0 | 0 | 0 | 0 |

F_bond | 0.0019 | 0.0035 | 0.0047 | 0.0077 | 0.0090 |

C_bond | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 |

Fund | 0.2000 | 0.2000 | 0.2000 | 0.2000 | 0.2000 |

Stock | 0.0981 | 0.0986 | 0.0989 | 0.0996 | 0.1000 |

Overseas | 0.1500 | 0.1479 | 0.1463 | 0.1427 | 0.1410 |

var (10^{−4}) | 12 | 12 | 12 | 12 | 12 |

In addition, since the mean-variance-CVaR model considers two risk measures, we can find that its optimal portfolios will have neither excessive variance nor excessive CVaR. In other words, it takes into account both the regulators’ demand for controlling the left-tail risk and investors’ demand for minimizing overall volatility. And the solutions derived from the model are generally discarded by both mean-variance ( c = c d * , max ) and mean-CVaR ( c = c d * , min ) model. It does not mean that mean-variance or mean-CVaR models are dismissed, but on the contrary, the mean-variance-CVaR model reinforces them by providing decision-makers more choices.

Motivated by Roman et al. (2007) [

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) [

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.

This work was partially supported by National Natural Science Foundation of China (71671104), Project of Humanities Social Sciences of Research of the Ministry of Education, China (16YJA910003, 18YJC630220) and Special Funds of Taishan Scholar Project (tsqn20161041).

The authors declare no conflicts of interest regarding the publication of this paper.

Shi, Y., Zhao, X. and Yan, X. (2019) Optimal Asset allocation for a Mean-Variance-CVaR Insurer under Regulatory Constraints. American Journal of Industrial and Business Management, 9, 1568-1580. https://doi.org/10.4236/ajibm.2019.97103