Bayesian mean-variance analysis: Optimal portfolio selection under parameter uncertainty

Abstract

The paper solves the problem of optimal portfolio choice when the parameters of the asset returns distribution, like the mean vector and the covariance matrix are unknown and have to be estimated by using historical data of the asset returns. The new approach employs the Bayesian posterior predictive distribution which is the distribution of the future realization of the asset returns given the observable sample. The parameters of the posterior predictive distributions are functions of the observed data values and, consequently, the solution of the optimization problem is expressed in terms of data only and does not depend on unknown quantities. In contrast, the optimization problem of the traditional approach is based on unknown quantities which are estimated in the second step leading to a suboptimal solution.We also derive a very useful stochastic representation of the posterior predictive distribution whose application leads not only to the solution of the considered optimization problem, but provides the posterior predictive distribution of the optimal portfolio return used to construct a prediction interval. A Bayesian efficient frontier, a set of optimal portfolios obtained by employing the posterior predictive distribution, is constructed as well.Theoretically and using real data we show that the Bayesian efficient frontier outperforms the sample efficient frontier, a common estimator of the set of optimal portfolios known to be overoptimistic.

Smart network based portfolios

“…As a robustness check we consider two large-dimensional portfolios with different characteristics. The investment universe of the first portfolio is composed by 266 among largest banks and insurance companies in the world 3 .…”

“…Alternative approaches deal with the problem of optimal portfolio choice by employing a Bayesian methodology to estimate unknown mean-variance parameters reducing the estimation errors. In this context, one of the most prominent is the Bayes-Stein approach based on the idea of shrinkage estimation ( [22,23,3]). The authors in [26] propose the shrinkage estimator toward the constant correlation, while in [31] this approach has been extended to higher moments such as skewness and kurtosis.…”

Bayesian Filtering for Multi-period Mean-Variance Portfolio Selection

“…They derive posterior distribution for the weights using various standard priors for mean vector and covariance matrix. Recently, Bauder et al [3] assumed returns to be infinitely exchangeable and multivariate centered spherically symmetric for unknown mean vector and covariance matrix. They derive posterior predictive distributions of returns which is used to obtain optimal portfolio weights.…”