A test on the location of the tangency portfolio on the set of feasible portfolios

Abstract

Due to the problem of parameter uncertainty, specifying the location of the tangency portfolio (TP) on the set of feasible portfolios becomes a challenging task. The set of feasible portfolios is a parabola in the mean-variance space with optimal portfolios lying on its upper part. Using statistical test theory, we want to decide if the tangency portfolio is mean-variance efficient, i.e. if it belongs to the upper limb of the efficient frontier. In the opposite case, the investor would prefer to invest into the risk-free asset or into the global minimum variance portfolio which lies in the vertex of the set of feasible portfolios. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we suggest a test on the location of the tangency portfolio on the set of feasible portfolios. The distribution of the test statistic is derived under both hypotheses, which we use to assess the power of the test and construct a confidence interval. Moreover, out-of-sample performance of the test is evaluated based on real data. The robustness to the assumption of normality is investigated via an extensive simulation study where we show that the new test is robust to the violation of the normality assumption and can also be used for heavy-tailed stochastic models. Moreover, in an empirical study we apply the developed theory to real data. We find that when the sample size is relatively large and a stable period is present on the market, then the mean-variance efficiency of the tangency portfolio can be statistically justified.

A test on mean-variance efficiency of the tangency portfolio in high-dimensional setting

“…In the case of non rejection of H 0 , the investor cannot be sure of the optimality of the TP and the investment in the risk-free rate could be considered as a suitable alternative. Muhinyuza et al (2017) proposed the following test statistics for (11)…”

“…No further relation is imposed between k and n. Note that, under high-dimensional setting, the usual estimators for the precision matrix (Inverse of the covariance matrix) performs poorly and are not consistent anymore (Bodnar et al (2016a)). Therefore, it is worth to study the behaviour of the test statistic developed by Muhinyuza et al (2017) under highdimensional environment and propose an alternative test which takes into account the correction of the estimated precision matrix.…”

“…In this section we examine the performance of developed test by comparing their asymptotic powers and we also study how robust are the two approaches to the violation of normality assumption. we firstly compare the performance of the power function of the test derived in Section 2.1 and the power function of the statistical test developed in Muhinyuza et al (2017) for k/n → c ∈ (0, 1) as n → ∞. Results of both Theorem 2 and Muhinyuza et al (2017, Proposition 1) show that the power functions of the tests depend on the mean vector and the covariance matrix only through the slope parameter s of the efficient frontier and the Sharpe ratio S GM V of the GMVP.…”

“…Since the introduction of mean-variance theory by Markowitz (1952), a large number of papers devoted to the optimal portfolio selection have been published and brought remarkable contributions in different avenues of finance, be it in research or in practice. Since then, the problem of testing the efficiency of a given portfolio has gained a lot of attention (Bodnar and Schmid (2008); Bodnar et al (2019a); Gibbons et al (1989); Glombek (2014); Britten-Jones (1999); Muhinyuza et al (2017)) to just name a few. In this regard, the aim of the investor is to find an optimal portfolio that minimizes the risk, i.e, the variance of the portfolio for a given level of the expected return.…”

Higher order moments of the estimated tangency portfolio weights

“…Kotsiuba and Mazur [34] derived the approximate density function formula for the weights, which is based on the Gaussian integral and the third-order Taylor expansion. A test on the location of the TP on the set of feasible portfolios is developed by Muhinyuza et al [42]. Bodnar et al [15] extended the results by Bodnar and Okhrin [16] in the setting when both the population and the sample covariance matrices are singular.…”