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.

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ایده به زبان ساده

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.

روش محاسبه

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

نحوه استفاده

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.

نتیجه‌گیری

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

مثال

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.

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مراقب این‌ها باشید

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.

کمی تاریخچه

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).

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