Portfolio Management of Copula-Dependent Assets Based on P ( Y < X ) Reliability Models: Revisiting Frank Copula and Dagum Distributions

Modern portfolio theory indicates that portfolio optimization can be carried out based on the mean-variance model, where returns and risk are represented as the average and variance of the historical data of the stock’s returns, respectively. Several studies have been carried out to find better risk proxies, as variance was not that accurate. On the other hand, fewer papers are devoted to better model/characterize returns. In the present paper, we explore the use of the reliability measure P ( Y < X ) to choose between portfolios with returns given by the distributions X and Y . Thus, instead of comparing the expected values of X and Y , we will explore the metric P ( Y < X ) as a proxy parameter for return. The dependence between such distributions shall be modelled by copulas. At first, we derive some general results which allows us to split the value of P ( Y < X ) as the sum of independent and dependent parts, in general, for copula-dependent assets. Then, to further develop our mathematical framework, we chose Frank copula to model the dependency between assets. In the process, we derive a new polynomial representation for Frank copulas. To perform a study case, we considered assets whose returns’ distributions follow Dagum distributions or their transformations. We carried out a parametric analysis, indicating the relative effect of the dependency of return distributions over the reliability index P ( Y < X ) . Finally, we illustrate our methodology by performing a comparison between stock returns, which could be used to build portfolios based on the value of the the reliability index P ( Y < X ) .