Portfolio Optimization Model Incorporating the Investor's View.

Student Name: Priya G. M.

In a portfolio, deciding the proportion to be invested in an individual security component is a critical decision, especially where large sum of investment is involved. The most common method which facilitates the decision-making in such cases is, using historical data for each security in the portfolio for estimating the first and second moments of the returns, which are then used as inputs in the Markowitz or Black-Litterman portfolio optimization model yielding the optimal portfolio weights. Now the historical returns of the securities in a portfolio are typically auto as well as cross correlated. Thus an appropriate way of capturing this stylistic feature would be to use a Vector Auto Regressiove (VAR) model for the return-vector of the securities in the portfolio, which can then be subsequently used for computing the expectation and the variance-covariance matrix of the return-vector. However there are situations where an investor has some prior idea about the value of the returns of the securities involved in the portfolio. In such a situation the usual Maximum Likelihood or Least Square analysis of the VAR model does not allow inclusion of such prior information. These prior views can be easily accommodated using a Bayesian approach to VAR modeling which incorporates the historical data as well as the prior view of the investor. Typically the prior view is expressed in terms of the investor's guess about the expected return and the uncertainty the investor has in this guessed value, which is expressed in terms of the standard deviation of the expected return, assuming that the prior of the vector of expected returns is multivariate Normal. However note that a general VAR(p) model for a k× 1 return-vector has a k× 1 constant-vector parameter, say B0, p k×k AR coefficient matrices and a k×k variance convariance matrix of the innovation vector. A semi-non-informative joint Wishart-Multivariate prior is assumed for the p k×k AR coefficient matrices and the k×k variance convariance matrix of the innovation vector. These together with the investors prior specification on the k× 1 vector of expected returns are used to provide a prior for B0. Then these priors are used for a Bayesian VAR modeling of the historical returns. The resulting posterior distribution of these VAR parameters are then used to compute the posterior means of the expectation and variance-covariance matrix of the return-vector (as determined by the fitted VAR model) using Monte-Carlo simulation. Finally these posterior mean values are fed into Markowitz portfolio optimization model for determining the optimal portfolio weights.