Portfolio construction
Equal risk contribution (ERC)
The true risk-parity portfolio: every ticker contributes exactly the same share of the portfolio's risk, correlations included.
The idea in plain words
A ticker's contribution to risk is its weight times how much it adds to the portfolio's volatility at the margin, which depends on its correlations with the rest. ERC sets the weights so that these contributions are equal. Tickers that hedge others get more money; tickers that duplicate others get less.
How it is calculated
Risk contribution of ticker i: wᵢ × (Σw)ᵢ ÷ σₚ; they add up to σₚ. The optimiser finds equal contributions by cyclical coordinate descent: each weight in turn is set to the root of a quadratic, repeated until nothing changes.
RCᵢ = wᵢ (Σw)ᵢ ÷ √(wᵀΣw), Σᵢ RCᵢ = σₚ
ERC: RCᵢ = σₚ ÷ n for every i
Update: xᵢ = [−cᵢ + √(cᵢ² + 4 Σᵢᵢ bᵢ σₚ)] ÷ (2 Σᵢᵢ), cᵢ = (Σx)ᵢ − Σᵢᵢ xᵢHow to use it
- The best "no forecasts" balanced portfolio when correlations matter.
- Check the Risk shares chart: all bars equal is the signature of ERC.
- Add Volatility targeting if you want a higher or lower total risk.
Drawing conclusions
- Compared with risk parity, money moves away from correlated groups and towards hedges.
- Like risk parity, it is long only and holds every ticker.
Example
Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). Weights 17% US500, 15% GER40, 24% gold, 45% EURUSD: each carries exactly 25% of the risk. Against simple risk parity, gold gains 3 points and the two correlated indices lose 2 between them. Return 4.5%, volatility 8.5%, diversification ratio 1.50.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- Still puts a lot of money in calm tickers; the return is moderate unless leveraged.
- Contributions are equal on past correlations; in a crisis they can converge.
A little history
Maillard, Roncalli and Teïletche, "The Properties of Equally Weighted Risk Contribution Portfolios" (Journal of Portfolio Management, 2010); the coordinate-descent algorithm used here is from Griveau-Billion, Richard and Roncalli (2013).
Educational content only. No portfolio model knows the future: every one of them works on estimates from the past, and nothing here is investment advice.