Risk-Constrained Kelly Portfolios Under Alpha-Stable Laws

Abstract

This paper provides a detailed framework for modeling portfolios, achieving the highest growth rate under risk constraints such as value at risk (VaR) and expected shortfall (ES) in the presence of $$\alpha $$-stable laws. Although the maximization of the expected logarithm of wealth induces outperforming any other significantly different strategy, the Kelly criterion implies larger bets than a risk-averse investor would accept. Restricting the Kelly optimization by spectral risk measures, the authors provide a generalized mapping for different measures of growth and risk. Analyzing over 30 years of S&P 500 returns for different sampling frequencies, the authors find evidence for leptokurtic behavior for all respective sampling frequencies. Given that lower sampling frequencies imply a smaller number of data points, this paper argues in favor of $$\alpha $$-stable laws and its scaling behavior to model financial market returns for a given horizon in an i.i.d. world. Instead of simulating from the class of elliptically $$\alpha $$-stable distributions, a semiparametric scaling approximation, based on hourly NASDAQ data, is proposed. Our paper also uncovers that including long put options into the portfolio optimization, improves portfolio growth for a given level of VaR or ES, leading to a new Kelly portfolio providing the highest geometric mean.