Statistical Estimation of Composite Risk Functionals and Risk Optimization Problems

We address the statistical estimation of composite functionals which may be nonlinear in the probability measure. Our study is motivated by the need to estimate coherent measures of risk, which become increasingly popular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit formulae for composite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal values when an estimator of the risk functional is used. While the mathematical structures accommodate commonly used coherent measures of risk, they have more general character, which may be of independent interest

The gradient allocation principle based on the higher moment risk measure

According to the gradient allocation principle based on a positively homogeneous and subadditive risk measure, the capital allocated to a sub-portfolio is the Gâteaux derivative, assuming it exists, of the underlying risk measure at the overall portfolio in the direction of the sub-portfolio. We consider the capital allocation problem based on the higher moment risk measure, which, as a generalization of expected shortfall, involves a risk aversion parameter and a confidence level and is consistent with the stochastic dominance of corresponding orders. As the main contribution, we prove that the higher moment risk measure is Gâteaux differentiable and derive an explicit expression for the Gâteaux derivative, which is then interpreted as the capital allocated to a corresponding sub-portfolio. We further establish the almost sure convergence and a central limit theorem for the empirical estimate of the capital allocation, and address the robustness issue of this empirical estimate by computing the influence function of the capital allocation. We also explore the interplay of the risk aversion and the confidence level in the context of capital allocation. In addition, we conduct intensive numerical studies to examine the obtained results and apply this research to a hypothetical portfolio of four stocks based on real data