关键词: 非对称Laplace分布；在险价值；投资组合；凸优化

Abstract: 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.

Key words: asymmetric Laplace distribution; value-at-risk; investment portfolio; convex optimization