非对称Laplace分布下的均值-VaR模型

Mean-VaR Model Based on the Asymmetric Laplace Distribution

非对称Laplace分布可以描述分布的尖峰厚尾和有偏特征,被许多学者用来拟合金融资产的历史收益率数据,进而测算金融资产的尾部风险,然而非对称Laplace分布下的投资组合研究尚不成熟。因此本文在非对称Laplace分布设定下给出VaR的解析表达式,并建立均值-VaR模型研究投资组合选择问题。在理论上我们证明该模型是凸优化问题,可以转化为二次规划问题进行求解,从而可得到模型全局最优的解析解。进一步地,我们分别得到存在无风险资产和不存在无风险资产时投资组合前沿的解析式。最后基于上证50指数及其成份股的历史数据进行实证分析,研究结果表明本文构建的模型在实践中的投资表现良好。

As a popular risk measurement, the Value at Risk(VaR) has been widely concerned by the industry and academia since it was proposed by Morgan in 1994. While VaR is intuitive in risk management practice and simple in its definition, it has several notorious limitations and drawbacks such as its insensitivity to the magnitude of losses beyond VaR, non-qualification as a coherent risk measure and its non-convexity with respect to the portfolio weights which results in computational difficulties under the general distribution. At present, a good analytical solution of VaR-based model is mostly obtained under the normal distribution. However, the financial time series generally represent asymmetric leptokurtic features, so that the normal distribution hypothesis will increase the systematic biases of risk estimation. Among many probability distributions, the asymmetric Laplace distribution(ALD) allows the heavy tail and asymmetry, and has been widely used to fit the distribution of financial assets’ return and measure tail risk, but its application in the portfolio selection has not been mature. In view of this, supposing assets’ return follows ALD, the VaR’s analytical formula is obtained and a novel mean-VaR model is proposed. In theory, it is proved that the model is a convex optimization problem and can be transferred into a quadratic programming problem. Furthermore, the global optimal analytical solution of the model can be easily obtained and then the analytical formula of portfolio frontier can be derived with and without a risk-free asset respectively.Finally, 308 weekly yield data of SSE50 and its components from January 2012 to December 2017 are collected for empirical analysis, which shows that the mean-VaR model based on ALD performs better than indexation investment. In a word, there are two main conclusions in this paper. On the one hand, by comparing the VaR under two different distributions, it is found that with the increase of the mean value of portfolio return, the tail risk under ALD is smaller than that under normal distribution. On the other hand, because the analytical expression of the VaR under ALD is a convex function of portfolio position, the global optimal analytical solution of the mean-VaR model can also be easily obtained. Therefore, the research achievement of this paper can enrich and deepen the theories of asset allocation and risk management;in practice, our works can improve investors’ trade strategies and provide more accurate risk measure techniques to prevent financial market risks for enterprises and governments.