# American Institute of Mathematical Sciences

November  2020, 16(6): 3065-3081. doi: 10.3934/jimo.2019094

## Mean-CVaR portfolio selection model with ambiguity in distribution and attitude

 1 Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Fujian, China 2 Sun Yat-sen Business School, Sun Yat-sen University, Guangzhou, China

* Corresponding author: Zhongfei Li

Received  November 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author was supported in part by the Program for Innovative Research Team in Science and Technology in Fujian Province University, and Quanzhou High-Level Talents Support Plan (No. 2017ZT012), and the Scientific Research Foundation of Huaqiao University (No. 18BS311). The third author was supported by the National Natural Science Foundation of China (No. 71721001) and the Natural Science Research Team of Guangdong Province of China (No. 2014A030312003)

In this paper, we develop $\alpha$-robust (maxmin) models, where the Conditional Value-at-Risk (CVaR) is to be optimized under ambiguity in distribution, mean returns, and covariance matrix. Our models allow the investor to distinguish ambiguity and ambiguity attitude with different levels of ambiguity aversion. For the case when there is a risk-free asset and short-selling is allowed, we obtain the analytic solution for the $\alpha$-robust CVaR optimization model subject to a minimum mean return constraint. Moreover, we also derive a closed-form portfolio rule for the $\alpha$-robust mean-CVaR optimization problem in a market without the risk-less asset. The results obtained from solving the numerical example show that if an investor is more ambiguity-averse, his investment strategy will always be more conservative.

Citation: Zhilin Kang, Xingyi Li, Zhongfei Li. Mean-CVaR portfolio selection model with ambiguity in distribution and attitude. Journal of Industrial & Management Optimization, 2020, 16 (6) : 3065-3081. doi: 10.3934/jimo.2019094
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The left panel shows that $k(\alpha)$ and $b(\alpha)$ are decreasing in $\alpha$. The efficient frontier lines for $\alpha$-robust CVaR model are shown in the right panel. The steepest line (Dash-dot line, black) and flattest line (Solid line, blue) correspond to the cases $\alpha = 0.5$ and $\alpha = 1$, respectively. ($H = 0.4722$, $r_f = 1.01$, $\gamma_1 = 0.0001$, $\gamma_2 = 1.2$, $\beta = 0.95$)
Efficient frontiers of the $\alpha$-maxmin mean-CVaR model with different parameter $\alpha$. The $\alpha$-maxmin portfolio CVaR in the $x$-axis ($\alpha$-maxmin portfolio return in the $y$-axis) is a convex mixture between the worst-case and best-case values of CVaR risk measures (expected return)
Effects of $\alpha$ (the level of ambiguity aversion) on the $\alpha$-maxmin portfolio return and $\alpha$-maxmin portfolio CVaR
Effect of $\alpha$ (the level of ambiguity aversion) on the composition of efficient portfolios from the $\alpha$-maxmin mean-CVaR model. The percentage allocation of assets 1-3 in the optimal allocation $x^*$ have been illustrated in different colors
The variations of optimal portfolio strategies under different levels of ambiguity $\gamma_1$ (for a given $\gamma_2 = 1.2$) and $\gamma_2$ (for a given $\gamma_1 = 0.0001$). The percentage allocation of assets 1-3 in the optimal allocation $x^*$ have been illustrated in different colors
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