Aggregate Portfolio Risk Approximation under Bayesian Setting

In portfolio management, it is too important to consider non-sampling information. In this problem, the non-sampling information may be belief of investor about a special asset obtained of historical data of past economical performance of specified asset. This information forms a prior probability regard keeping the asset in portfolio or dropping it. Therefore, for each asset a binary random variable is induced to the problem which is one if the asset will be kept in portfolio and zero if it will be dropped based on investor prior belief about the asset before observing the actual risk and return. These variables are correlated come from a Dirichlet distribution. Hence, the Bayesian setting is a suitable framework to study this problem. In this paper, the Monte Carlo method is applied to approximate the posterior distribution using Monte Carlo Markov Chain (MCMC) method of binary variables given the past returns which indicates the tendency of investor to keep or drop an asset by using the non-sampling and sampling information simultaneously. ModelRisk software is used to derive the analytical results. Bayesian CAPM and APT are proposed. Stochastic approximations are given.