Risk Parity Optimality Even with Negative Sharpe Ratio Assets

Abstract

A stopped clock is right twice a day. Similarly, any portfolio allocation is likely to be optimal at least at some point. Risk parity is no stopped clock. The authors derive a general result regarding when and why risk parity is Sharpe ratio optimal, even with negative-Sharpe-ratio assets. This derivation goes beyond the simple observation that risk parity is Sharpe ratio optimal when asset correlations and Sharpe ratios are identical. Based on the analytical result, the authors develop an indicator to describe when risk-parity strategies are likely to be more or less optimal. They also explain how negative-Sharpe-ratio assets can still be an important part of a portfolio—whether it is a risk-parity portfolio or not. Although risk parity and risk balancing in general do not require assumptions about returns, the authors provide guidance regarding how to infer returns that are consistent with the portfolios built from targeting risk.

TOPICS: Statistical methods, portfolio theory, portfolio construction

Key Findings

• • Conditions that are generally sufficient for risk-parity portfolios to be Sharpe ratio optimal are provided, beyond the special case of identical cross-asset correlations and Sharpe ratios. From this, an indicator of how likely risk parity is to be Sharpe ratio optimal can be derived.

• • Based on diversification properties of assets, the generally sufficient conditions for risk-parity portfolios to be Sharpe ratio optimal explain why even negative-return assets can be included in a diversified portfolio.

• • A method is provided to translate risk targets into a rank ordering of asset returns. With an additional set of assumptions, these rank orderings can be translated into returns for a coherent set of risk targets, risks, and returns.