  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.
