PORTFOLIO OPTIMIZATION IN STATISTICAL FINANCE

Abstract

In this thesis we will find the optimum portfolio for a given set of assets. Since the return is a random quantity we would like to maximize its expected value. Instead of assuming anything about the distribution of the assets, we will estimate the expected value of the return and maximize the estimator. This will be done using a predictive distribution of the assets based on their history. To find the predictive distribution we can use an ICA based model or a similar randomization scheme but we will not address the prediction problem in this thesis. We will invest in terms of probability. The constraints of the maximization problem will be that the probability of the return to be less than a given value ? not to fall below a given probability q denote the probability of the return to be less than ? by Fn+1,?. In the code for the brute force approach, which will give us the true value of the optimum portfolio and return, we will use a non-smooth approximation of Fn+1,?. For the interior point method we will approximate Fn+1,? by a smooth function F� ?. We will do this by convoluting F�? with the pdf of a normal distribution with zero mean and a very small variance. This approach is non-parametric, hence more robust. To solve the maximization problem we will apply a log-barrier method. We will give an interior point method and also a brute force approach which will solve the maximization problem. The brute force approach will be used only for the cases of k ? 3 assets. The role of the brute force approach will be mainly to give us the true solution to the maximization problem with which we will compare the solution given by the interior point method. In this way we will be able to validate the solution given by the interior point method for the cases of k ? 3 ts and to rely only on this method for the cases of k ? 4 assets.