What is equal risk contribution (ERC) in risk budgeting, and how do you calculate each asset's risk contribution?

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AcadiFi · Accepted Answer

Equal risk contribution (ERC) is a risk budgeting approach where each asset contributes the **same amount** to total portfolio risk. It is a practical alternative to mean-variance optimization that avoids the need for expected return forecasts.

**Risk Contribution Decomposition**

For asset i in a portfolio, the **marginal contribution to risk (MCTR)** and **total risk contribution (TRC)** are:

- **MCTR_i = (Sigma * w)_i / sigma_p** (the partial derivative of portfolio sigma with respect to w_i)
- **TRC_i = w_i * MCTR_i**

The key identity: **Sum of all TRC_i = sigma_p** (total risk contributions sum to portfolio volatility).

For **equal risk contribution**: TRC_i = sigma_p / n for all assets.

**Worked Example — Dalton Three-Asset ERC Portfolio:**

Assets: Greystone Equity (sigma = 18%), Millbrook Bonds (sigma = 5%), Ashford Commodities (sigma = 22%).
Correlations: rho(GE, MB) = 0.15, rho(GE, AC) = 0.35, rho(MB, AC) = 0.05.

After iterative optimization (ERC requires numerical solving), the weights are:

| Asset | Weight | Sigma_i | MCTR | TRC | % of Risk |
|---|---|---|---|---|---|
| Greystone Equity | 18.7% | 18% | 12.4% | 2.32% | 33.3% |
| Millbrook Bonds | 62.6% | 5% | 3.72% | 2.32% | 33.3% |
| Ashford Commodities | 18.7% | 22% | 12.4% | 2.32% | 33.3% |
| **Portfolio** | **100%** | | | **6.96%** | **100%** |

Each asset contributes exactly one-third of total portfolio risk (2.32% each, summing to 6.96%).

**Comparison With Other Approaches:**

| Approach | Weights | Risk Contribution |
|---|---|---|
| Equal weight (1/n) | 33.3% / 33.3% / 33.3% | Dominated by equity and commodities |
| Min variance | ~5% / 85% / 10% | Bonds dominate, others minimal |
| **ERC** | **18.7% / 62.6% / 18.7%** | **Equal across all assets** |
| Risk parity | Same as ERC (for this case) | Equal |

**Why ERC Is Attractive:**

1. **Diversification guarantee:** No single asset dominates portfolio risk.
2. **No return inputs:** Like GMV, ERC depends only on the risk model (covariance matrix).
3. **Intuitive:** Each dollar of risk budget is allocated equally.
4. **Robust:** Less sensitive to estimation error than mean-variance optimization.

**Solving for ERC Weights:**

ERC weights cannot generally be solved in closed form (except in special cases). The typical approach is to minimize:

Sum over all pairs (i,j): [TRC_i - TRC_j]^2

This is a convex optimization problem solvable with standard numerical methods.

**Exam Points:**

1. Know the formulas: MCTR, TRC, and the sum-to-sigma identity.
2. ERC is the specific case of risk budgeting where all budgets are equal.
3. Unlike 1/n weighting, ERC considers correlations and volatilities — assets with higher risk get lower weights.
4. ERC and risk parity are often used interchangeably, though risk parity sometimes implies leverage to target a desired return.

Explore risk budgeting in our CFA Level II Portfolio Management module.

What is equal risk contribution (ERC) in risk budgeting, and how do you calculate each asset's risk contribution?

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