Portfolio risk calculation and stochastic portfolio optimization by a copula based approach

Abstract

In this study we used copulas to calculate the risks of stock portfolios and developed a stochastic portfolio optimization model using copulas to find optimal portfolios. Copula is a multivariate distribution function supported in the unit hypercube. The main advantage of copula is that one can separate the marginals of a multivariate distribution from their dependence structure. Thus it is able to model the marginals separately and choose a copula to represent the dependence structure between them. Since the portfolio return is a multivariate distribution of individual asset returns, the portfolio return distribution can be modeled by copulas. With this aim, we selected 15 stocks from New York Stock Exchange and constructed different portfolios. Then we modeled the distributions of individual stock returns and fitted a set of copulas to the joint return data. We found that Student-t and Generalized Hyperbolic distributions are very nice models for modeling individual asset returns. We also found that the t-copula is the best copula to represent the dependence structure between stock returns. Therefore we used this model to calculate the risks of portfolios and compared the results of this model with the results of the classical portfolio risk calculation methods. After the risk calculation, we adopted the copula model to the classical Markowitz portfolio selection problem since the Markowitz optimal portfolio would no longer be optimal. Therefore we transformed the classical quadratic optimization problem into a stochastic optimization problem. We used Nelder-Mead simplex search algorithm to solve this problem and compared our findings with the solution of the classical model.