Power-law Portfolios

TL;DR

This paper introduces power-law portfolios that optimize asset allocation based on arbitrary risk penalties, leading to improved diversification and a new class of portfolios that generalize Kelly strategies.

Contribution

It develops a framework for constructing portfolios with risk penalties proportional to higher moments of returns, revealing a power-law relationship and a limit case of perfect diversification.

Findings

Component weights scale as a power-law with returns.

Power-law portfolios outperform Kelly portfolios in diversification.

Infinite order portfolios achieve perfect diversification.

Abstract

Portfolio optimization methods suffer from a catalogue of known problems, mainly due to the facts that pair correlations of asset returns are unstable, and that extremal risk measures such as maximum drawdown are difficult to predict due to the non-Gaussianity of portfolio returns. \\ In order to look at optimal portfolios for arbitrary risk penalty functions, we construct portfolio shapes where the penalty is proportional to a moment of the returns of arbitrary order $p>2$. \\ The resulting component weight in the portfolio scales sub-linearly with its return, with the power-law $w \propto \mu^{1/(p-1)}$. This leads to significantly improved diversification when compared to Kelly portfolios, due to the dilution of the winner-takes-all effect.\\ In the limit of penalty order $p\rightarrow\infty$, we recover the simple trading heuristic whereby assets are allocated a fixed positive weight when their return exceeds the hurdle rate, and zero otherwise. Infinite order power-law portfolios thus fall into the class of perfectly diversified portfolios.