Multi-Moments Method for Portfolio Management: Generalized Capital Asset Pricing Model in Homogeneous and Heterogeneous markets

TL;DR

This paper introduces a new set of risk measures based on semi-invariants, derives generalized efficient frontiers and CAPM models for both homogeneous and heterogeneous markets, and provides exact formulas for portfolio moments using Weibull distributions.

Contribution

It develops a generalized CAPM framework and risk measures based on semi-invariants, extending previous models to heterogeneous assets with different Weibull exponents and dependence structures.

Findings

New risk measures emphasize tail risks.

Exact formulas for portfolio moments with Weibull distributions.

Conditions for achieving higher returns with lower large risks.

Abstract

We introduce a new set of consistent measures of risks, in terms of the semi-invariants of pdf's, such that the centered moments and the cumulants of the portfolio distribution of returns that put more emphasis on the tail the distributions. We derive generalized efficient frontiers, based on these novel measures of risks and present the generalized CAPM, both in the cases of homogeneous and heterogeneous markets. Then, using a family of modified Weibull distributions, encompassing both sub-exponentials and super-exponentials, to parameterize the marginal distributions of asset returns and their natural multivariate generalizations, we offer exact formulas for the moments and cumulants of the distribution of returns of a portfolio made of an arbitrary composition of these assets. Using combinatorial and hypergeometric functions, we are in particular able to extend previous results to the case where the exponents of the Weibull distributions are different from asset to asset and in the presence of dependence between assets. In this parameterization, we treat in details the problem of risk minimization using the cumulants as measures of risks for a portfolio made of two assets and compare the theoretical predictions with direct empirical data. Our extended formulas enable us to determine analytically the conditions under which it is possible to ``have your cake and eat it too'', i.e., to construct a portfolio with both larger return and smaller ``large risks''.