A "balanced" 60/40 stock–bond portfolio is not balanced in risk: equities are several times as volatile as bonds, so they supply almost all of its variance. The same is true of any equal-weight or cap-weight allocation across assets of different volatility. Equal risk contribution (ERC), the precise version of what the industry calls risk parity, chooses weights so that every asset contributes the same share of portfolio risk. It uses only the covariance matrix, not expected returns, so it avoids the input that makes mean–variance optimisation fragile. This note defines the ERC portfolio using the risk contributions derived in Risk Decomposition and Budgeting, solves it in the cases that have closed forms, shows how to compute it in general, and proves that it sits between the minimum-variance and the equal-weight portfolio. It closes with the evidence and with when ERC is, and is not, optimal.
Learning objectives. By the end of this note you should be able to:
- state the ERC condition and contrast it with the conditions that define the and minimum-variance portfolios;
- derive the ERC weights for two assets and for assets with a common correlation, and the implicit inverse-beta characterisation in general;
- compute ERC weights numerically from the log-barrier formulation, and explain why the solution is unique;
- prove the volatility ordering ;
- state when ERC coincides with the maximum-Sharpe portfolio, and relate it to the risk-parity sizing rule used by fundamental PMs.
In this lesson
- Three heuristics without expected returns
- Closed forms
- Computing ERC weights
- ERC sits between minimum variance and
- When is ERC optimal?
- The evidence
- Code: ERC, minimum variance and
Before you start
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