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.

How to efficiently allocate capital lies at the heart of financial decision making. Portfolio theory, as developed by [

In this paper, we consider the tangency portfolio (TP) which is one of the most important portfolios in the financial literature. The TP weights determine what proportions of the capital to invest in each asset and are obtained by maximizing the expected quadratic utility function. For a portfolio of

This value represents how willing an investor is to accept upward and downward risks on their investment. It can be determined through, e.g., qualitative assessment, such as interview questions posed to the investor.

Naturally, the TP weights

It is worth to mention that a similar structure appears in the discriminant analysis. Namely, the coefficients of a discriminant function that maximizes the discrepancy between two datasets are expressed as a product of the inverse sample covariance matrix and the sample mean vector (see, for example, [

In the Bayesian setting, the posterior distribution of TP weights is expressed as a product of the (singular) Wishart matrix and Gaussian vector. Statistical properties of those products are studied by [

The common scenario considered is that the number of observations available for the estimation, denoted by

This issue can be remedied by estimating

Instead of using the Moore–Penrose inverse, one can consider regularization techniques such as the ridge-type method [

The expectation and variance of an estimator are key quantities to describe its statistical properties. With the standard assumption of normally distributed asset returns, the stochastic components of

The rest of this paper is organized as follows. Section

Let

In the following, let

When

Since

A direct consequence of Theorem

This section aims to provide upper and lower bounds for the expected value of

The result follows directly from the element-wise bounds in Lemma

Note that when

The following result provides two upper bounds for the variance of the TP weights estimate

We are interested in bounds for the quantity

Regarding general

Also define

From Theorem 3.1 in [

In [

An alternative to using the Moore–Penrose inverse

The first result follows directly from Corollary 2.3 in [

An obvious drawback of

The aim of this section is to compare the bounds on the moments of

In the following, we will study simulations of (

Generate

Generate

Independently generate

Compute

Repeat steps 3) and 4) above

Based on the

Given

The computation time for each set of simulations for

The TP is an important portfolio in mean-variance asset optimization framework of [

In this paper, we provide bounds on the mean and variance of the TP weights estimator in the singular case. Further, we present approximate results, as well as exact moment results in the case when the population covariance is equal to the identity matrix. We also provide exact moment results when the reflexive generalized inverse is applied in the TP weights equation.

Moreover, we investigate the properties of the derived bounds, and the estimator based on the reflexive generalized inverse, in a simulation study. The difference between the various bounds and the sample counterparts are measured by several quantities, and studied for numerous dimensions, sample sizes and levels of dependencies of the population covariance matrix. The results suggest that many of the derived bounds are closest to the sample moments when the population covariance matrix implies low dependency between the considered assets. Finally, the study implies that in some cases the moments of TP weights based on the reflexive generalized inverse can be used as a rough approximation for the moments of TP weights based on the Moore–Penrose inverse. For future studies, it would be relevant, for example, to perform a sensitivity analysis on how fluctuations in the population covariance matrix affect the estimated TP weights.

First note that in accordance with Theorem 3.2 and Theorem 3.3 of [

Moreover, note that every principal submatrix of a positive definite matrix is also positive definite. Combined with (

Now, first assume

The results in Lemma

First, we have that

In the following, let

In accordance with page 130 in [

First, let

Let

We would like to thank Prof. Yuliya Mishura, the Associate Editor and the two anonymous referees for helping to improve the paper. We are also grateful to Andrii Dmytryshyn and Mårten Gulliksson for helpful remarks on matrix inequalities.