This paper investigates the coherent risk measure of a class of normal mixture distributions which are widely-used in finance. The main result shows that the mean-risk portfolio optimization problem with these normal mixture distributions can be reduced to a quadratic programming problem which has closed form of solution by fixing the location parameter and skewness parameter. In addition, we show that the efficient frontier of the portfolio optimization problem can be extended to three dimensions in this case. The worst-case value-at-risk in the robust portfolio optimization can also be calculated directly. Finally, the conditional value-at-risk (CVaR) is considered as an example of coherent risk measure. We obtain the marginal contribution to risk for a portfolio based on the normal mixture model.

中文翻译：

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在投资组合优化中应用的连贯风险度量和正态混合分布

本文研究了金融中广泛使用的一类正态混合分布的相干风险测度。主要结果表明，通过固定位置参数和偏度参数，可以将具有这些正态混合分布的平均风险投资组合优化问题简化为具有封闭解形式的二次规划问题。此外，我们表明，在这种情况下，投资组合优化问题的有效边界可以扩展到三个维度。稳健投资组合优化中的最坏情况风险价值也可以直接计算。最后，条件风险价值（CVaR）被认为是连贯风险度量的一个例子。我们基于正态混合模型获得投资组合对风险的边际贡献。