Portfolio Diversification under Local, Moderate and Global Deviations from Power Laws

Abstract

This paper focuses on the analysis of portfolio diversification for a wide class of nonlinear transformations of heavy-tailed risks. We show that diversification of a portfolio of nonlinear transformations of thick-tailed risks increases riskiness if expectations of these functions are infinite. In addition, coherency of the value at risk measure is always violated for such portfolios. On the contrary, for nonlinearly transformed heavy-tailed risks with finite expectations, the stylized fact that diversification is preferable continues to hold. Moreover, in the latter setting, the value of risk is a coherent measure of risk. The framework of transformations of long-tailed random variables includes many models with Pareto-type distributions that exhibit local, moderate and global deviations from power tails in the form of additional slowly varying or exponential factors. This leads to a refined understanding of under what distributional assumptions diversification increases riskiness.