The paper solves the problem of optimal portfolio choice when the parameters of the asset returns distribution, for example the mean vector and the covariance matrix, are unknown and have to be estimated by using historical data on asset returns. Our new approach employs the Bayesian posterior predictive distribution which is the distribution of future realizations of asset returns given the observable sample. The parameters of 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. By contrast, the optimization problem of the traditional approach is based on unknown quantities which are estimated in the second step, and lead to a suboptimal solution. We also derive a very useful stochastic representation of the posterior predictive distribution whose application not only gives the solution of the considered optimization problem, but also provides the posterior predictive distribution of the optimal portfolio return which can be used to construct a prediction interval. A Bayesian efficient frontier, the 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 which is known to be overoptimistic.

In this paper, using the shrinkage-based approach for portfolio weights and modern results from random matrix theory we construct an effective procedure for testing the efficiency of the expected utility (EU) portfolio and discuss the asymptotic behavior of the proposed test statistic under the high-dimensional asymptotic regime, namely when the number of assets p increases at the same rate as the sample size n such that their ratio p/n approaches a positive constant c is an element of (0, 1) as n -> infinity. We provide an extensive simulation study where the power function and receiver operating characteristic curves of the test are analyzed. In the empirical study, the methodology is applied to the returns of S&P 500 constituents.

We consider the estimation of the multi-period optimal portfolio obtained by maximizing an exponential utility. Employing the Jeffreys non-informative prior and the conjugate informative prior, we derive stochastic representations for the optimal portfolio weights at each time point of portfolio reallocation. This provides a direct access not only to the posterior distribution of the portfolio weights but also to their point estimates together with uncertainties and their asymptotic distributions. Furthermore, we present the posterior predictive distribution for the investor's wealth at each time point of the investment period in terms of a stochastic representation for the future wealth realization. This in turn makes it possible to use quantile-based risk measures or to calculate the probability of default, i.e the probability of the investor wealth to become negative. We apply the suggested Bayesian approach to assess the uncertainty in the multi-period optimal portfolio by considering assets from the FTSE 100 in the weeks after the British referendum to leave the European Union. The behaviour of the novel portfolio estimation method in a precarious market situation is illustrated by calculating the predictive wealth, the risk associated with the holding portfolio, and the probability of default in each period.

We derive new results related to the portfolio choice problem for power and logarithmic utilities. Assuming that the portfolio returns follow an approximate log-normal distribution, the closed-form expressions of the optimal portfolio weights are obtained for both utility functions. Moreover, we prove that both optimal portfolios belong to the set of mean-variance feasible portfolios and establish necessary and sufficient conditions such that they are mean-variance efficient. Furthermore, we extend the derived theoretical finding to the general class of the log-skew-normal distributions. Finally, an application to the stock market is presented and the behaviour of the optimal portfolio is discussed for different values of the relative risk aversion coefficient. It turns out that the assumption of log-normality does not seem to be a strong restriction.

In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix-variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large-dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n -> c is an element of [0, + infinity) when the sample covariance matrix does not need to be invertible and p/n -> c is an element of [0,1) otherwise.

In this paper, we construct two tests for the weights of the global minimum variance portfolio (GMVP) in a high-dimensional setting, namely, when the number of assets p depends on the sample size n such that p/n -> c is an element of (0, 1) as n tends to infinity. In the case of a singular covariance matrix with rank equal to q we assume that q/n -> <(c)over tilde is an element of (0,1) as n -> infinity. The considered tests are based on the sample estimator and on the shrinkage estimator of the GMVP weights. We derive the asymptotic distributions of the test statistics under the null and alternative hypotheses. Moreover, we provide a simulation study where the power functions and the receiver operating characteristic curves of the proposed tests are compared with other existing approaches. We observe that the test based on the shrinkage estimator performs well even for values of c close to one.