Comparing Mean-Variance and CVaR optimal portfolios, assuming bivariate skew-t distributed returns

In this paper we are building portfolios consisting of the S&P 500 index and a T-bond index. The portfolio weights are chosen in such a way that the risk for the portfolio is minimized. To be able to minimize the risk for a portfolio, we first have to specify how to measure the portfolios risk. There are several ways of measuring the risk for a portfolio. In this paper we are investigating how the portfolio weights differ whether we measure the portfolios risk by the variance or by the Conditional Value-at-Risk (CVaR). To measure the risk for the portfolios we first estimated a two-dimensional density function for the returns of the assets, using a skew student-t distribution. The time horizon for each portfolio was one week. The result shows that the weights in the S&P 500 index always were lower for the portfolios constructed by minimizing CVaR. The reason for this is that the distribution for the returns of the S&P 500 index exhibits a negative skewness and has fatter tails than the returns of the T-bond index. This fact isn't taken care of when choosing weights according to the variance criteria, which leads to an underestimation of the risk associated with the S&P 500 index. The underestimation of the risk leads to an overestimation of the optimal weights in the S&P 500 index.