5.4.2 Risk Contributions

Basic Prerequisites: Calculus (homogeneous functions, Euler’s theorem) and Linear algebra (quadratic forms, matrix–vector products).

1. Intuition & Practical Objective

Before we can equalize risk contributions we have to define them sharply. This page builds the exact, unambiguous decomposition of portfolio risk into per-asset pieces - the same object used by risk teams under the label risk budgeting. The objective is three well-defined quantities:

marginal MRC_{i} \times w_{i} total R C_{i} \div σ (w) percentage R C_{i} / σ (w)

Marginal risk contribution (MRC $_{i}$ ): if you add a tiny amount more of asset $i$ (say one more basis-point of weight, financed by cash), by how much does the portfolio’s volatility change? This is simply the partial derivative $\partial σ / \partial w_{i}$ .
Total risk contribution (RC $_{i}$ ): the slice of the portfolio’s total volatility that is attributable to holding asset $i$ - holding $w_{i}$ of an asset whose marginal effect is MRC $_{i}$ . RC $_{i} = w_{i} \times MRC_{i}$ .
Percentage contribution: RC $_{i} / σ (w)$ - the fraction of total risk coming from asset $i$ . Crucially, these $N$ percentages always sum to 100%. That is the “risk budgets do add up” result that Qian (2006) proved is more than a bookkeeping identity: it is (approximately) the expected contribution to a portfolio loss.

The practical payoff: once risk is decomposable, ETFs, pension books and hedge funds can look inside a portfolio and see which position is going to be responsible for the next big loss - and can then set the decomposition intentionally (that is risk budgeting, 04).

2. Mathematical Ground Truth & Derivations

Setup. Portfolio weights $w$ (summing to 1), covariance matrix $Σ$ , portfolio variance $w^{⊤} Σ w$ and volatility

σ (w) = w^{⊤} Σ w .

Verify the homogeneity claim first: $σ (λ w) = (λ w)^{⊤} Σ (λ w) = λ w^{⊤} Σ w = λ σ (w)$ , a homogeneous function of degree 1.

Marginal risk contribution.

MRC_{i} = \frac{\partial σ}{\partial w _{i}} = \frac{1}{2 w ^{⊤} Σ w} \cdot 2 (Σ w)_{i} = \frac{( Σ w ) _{i}}{w ^{⊤} Σ w} = \frac{( Σ w ) _{i}}{σ ( w )},

where $(Σ w)_{i}$ is the $i$ -th entry of the vector $Σ w$ - i.e. $cov (r_{i}, w^{⊤} r)$ , the covariance of asset $i$ with the whole portfolio. This is a beautiful object: the marginal risk of an asset is not its own volatility but its covariance with the portfolio.

Total risk contribution & the Euler decomposition. Euler’s theorem for an $R$ -homogeneous-function-of-degree-1 says $σ (w) = \sum_{i} w_{i} \frac{\partial σ}{\partial w _{i}}$ , hence

σ (w) = i = 1 \sum N R C_{i}, R C_{i} := w_{i} \frac{( Σ w ) _{i}}{σ ( w )}

Percentage contribution. Dividing through by $σ (w)$ gives $1 = \sum_{i} \frac{R C _{i}}{σ ( w )}$ . Each term $R C_{i} / σ (w)$ is the fraction of total risk from asset $i$ . Two equivalent formulas for it (both useful):

\frac{R C _{i}}{σ ( w )} = \frac{w _{i} ( Σ w ) _{i}}{σ ( w ) ^{2}} = \frac{cov ( r _{i} , w ^{⊤} r )}{var ( w ^{⊤} r )},

which is exactly the beta of asset $i$ against the portfolio: the $N$ betas sum to 1, and the percentage risk contribution of asset $i$ is its portfolio-beta. This beta-reading is the cleanest way to see why a 60/40 stock/bond portfolio is ~90% equity risk - the equity leg’s beta to the whole is ~0.9.

What does RC actually mean? (Qian 2006). RC is not a gratuitous decomposition. For a portfolio that suffers a loss $L$ , the expected fraction of that loss attributable to asset $i$ is, under normality (Qian App. A):

c_{i} = \frac{E [ w _{i} r _{i} ∣ w ^{⊤} r = L ]}{L} = p_{i} + \frac{D _{i}}{L}, D_{i} = w_{i} μ_{i} - p_{i} μ_{R}, μ_{R} = j \sum w_{j} μ_{j},

i.e. $c_{i} \approx p_{i}$ (the risk budget) plus a correction that vanishes for (a) zero expected returns, (b) mean-variance-optimal portfolios, or (c) large losses $L$ relative to the sub-optimality term. In words: the risk contribution is a good predictor of the loss contribution, and it gets better as the loss gets large - exactly when it matters. Risk teams can therefore budget risk before the loss and be confident it tracks who pays after.

3. Computational Implementation - decompose a real worked universe

On Maillard et al. (2010)‘s four-asset universe (vols 10/20/30/40%, $ρ_{12} = 0.8$ , $ρ_{34} = - 0.5$ ), compute MRC, RC and percentage RC for the naive 1/n portfolio, and verify the Euler decomposition sums exactly to $σ$ .

Read the pct line. The 1/n portfolio holds 25% of capital in every asset, yet asset 4 (the 40%-vol commodity, negatively correlated with asset 3) is responsible for 47.2% of all risk, while asset 1 (10% vol) is only 12.3%. Capital-equal is risk-concentrated - the exact phenomenon risk parity exists to fix. And the bottom line confirms the construction is exact: the four contributions sum to $σ$ to machine precision.

4. Failure Modes & First-Principles Breakdowns

Risk is non-additive in capital but additive in contribution. A common objection (“risks don’t add up”) conflates $w_{i} σ_{i}$ with $R C_{i}$ . Standard deviation is not additive in weights - but RC is defined precisely so that the decomposition holds. Qian (2006) exists to defuse exactly this confusion.
RC is a volatility statement, not a downside statement. The RC decomposition uses $σ$ ; when returns are far from normal, the small-loss contributions are not well captured - only tails/large losses are (that is where VaR/CVaR contributions from Quantitative Risk step in, and where Qian shows Cornish–Fisher VaR contributions do better).
Betas are interior, not exogenous. Percentage RC = portfolio-beta, but that beta is a function of $w$ itself. You cannot read an asset’s “fair” risk share off the standalone covariance matrix; the decomposition depends on the very portfolio it describes.

5. References

Qian, Edward: On the Financial Interpretation of Risk Contribution: Risk Budgets Do Add Up, Journal of Investment Management 4(4) (2006)
Maillard, Roncalli & Teïletche (2010)
Hallerbach, Winfried: Decomposing Portfolio Value-at-Risk: A General Analysis, Journal of Risk 5(2) (2003)

6. Connected Graph Bridges

Back: 01 · From Zero · Index Hub
Forward: 03 · Equal Risk Contribution · 04 · Risk Budgeting
Base: Calculus, CVaR)

Kwant Atlas

Explorer