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

Drin, Svitlana; Mazur, Stepan; Muhinyuza, Stanislas

doi:10.15559/24-VMSTA261

Modern Stochastics: Theory and Applications

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

Svitlana Drin Stepan Mazur Stanislas Muhinyuza

https://doi.org/10.15559/24-VMSTA261

Pub. online: 11 July 2024 Type: Research Article

Open Access

Received
20 February 2024

Revised
20 June 2024

Accepted
21 June 2024

Published
11 July 2024

Abstract

The test for the location of the tangency portfolio on the set of feasible portfolios is proposed when both the population and the sample covariance matrices of asset returns are singular. The particular case of investigation is when the number of observations, n, is smaller than the number of assets, k, in the portfolio, and the asset returns are i.i.d. normally distributed with singular covariance matrix Σ such that $rank(\boldsymbol{\Sigma })=r\lt n\lt k+1$. The exact distribution of the test statistic is derived under both the null and alternative hypotheses. Furthermore, the high-dimensional asymptotic distribution of that test statistic is established when both the rank of the population covariance matrix and the sample size increase to infinity so that $r/n\to c\in (0,1)$. Theoretical findings are completed 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. To get a better understanding of the developed theory, an empirical study with data on the returns on the stocks included in the S&P 500 index is provided.

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Keywords

Tangency portfolio Hypothesis testing Singular Wishart distribution Singular covariance matrix Moore–Penrose inverse High-dimensional asymptotics

MSC2010

62P05 62P99 91B06 91G10 62F03 62F05

Funding

Svitlana Drin acknowledges financial support from the Knowledge Foundation Grant “Forecasting for Supply Chain Management” (Dnr: 20220115). Stepan Mazur acknowledges financial support from the project “Improved Economic Policy and Forecasting with High-Frequency Data” (Dnr: E47/22) funded by the Torsten Söderbergs Foundation and the internal research grants at Örebro University.

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