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.
The beta coefficient plays a crucial role in finance as a risk measure of a portfolio in comparison to the benchmark portfolio. In the paper, we investigate statistical properties of the sample estimator for the beta coefficient. Assuming that both the holding portfolio and the benchmark portfolio consist of the same assets whose returns are multivariate normally distributed, we provide the finite sample and the asymptotic distributions of the sample estimator for the beta coefficient. These findings are used to derive a statistical test for the beta coefficient and to construct a confidence interval for the beta coefficient. Moreover, we show that the sample estimator is an unbiased estimator for the beta coefficient. The theoretical results are implemented in an empirical study.
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