We introduce a framework for robustifying portfolio selection problems with respect to ambiguity in the distribution of the random asset losses. In particular, we are interested in convex, version independent risk measures. To robustify these risk measures, we use an ambiguity set which is defined as a neighborhood around a reference probability measure which represents the investors beliefs about the distribution of asset losses. The robustified risk measures are defined as the worst case portfolio risk over the ambiguity set of loss distributions. We demonstrate that under mild conditions, the infinite dimensional optimization problem of finding the worst case risk can be solved analytically and consequently closed form expressions for the robust risk measures are obtained. We use these results to derive robustified version for several examples of risk measures among them the standard deviation and the Conditional Value-at-Risk and the general class of distortion functionals. The resulting robust policies are computationally of the same complexity as their non-robust counterparts. We conclude with a numerical study that shows that in most instances the robustified risk measures perform significantly better out-of-sample than their non-robust variants in terms of risk, expected losses as well as turnover.

## Citation

Technical Report, 11/2011

## Article

View Robustifying Convex Risk Measures: A Non-Parametric Approach