Building multi-scale portfolios and efficient market frontiers using fractal regressions

Abstract

The risk-return relationship remains a hot research topic, since it continues to have some non-explored issues. For example, despite the existence of several limitations, the Capital Asset Pricing Model (CAPM) continues to be used and studied in the literature. Recently, CAPM was estimated based on fractal regressions, making it possible to distinguish between short and long-time scales, with the possibility of supporting the fractal market hypothesis. In this paper, we use this approach and extend it by building a multi-scale portfolio and efficient frontiers and considering three different market stages: pre-crisis, crisis and post-crisis, applied to the Dow Jones Index and its components. The main results show different behaviors along the different market stages, with higher scale dependence in the crisis period, when compared, for example, with the post-crisis one. Furthermore, when analyzing the efficient frontiers, it is possible to identify an inverted U-shaped relation between return and risk in the pre-crisis period, a negative relationship in the crisis period and a positive (and expected) relationship in the post-crisis period. These results show the importance of this relationship for investors, in order to build their portfolios, but also for financial authorities, because the continuous monitoring of markets could help to prevent possible crisis.