On the mean and variance of the estimated tangency portfolio weights for small samples
Volume 9, Issue 4 (2022), pp. 453–482
Pub. online: 2 September 2022
Type: Research Article
Open Access
Open Access
Received
13 December 2021
13 December 2021
Revised
24 May 2022
24 May 2022
Accepted
29 July 2022
29 July 2022
Published
2 September 2022
2 September 2022
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.
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