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,

Modern portfolio theory, introduced by Harry Max Markowitz in [

The above-mentioned papers focus on the case when the number of assets,

Let’s now discuss closely the setting with two sources of singularity. Let

Thus, the present paper assumes that the asset returns

The rest of the paper is organized as follows. In Section

Let

The optimal portfolios as proposed by Markowitz’s theory lie on the upper part of the parabola in the mean-variance space. This parabola is known as the efficient frontier (EF) and, if

On the other hand, if

If there is a possibility to invest in a risk-free asset, one may choose to put a portion of his/her investment into a risk-free asset, henceforth, the efficient frontier becomes a straight line in the mean-variance space passing through the return of the risk-free asset and tangent to the parabola in (

Location of the tangency portfolio on the set of feasible portfolios in the two cases: (a)

The rejection of the null hypothesis suggests that the TP lies on the upper part of the efficient frontier as shown in Figure

Assuming positive definiteness of the population covariance matrix,

The following theorem provides distribution of

The density function of the test statistic

In Theorem

Using Theorem

Theorem

Power of the test statistic

Since a statistical test and interval estimation are related, we can construct a one-sided

In this section, we derive the high-dimensional asymptotic distribution of test statistic given in (

In the following theorem, we derive the high-dimensional asymptotic distribution of the test statistic

From the proof of Theorem

Now, it holds that

From the proof of Theorem 5 in [

Having the high-dimensional asymptotic distribution of test statistic in Theorem

In this section, we compare the power functions of the exact test and the high-dimensional asymptotic test which are delivered in Theorems

In Figure

Powers of the exact test and the high-dimensional asymptotic test as a function of

To better understand the results obtained in the previous sections, we apply the derived theoretical results to real data. The empirical study highlights the effect of the singularity of the covariance matrix and provides insight into the challenges posed by the high-dimensionality of financial data combined with distributional and dependence structure assumptions. This study also shows how the results can be used and how the presence of the singularity affects the inference of the TP efficiency.

The derivation of the theoretical results in this paper is based on the assumption of i.i.d. multivariate normal asset returns. However, in practice, day-to-day dependence cannot be ignored and the assumption of normality assumption is often violated [

In reality, the clean case of the singularity of the population covariance matrix

We consider weekly averages of the daily log returns data from the S&P 500 of 368 stocks for the period from the 15th of April, 2014 to the 17th of April, 2024. In addition, we use the weekly return on the three-month US Treasury bill as the risk-free rate. The risk aversion coefficient

In Figure

The rolling window estimation for the rank of the covariance matrix with the estimation window of 300 weeks

In Figure

The role of the TP has become indispensable for both researchers and practitioners in finance. Hence, having complete comprehension of the TP properties under all possible scenarios is vital for any financial strategist. In this paper, we deal with the test on the mean-variance efficiency of the TP when both the population and sample covariance matrices are singular. Under these conditions, we deliver the finite sample test statistic and its distribution under both the null and alternative hypotheses. We also derive the high-dimensional asymptotic distribution of the considered test statistic under the null hypothesis as well as for the alternative hypothesis. Through the simulation study, we observe a good quality of the asymptotic approximation of the finite sample statistics, that is, the high-dimensional asymptotic test is properly sized for all values of

The authors are thankful to Prof. Yuliya Mishura and two anonymous reviewers for careful reading of the manuscript and for their suggestions which have improved an earlier version of this paper.