On the mean and variance of the estimated tangency portfolio weights for small samples

Abstract

In this paper, a sample estimator of the tangency portfolio (TP) weights is considered. The focus is on the situation where the number of observations is smaller than the number of assets in the portfolio and the returns are i.i.d. normally distributed. Under these assumptions, the sample covariance matrix follows a singular Wishart distribution and, therefore, the regular inverse cannot be taken. In the paper, bounds and approximations for the first two moments of the estimated TP weights are derived, as well as exact results are obtained when the population covariance matrix is equal to the identity matrix, employing the Moore–Penrose inverse. Moreover, exact moments based on the reflexive generalized inverse are provided. The properties of the bounds are investigated in a simulation study, where they are compared to the sample moments. The difference between the moments based on the reflexive generalized inverse and the sample moments based on the Moore–Penrose inverse is also studied.

Smart betas, return models and the tangency portfolio weights

“…See e.g. [ 10 , 11 ] for an active search for the tangency portfolio, and [ 12 – 14 ] or [ 15 ] for estimation and analysis of statistical properties of the tangency portfolio in both low and high dimensional settings.…”

“…� � > 0, then by Eq (12) we have that the relaxed tangency portfolio weights given the CAPM return model are given by weights…”

Tangency portfolio weights under a skew-normal model in small and large dimensions

“…Javed et al (2021) obtained analytical expressions for the higher order moments of the estimated TP weights. In Alfelt and Mazur (2020), the mean and variance of the estimated TP weights are studied when the sample covariance matrix is singular.…”