5.4.3 Equal Risk Contribution (ERC)

Basic Prerequisites: 02 · Risk Contributions and Calculus & KKT Convex Optimization.

1. Intuition & Practical Objective

Now we put a target on the decomposition of page 02: find the weights $w$ such that every asset contributes the same amount of risk.

R C_{1} (w) = R C_{2} (w) = \dots = R C_{N} (w) = \frac{σ ( w )}{N} .

This is the Equal Risk Contribution (ERC) portfolio. It is the disciplined, covariance-aware form of risk parity - and, under a constant-correlation $Σ$ , it is the naive inverse-vol risk parity. The practical objective of this page is threefold: (1) state the ERC condition and why it is well-posed (existence & uniqueness), (2) give the closed forms that exist in special cases, and (3) present the convex optimization / cyclical-coordinate-descent solver that handles every other case.

Why ERC and not the minimum-variance (MV) or $1/ N$ portfolios? Recall the “menu” (Maillard §3.4):

x_{i} = x_{j} (1/ N), \partial_{x_{i}} σ = \partial_{x_{j}} σ (MV), x_{i} \partial_{x_{i}} σ = x_{j} \partial_{x_{j}} σ (ERC) .

MV equalizes marginal contributions - it is the least-volatile point on the frontier but concentrates both weights and risk in a handful of assets. $1/ N$ equalizes weights - diversified in capital but wildly unbalanced in risk (47% of risk in one of four assets on page 02). ERC is the middle ground: diversified in risk (which is what matters), not maximally cheap, and never zeroing any position because $w_{i} \to 0$ would send its risk contribution to zero. The volatility ordering is strict:

σ_{m v} \leq σ_{erc} \leq σ_{1/ N} .

(Maillard App. A.3 proves this by embedding all three in one family $min w^{⊤} Σ w s.t. \sum_{i} ln w_{i} \geq c$ with $c = - \infty$ → MV, $c$ = ERC level, $c = - N ln N$ → $1/ N$ .)

The one-sentence essence. “ERC is the portfolio that divides the risk pie, not the capital pie, into $N$ exactly equal slices - the unique, correlation-aware sweet spot between $1/ N$ and minimum variance.”

2. Mathematical Ground Truth & Derivations

The condition. From page 02, $R C_{i} = w_{i} (Σ w)_{i} / σ (w)$ . ERC demands equal slices:

w_{i} (Σ w)_{i} = w_{j} (Σ w)_{j} \forall i, j ⟺ w_{i} (Σ w)_{i} = const .

Multiplying by the budget $\sum_{i} w_{i} = 1$ normalizes. Well-posedness: Maillard et al. (2010) show the ERC portfolio exists and is unique as long as $Σ$ is positive-definite and the weights are long-only, by embedding it in a strictly convex program (below).

Closed form - two assets. Let $w_{1} = w$ , $w_{2} = 1 - w$ . The ERC condition $w_{1}^{2} σ_{1}^{2} = (1 - w)^{2} σ_{2}^{2}$ gives the unique solution in $[0, 1]$ :

w_{1} = \frac{σ _{1}^{- 1}}{σ _{1}^{- 1} + σ _{2}^{- 1}} = \frac{σ _{2}}{σ _{1} + σ _{2}}, w_{2} = \frac{σ _{2}^{- 1}}{σ _{1}^{- 1} + σ _{2}^{- 1}} = \frac{σ _{1}}{σ _{1} + σ _{2}}

- and it does not depend on the correlation $ρ$ at all.

Closed form - constant correlation ( $ρ_{ij} = ρ$ ). With $Σ_{ij} = ρ σ_{i} σ_{j}$ ( $i \neq = j$ ) the ERC condition reduces to $w_{i} σ_{i} = w_{j} σ_{j}$ , giving

w_{i} = \frac{σ _{i}^{- 1}}{\sum _{j = 1}^{N} σ _{j}^{- 1}}

i.e. constant correlation ⇒ ERC = inverse-volatility (naive) risk parity. This is the cleanest statement of when the simple rule is exactly right.

Closed form - beta reading (any $Σ$ ). Define $β_{i} = (Σ w)_{i} / σ (w)^{2}$ (the portfolio beta of asset $i$ ). Then $R C_{i} = w_{i} β_{i} σ (w)$ , and ERC implies

w_{i} \propto β_{i}^{- 1},

i.e. weight is inversely proportional to portfolio beta - high-beta, highly-correlated assets get a small capital weight. This is endogenous ( $β_{i}$ depends on $w$ ), so it is interpretive, not a closed form.

Convex formulation (the general solver). Because $σ$ is homogeneous of degree 1, ERC = risk budgeting with equal budgets $b_{i} = 1/ N$ . The standard convex embedding: minimize a quadratic subject to a log-barrier:

w > 0 min \frac{1}{2} w^{⊤} Σ w s.t. \sum_{i = 1}^{N} b_{i} ln w_{i} \geq c,

whose KKT stationarity $(Σ w)_{i} = λ b_{i} / w_{i}$ forces $w_{i} (Σ w)_{i} \propto b_{i}$ - i.e. $R C_{i} \propto b_{i}$ . Setting $b_{i} = 1/ N$ recovers ERC. Equivalently one may minimize $f (w) = \frac{1}{2} w^{⊤} Σ w - \sum_{i} b_{i} ln w_{i}$ for any fixed barrier weight, then renormalize - the road the computational implementation takes.

3. Computational Implementation - ERC by cyclical coordinate descent, verified

The general case (non-constant $Σ$ ) has no closed form, so we solve it numerically. Below is a cyclical coordinate descent (CCD) on the convex barrier $f (w) = \frac{1}{2} w^{⊤} Σ w - \sum_{i} b_{i} ln w_{i}$ : for each coordinate $i$ in turn, with the others held fixed, stationarity $\partial f / \partial w_{i} = 0$ becomes a simple quadratic in $w_{i}$ ,

S_{ii} w_{i}^{2} + (\sum_{j \neq = i} S_{ij} w_{j}) w_{i} - b_{i} = 0,

whose positive root is taken in closed form, then the vector is renormalized to $\sum w = 1$ . On Maillard’s universe it reproduces the paper’s ERC weights exactly - and we verify the equal contributions and the exact Euler sum.

The bottom two rows are the entire argument for “ERC over naive risk parity”: the inverse-vol portfolio (the marketing-default of many risk-parity funds) puts 39.1% of risk in each of assets 1 and 2 and only 10.9% in each of 3 and 4 - it is risk parity only by name. Because assets 1 and 2 are highly correlated ( $ρ = 0.8$ ) and asset 3 is negatively correlated with asset 4 ( $ρ = - 0.5$ ), naive inverse-vol double-counts the redundant correlation among 1–2 and ignores the free diversification of 3–4. ERC’s CCD solution fixes exactly that: 25.0% of risk, each, exactly.

4. Failure Modes & First-Principles Breakdowns

“Inverse volatility” is not always ERC. It is ERC iff the correlation matrix is constant (all $ρ_{ij}$ equal). With real covariance structure, inverse-vol concentrates risk exactly where ERC would not - the worked example above proves it numerically.
ERC is long-only by construction. Relaxing the short constraint destroys uniqueness: multiple weight vectors can satisfy $w_{i} (Σ w)_{i} = const$ when shorting is allowed (Maillard §3.3). The solver above implicitly forces $w > 0$ ; a short-allowed copy of the problem is a different animal.
The solution is a function of $Σ$ , full stop. ERC takes no expected-return input - its robustness to return estimation is real, but it inherits every error in $Σ$ . A covariance estimated on too-short a window, or un-denoised, quietly mismeasures “risk balance.” (See 05 · Failure Modes and Covariance Shrinkage & RMT.)
Convex ≠ cheap for huge $N$ . CCD is $O (N^{2})$ per sweep and excellent for the $N \sim 10$ -100 asset-class case; at security-level $N > 1000$ the same machinery is heavier, which is part of HRP’s appeal (Hierarchical Risk Parity).

5. References

Maillard, Roncalli & Teïletche (2010), JPM 36(4)
Qian, Edward (2005): Risk Parity Portfolios
Griveau-Billion, T., Richard, J.-C. & Roncalli, T.: A Fast Algorithm for Computing High-Dimensional Risk Parity Portfolios (2013)
Roncalli, Thierry: Introduction to Risk Parity and Budgeting, CRC (2013)

6. Connected Graph Bridges

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Forward: 04 · Risk Budgeting · 05 · Failure Modes
Base: KKT Convex Optimization · Min-Variance & Constraints

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