Coherent Risk Measure and Normal Mixture Distributions with Application in Portfolio Optimization and Risk Allocation

This paper investigates the coherent risk measure of normal mixture distributions. The main result shows that the mean-risk portfolio optimization problem of some widely-used 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, the worst-case value at risk in the robust portfolio optimization can also be calculated directly. Finally the conditional value at risk is considered as an example of coherent risk measure; and we obtain the marginal contribution to risk for a portfolio based on normal mixture model