Due to the problem of parameter uncertainty, specifying the location of the tangency portfolio (TP) on the set of feasible portfolios becomes a challenging task. The set of feasible portfolios is a parabola in the mean-variance space with optimal portfolios lying on its upper part. Using statistical test theory, we want to decide if the tangency portfolio is mean-variance efficient, i.e. if it belongs to the upper limb of the efficient frontier. In the opposite case, the investor would prefer to invest into the risk-free asset or into the global minimum variance portfolio which lies in the vertex of the set of feasible portfolios. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we suggest a test on the location of the tangency portfolio on the set of feasible portfolios. The distribution of the test statistic is derived under both hypotheses, which we use to assess the power of the test and construct a confidence interval. Moreover, out-of-sample performance of the test is evaluated based on real data. The robustness to the assumption of normality is investigated via an extensive simulation study where we show that the new test is robust to the violation of the normality assumption and can also be used for heavy-tailed stochastic models. Moreover, in an empirical study we apply the developed theory to real data. We find that when the sample size is relatively large and a stable period is present on the market, then the mean-variance efficiency of the tangency portfolio can be statistically justified.

In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation is derived for this product, using which the characteristic function of the product and its asymptotic distribution under the double asymptotic regime are established. The application of obtained stochastic representation speeds up the simulation studies where the product of a singular Wishart random matrix and a singular normal random vector is present. We further document a good performance of the derived asymptotic distribution within a numerical illustration. Finally, several important properties of the singular Wishart distribution are provided.ASM Classification: 60E05, 60E10, 60F05, 62H10, 62E20

In this paper we derive the asymptotic distribution of the test of the efficiency of the tangency portfolio in high-dimensional settings, namely when both the portfolio dimension and the sample size grow to infinity. Moreover, we propose a new test based on the estimator for the slope parameter of the efficient frontier in the mean-variance space when there is a possibility in investing into the riskless asset, and derive the asymptotic distribution of that test statistic under both the null and alternative hypotheses. Additionally, we study the finite sample performance of the derived theoretical results via simulations.