Minimum VaR and minimum CVaRoptimal portfolios: Estimators, confidence regions, and tests

In this paper, we consider the sample estimators for the expected return, the variance, the value-at-risk (VaR), and the conditional VaR (CVaR) of the minimum VaR and the minimum CVaR portfolio. Their exact distributions are derived. These expressions are used for studying the distributional properties of the estimated characteristics. We prove that the expectation does not exist for the estimated variance, while the second moment does not exist for the estimated expected return. Moreover, expressions for the joint densities and the corresponding dependence measures between the estimators for the expected return and the variance as well as between the estimated expected return and the estimated VaR (CVaR) are derived. Finally, we present a confidence region for the minimum VaR portfolio and the minimum CVaR portfolio in the mean-variance space as well as in the mean-VaR (mean-CVaR) space. The obtained results are illustrated in an empirical study throughout the paper.