Component Risk Contributions Are Not Capital Weights

What is being allocated

Capital weights describe how a portfolio's funds are distributed. Component risk contributions describe how a specified risk estimate is attributed across those positions. They answer different questions. This guide uses portfolio volatility, computed as the square root of w' Sigma w, where Sigma is a return covariance matrix. The attribution is conditional on both the weights and that matrix; it is not a permanent characteristic of an asset. A holding with a small standalone volatility can reinforce the rest of the portfolio, while a more volatile holding can partially offset it. Its role emerges from joint behavior rather than capital size alone.

Distinguish marginal, component and relative quantities

For positive portfolio volatility sigmaP, the marginal contribution of asset i is (Sigma w)_i divided by sigmaP. It measures the local derivative of volatility with respect to that weight, holding the other weights fixed. Multiplying by w_i gives the component contribution. Dividing again by sigmaP gives its relative share. Euler's identity makes component contributions sum to total volatility, while relative shares sum to one. These are three different scales: sensitivity per unit of weight, an allocated amount of volatility, and a fraction of total volatility. A table headed only risk contribution leaves the reader guessing which quantity is being reported.

A long holding can have negative contribution

The synthetic example gives A and B equal capital weights, annual volatilities of 20% and 10%, and correlation of -0.75. Portfolio variance is 0.005, so volatility is approximately 7.071068%. A's variance allocation is 0.00625 and B's is -0.00125. Dividing those allocations by portfolio volatility gives component contributions of 8.838835 and -1.767767 percentage points. Their relative shares are 125% and -25%. B's negative share means its covariance with this particular portfolio is negative. It does not mean B has negative standalone variance, produces guaranteed gains or will protect the portfolio under every possible market move.

Variance attribution has a factor-of-two trap

Define q_i = w_i times (Sigma w)_i. These quantities sum to portfolio variance and, divided by total variance, produce the same relative shares as volatility contributions. However, the derivative of variance with respect to w_i is 2 times (Sigma w)_i. Multiplying that derivative by w_i and summing gives twice the variance, because variance is homogeneous of degree two rather than one. Thus q_i is one half of the weighted variance derivative. A component-risk calculation must identify whether it returns q_i, the volatility contribution or a normalized share. A numerically plausible sum can otherwise conceal a mismatch between labels and formulas.

A marginal derivative is not a funded trade

Increasing one weight while holding all others constant changes the sum of weights. In a fully invested portfolio, buying more A commonly requires selling B or another holding. For a small transfer epsilon from B to A, the first-order volatility change is epsilon times the difference between their marginal contributions, not epsilon times A's marginal contribution alone. For a large transfer, the derivative is only a local approximation and the full quadratic must be recomputed. Removing a holding altogether is likewise a finite portfolio change, not the same operation as subtracting its original component contribution from total volatility.

Aggregation and numerical instability

Contributions calculated from one consistent portfolio and covariance matrix can be added across holdings to describe a sector or sleeve's allocation of total risk. That aggregate is not the sleeve's standalone volatility, which would use only its internal holdings and covariances. Attribution also becomes unstable when total variance is very small because relative shares divide by that small quantity. Large positive and negative shares can then offset almost exactly. At zero portfolio variance, the usual normalized formulas are undefined. These cases should not be repaired by replacing negative shares with zero, which destroys the reconciliation and hides the hedging relationships being measured.

Interpret an attribution as a conditional explanation

The component-risk card explains a modeled portfolio at a stated allocation and covariance estimate. Capital weights and contribution shares belong beside each other precisely because disagreement is informative. A concentration in one contribution may arise from its weight, volatility, correlation pattern or all three, and an alternative covariance scenario can reveal which explanation dominates. Volatility attribution remains different from drawdown attribution, loss attribution or a stress test involving nonlinear instruments. The sources establish the decomposition; the synthetic reconciliation establishes only that the numbers follow it. Neither turns a negative contribution into a durable hedge designation or an allocation recommendation.

Primary references

These sources support methods and conventions. The example values are original constructions, not observations extracted from these documents.

• Dirk Tasche: Capital Allocation to Business Units and Sub-Portfolios, the Euler Principle, section 3.1
• Daniel P. Palomar: Portfolio Optimization, section 11.3, Risk Contributions

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