When Equal Risk Contribution Can Be Mean-Variance Optimal

Summary

The document asks what return and volatility assumptions would make risk-based portfolios, especially equal risk contribution (ERC), optimal under mean-variance analysis. It contrasts ERC with minimum variance, which it says is optimal when expected returns are independent of asset volatilities. The central claim, attributed to an external asset-management write-up, is that an all-positive-weight ERC portfolio is mean-variance optimal when each asset has the same expected marginal Sharpe ratio relative to the ERC portfolio.

This is a question about the conditions behind that claim, not a derivation or demonstration of it. It offers no data, worked example, or proof; the author is specifically seeking a concise proof or heuristic. The stated condition should therefore be treated as a proposed characterization from the cited write-up, rather than as a result established within the document. The discussion also does not specify broader constraints or how the condition changes when short positions or other portfolio restrictions are allowed.

Key ideas

• The document asks when equal risk contribution can be optimal under mean-variance analysis.
• It presents equal expected marginal Sharpe ratios relative to the ERC portfolio as a claimed optimality condition.
• The claim assumes the ERC portfolio holds every asset with positive weight.
• Minimum variance is contrasted as optimal when expected returns are independent of asset volatilities.
• No proof, empirical evidence, or worked example is provided.

Tags

# Risk Parity/ Equal Risk Contribution - Optimality in a Mean Variance Sense


# Risk Parity/ Equal Risk Contribution - Optimality in a Mean Variance Sense












I am trying to wrap my head around what characteristics/assumptions on average returns and volatilities of assets would make the equal risk contribution portfolio optimal in a mean-variance sense. I am trying to do this for all kinds of risk-based optimization methods (mainly: minimum variance, max diversification, and risk-parity) so that I can understand what the implied views of each technique are.

For example, the minimum variance portfolio is mean-variance optimal if average returns are independent of volatilities - this makes sense intuitively as it suggests there is little to be gained by taking on excess volatility. However, I wanted to know under what conditions an equal risk contribution portfolio is optimal, and if a proof/heuristic sketch of a proof would be available.

In general this by write-up by ReSolve Asset Management states that:

> The Equal Risk Contribution portfolio will hold all assets in positive weight, and is mean-variance optimal when all assets are expected to contribute equal marginal Sharpe ratios (relative to the Equal Risk Contribution portfolio itself)

but I was wondering if anyone could provide a short proof of this statement.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.