Statistical inference for the tangency portfolio in high dimension

Abstract

In this paper, we study the distributional properties of the tangency portfolio (TP) weights assuming a normal distribution of the logarithmic returns. We derive a stochastic representation of the TP weights that fully describes their distribution. Under a highdimensional asymptotic regime, i.e., the dimension of the portfolio, k, and the sample size, n, approach infinity such that k/n → c ∈ (0, 1), we deliver the asymptotic distribution of the TP weights. Moreover, we consider tests about the elements of the TP and derive the asymptotic distribution of the test statistic under the null and alternative hypotheses. In a simulation study, we compare the asymptotic distribution of the TP weights with the exact finite sample density. We also compare the high-dimensional asymptotic test with an exact small sample test. We document a good performance of the asymptotic approximations except for small sample sizes combined with c close to one. In an empirical study, we analyse the TP weights in portfolios containing stocks from the S&P 500 index.

Smart betas, return models and the tangency portfolio weights

“…Furthermore, if we find w˚, a global unique solution to problem (15), and rescale the solution by the factor κ = (1 T w˚) −1 where k 2 R is positive, then the rescaled vector is the relaxed tangency portfolio weights since it maximizes the Sharpe ratio and sums to one.…”

“…We relax the constraint of optimization problem (15) to feature an inequality constraint, w T Sw � 1:…”

“…See e.g. [ 10 , 11 ] for an active search for the tangency portfolio, and [ 12 – 14 ] or [ 15 ] for estimation and analysis of statistical properties of the tangency portfolio in both low and high dimensional settings.…”

Tangency portfolio weights under a skew-normal model in small and large dimensions

“…In Muhinyuza et al (2020) and Muhinyuza (2020) the statistical test for the TP in small and large dimension is derived to deduce whether the portfolio is efficient or not. Karlsson et al (2020) delivered the high-dimensional asymptotic distribution of the estimated TP weights and high-dimensional asymptotic test on the linear combination of the elements of TP weights. Javed et al (2021) obtained analytical expressions for the higher order moments of the estimated TP weights.…”

On the mean and variance of the estimated tangency portfolio weights for small samples

“…3 [15] studied the TP weights in small and large dimensions when both the population and sample covariance matrix are singular. Analytical expressions of higher order moments of the estimated TP weights are derived in [29], while the article [31] presented the asymptotic distribution of the estimated TP weights as well as the asymptotic distribution of the statistical test on the elements of the TP under a high-dimensional asymptotic regime. [38] derived a test for the location of the TP, and [37] extended this result to the high-dimensional setting.…”