A test on the location of tangency portfolio for small sample size and singular covariance matrix

In this paper, we propose the test for the location of the tangency portfolio on the set of feasible portfolios when both the population and the sample covariance matrices of asset returns are singular. We derive the exact distribution of the test statistic under both the null and alternative hypotheses. Furthermore, we establish the high-dimensional asymptotic distribution of that test statistic when both the portfolio dimension and the sample size increase to infinity. We complement our theoretical findings by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented.