Risk contribution

A risk contribution attributes an aggregate risk measure to the components of a portfolio. It does not exist independently of the selected measure: a component may contribute to volatility, VaR, Expected Shortfall or a scenario loss, but those contributions are not interchangeable. The variables, units, horizon, confidence level, model and data date must also be stated.

This decomposition answers a different question from a capital weight. A security with a 10% weight may produce more or less than 10% of portfolio volatility depending on its standalone risk and covariance with the rest. Nor is a contribution a risk budget: the former is calculated from the current portfolio, while the latter is an objective chosen before optimization or control.

Marginal, component and incremental

Let R(w) be a risk measure applied to a vector of weights or exposures w. Three expressions that are often confused have different meanings:

The marginal derivative is local and its unit depends on wᵢ. The component contribution incorporates the current exposure. Incremental risk can capture nonlinearities that a first-order derivative misses. “Marginal,” “component” and “incremental” are therefore not synonyms.

When R is differentiable and positively homogeneous of degree one, Euler’s theorem provides full allocation:

A percentage share may then be written as:

These are identities under precise conditions, not trading rules. If the measure is not homogeneous, is not differentiable at the point, or is estimated through an incompatible procedure, the contributions may not add to the total. Data errors, rounding and numerical approximation can also create a residual that must be reconciled.

The volatility case

For linear returns, coherent weights and a covariance matrix Σ, portfolio volatility is:

When σₚ > 0, marginal and component contributions are:

It follows that ∑ RCᵢ = σₚ. The term (Σw)ᵢ contains both standalone variance and covariances with other components. A contribution is therefore not obtained by simply multiplying a weight by standalone volatility.

The formula is an identity within a linear volatility model. By itself, it does not capture nonlinear payoffs, gaps, defaults, market impact, funding or tail dependence. Options and complex portfolios may require full revaluation or a coherent factor-and-sensitivity decomposition.

Numerical example

Two components have weights of 60% and 40%, estimated annualized volatilities of 10% and 20%, and correlation of 0.25. Their covariance is 0.10 × 0.20 × 0.25 = 0.005. Estimated portfolio variance is:

Thus σₚ ≈ 11.14%. The product Σw equals 0.008 for the first component and 0.019 for the second. The contributions are about:

They add to 11.14%. As shares, the first component contributes about 38.7% and the second 61.3%. In this example and under this estimated matrix, 40% of the capital assigned to the second component produces more than 60% of portfolio volatility. Contributions change when weights, volatilities or correlation change.

Negative contributions and hedges

A contribution can be negative when a position locally reduces the selected risk measure because of its sign or dependence with the portfolio. That does not make the position risk-free or guarantee protection in every scenario. A hedge can add basis, liquidity, collateral and counterparty risk, and its estimated relationship may change.

When some contributions are negative, positive shares can exceed 100% because they are offset by negative ones. A dashboard should show signed values, units, the reconciled total and scenario results. Ranking absolute values alone removes hedging information; showing only signed values may hide high gross exposures.

Other measures and their conditions

VaR and Expected Shortfall are homogeneous with respect to position scale in standard formulations, but their contributions require appropriate model conditions and estimation methods. VaR contributions may be unstable near the quantile, while a historical simulation with few tail events is noisy. Expected Shortfall permits tail attribution but remains dependent on the sample, revaluation method and loss definition.

Historical maximum drawdown, by contrast, is path-dependent and may switch its peak and trough as a weight changes. A simple Euler decomposition must not be assumed. Scenario losses are additive only when revaluation and interactions are defined consistently; in nonlinear portfolios, a “with-versus-without” contribution can differ from the marginal result.

Correct use in a process

1. Select a measure suited to the decision and state its unit.
2. Reconcile weights, prices, currency, factors and matrix at the same date.
3. Calculate MRC, RC and shares without using their names interchangeably.
4. Confirm that contributions add to the total or explain the residual.
5. Compare contributions by position, strategy and factor through look-through.
6. Repeat under alternative estimates and scenarios, not only a central case.
7. Use the output to diagnose concentration or compare with a budget, not as an automatic buy or sell instruction.

Sources

• Dirk Tasche, Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle — foundations, conditions and applications of Euler allocation for risk measures.
• Edward E. Qian, On the Financial Interpretation of Risk Contribution: Risk Budgets Do Add Up — financial interpretation of contributions and their relationship to risk budgets.
• Harry Markowitz, Portfolio Selection, The Journal of Finance (1952) — basis for portfolio measurement through weights, variances and covariances.
• Dirk Tasche, Expected Shortfall and Beyond — properties of Expected Shortfall and attribution to portfolio components.
• Benjamin Bruder and Thierry Roncalli, Managing Risk Exposures Using the Risk Budgeting Approach — use of contributions in portfolios with stated risk budgets.

Links

• Risk allocation and sensitivities
• Aggregate risk
• Concentration risk
• Risk budgeting
• Risk parity