Encolhimento da matriz de covariâncias e índices de incerteza: a utilização do Bootstrap na seleção de portfólios

Abstract

Capital market agents, which aim to obtain return from investments, end up facing decisions that involve expected gains associated with the inherent risks. For a given type of investor, or portfolio manager, it is essential to obtain the mean and the covariance estimates of the returns on the financial assets to build efficient portfolios. Improving the portfolio selection process can come from both better inputs as much as investment policies. In this research, we used the bootstrap technique to promote improvements in this context. From the bootstrap of assets, that is, the random drawing of subsets of distinct assets, multiple projections of portfolios were obtained. In possession of such information, it was possible to create two innovations: a alternative way to shrink the covariance matrix and individualized uncertainty indexes per asset. Additionally, a systemic sensitivity index for the weights of the portfolios was proposed, as a function of the covariance matrix of the returns. A comparison of different forms of shrinkage of the covariance matrix was carried out and the results showed that good numerical conditioning is not always reflected in low sensitivity of the weights. There is a trade-off between the sensitivity of the weights, distance from the sample estimate and the numerical conditioning that must be taken into account when choosing the most appropriate shrinkage method. A comparison with strategies present in the literature was made using data related to the US financial market and, although none of them statistically exceeded the Sharpe ratio from the naive allocation (1/N), out-of-sample performance improvements were detected in relation to portfolios of minimum variance or mean-variance traditional approaches. The individualized uncertainty indices were used to modify a portfolio selection strategy, present in the literature, which allows the simultaneous incorporation of expected return, variance and uncertainty. In this case, no distinction was identified between the original version and the modified strategy. It was found that the process of combining multiple projections, chosen by the asset bootstrap, leads to the creation of "factor portfolios", driven to the intrinsic factors of the returns, without the prior estimation of how many or what these risk factors are necessarily. Such an approach proved to be an alternative to control the level of exposure to idiosyncratic risk in portfolios without substantial loss of Jensen’s alfa ("abnormal return").