Differentiated risk models in portfolio optimization: a comparative analysis of the degree of diversification and performance in the São Paulo Stock Exchange (BOVESPA)

Abstract

Faced with so many risk modeling alternatives in portfolio optimization, several questions arise regarding their legitimacy, utility and applicability. In particular, a question arises involving the adherence of the alternative models with regard to the basic presupposition of Markowitz's classical model, with regard to the concept of diversification as a means of controlling the relationship between risk and return within a process of optimization. In this context, the aim of this article is to explore the risk-differentiated configurations that entropy can provide, from the point of view of the repercussions that these have on the degree of diversification and on portfolios performance. The reach of this objective requires that a comparative analysis is made between models that include entropy in their formulation and the classic Markowitz model. In order to contribute to this debate, this article proposes that adaptations are made to the models of relative minimum entropy and of maximum entropy, so that these can be applied to investment portfolio optimizations. The comparative analysis was based on performance indicators and on a ratio of the degree of portfolio diversification. The portfolios were formed by considering a sample of fourteen assets that compose the IBOVESPA, which were projected during the period from January 2007 to December 2009, and took into account the matrices of covariance that were formed as from January 1999. When comparing the Markowitz model with two models that were constructed to represent new risk configurations based on entropy optimization, the present study concluded that the first model was far superior to the others. Not only did the Markowitz model present better accumulated nominal yields, it also presented a far greater predictive efficiency and better effective performance, when considering the trade-off between risk and return. However, with regards to diversification, the Markowitz model concentrated its weights in only five of the fourteen sample assets. Contrary to these two models, the maximum entropy model showed a level of diversification that was very close to the maximum level, which would be a situation that is far more in keeping with Markowitz's diversification precepts. However, these models showed the worst results in the comparative analysis of performance.