Equal risk contribution (ERC)

用通俗的话解释

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.

计算方法

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 xᵢ

如何使用

• 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.

得出结论

• Compared with risk parity, money moves away from correlated groups and towards hedges.
• Like risk parity, it is long only and holds every ticker.

注意事项

• 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.

一点历史

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).