Euler Allocation of Portfolio Risk

A portfolio's total risk is a single number, say a $4.2m 99% VaR. A risk manager needs to know how much of that $4.2m is "caused by" each of the twenty positions in the book, so limits can be set and the biggest contributor can be trimmed. The naive approach, compute each position's standalone VaR and rank by that, is wrong, because standalone risk ignores how a position interacts with everything else: a position can have huge standalone risk yet actually hedge the book, or small standalone risk yet be the single biggest driver of the total because of how it correlates with everything else. You need an allocation rule that respects the whole portfolio, not each piece in isolation, and that adds up exactly to the total.

Splitting a restaurant bill by what you actually ordered

Imagine four friends share a large group order and the total bill needs splitting, but not evenly, fairly, by who ordered what. The naive approach, look at what each person would have paid if they'd eaten alone at the restaurant's normal prices, gets the split wrong, because the group discount and shared appetizers change the true marginal cost each diner added. The right way is to ask, for each diner, "how much did the total bill go up because this specific person was at the table, given everyone else's order stays fixed," then use a mathematical fact about how the total scales with each person's order to guarantee those marginal contributions add up to exactly the actual bill, no leftover, no double-counting. Euler allocation is exactly this idea applied to portfolio risk: each position's contribution is its true marginal effect on the whole, and the pieces are guaranteed to sum to the total by construction.

The mathematical fact that makes it work

Portfolio VaR, viewed as a function of the position sizes $w_1, \ldots, w_n$ (dollar or weight amounts), has a special property: if you scale every position by the same factor $t$, portfolio VaR scales by exactly $t$ too. This is called positive homogeneity of degree one, and in words it just says doubling every position in the book doubles the risk, nothing subtler than that. Euler's theorem says that any function with this scaling property can always be written as

$\text{VaR}(w) = \sum_{i=1}^n w_i \frac{\partial \text{VaR}}{\partial w_i}.$

In words: the total is exactly the sum, over every position, of that position's size times how fast the total risk changes as you nudge that position. The term $\dfrac{\partial \text{VaR}}{\partial w_i}$ is called the position's marginal risk contribution, how much an extra dollar in position $i$ adds to total VaR holding everything else fixed, and $w_i$ times that marginal contribution is the Euler risk contribution of position $i$. Unlike an ad hoc allocation rule, this one is guaranteed by the mathematics to add up to the whole, no residual, no arbitrary fudge factor.

For a portfolio VaR built from a normal approximation, $\text{VaR} = z_\alpha \sigma_P = z_\alpha \sqrt{w^\top \Sigma w}$ where $\Sigma$ is the covariance matrix of the underlying assets, the marginal contribution has a clean closed form:

$\text{Contribution}_i = w_i \cdot z_\alpha \frac{(\Sigma w)_i}{\sigma_P}.$

In words: each position's contribution is proportional to its size, multiplied by its covariance with the whole portfolio (not its own standalone variance), scaled to match the units of VaR. A position that is small but highly correlated with the rest of the book can contribute far more risk than a large position that happens to be a natural hedge.

standalone VaRs (do not sum meaningfully) A+B+C+D+E alone: 190% ignores hedges and correlation

Worked example: two positions, by hand

A book holds Position A ($60m) and Position B ($40m). Daily volatilities: $\sigma_A = 1.5\%$, $\sigma_B = 2.0\%$, correlation $\rho = 0.4$. Portfolio variance:

$\sigma_P^2 = (60)^2(0.015)^2 + (40)^2(0.02)^2 + 2(60)(40)(0.4)(0.015)(0.02) = 0.81 + 0.64 + 0.576 = 2.026,$

(all in $m² units), so $\sigma_P = \sqrt{2.026} = 1.423$m. At 99%, $z = 2.326$, total VaR $= 2.326 \times 1.423 =$ $3.31m.

Marginal contributions use $(\Sigma w)_i$: for A, $(\Sigma w)_A = 60(0.015)^2 + 40(0.4)(0.015)(0.02) = 0.0135 + 0.0048 = 0.0183$ (per $m², times weight already folded in as dollar exposure above; working directly in dollar terms), giving Euler contribution of A as $2.326 \times 0.0183 \times 60 / 1.423 \approx$ $1.80m. By the guaranteed-sum property, B's contribution is $3.31 - 1.80 =$ $1.51m.

Check the intuition: A is 60% of the dollar exposure, but only about 54% of the risk ($1.80/3.31$), because B's higher standalone volatility pulls more than its dollar share of the total, even after accounting for the diversifying correlation.

Worked example: a hedge shows up as negative

Add Position C, a $20m short that has $-0.7$ correlation with A and negligible correlation with B, $\sigma_C = 1.8\%$. Recomputing the full covariance sum (steps omitted for space) typically shows C's Euler contribution as negative, say -$0.35m, meaning C is reducing total portfolio VaR even though its standalone VaR (about $2.326 \times 0.018 \times 20 =$ $0.84m) looks like meaningful risk on its own. A limit system built on standalone VaR would flag C as a risk to cut; a limit system built on Euler contributions correctly identifies it as a hedge worth keeping, or even growing.

What this means in practice

• Risk budgeting uses Euler contributions directly: allocate a target share of total risk to each strategy or desk, and Euler contribution is the only decomposition consistent with that budget actually summing to 100%. See Risk Budgeting.
• Hot spot identification. Ranking positions by Euler contribution, not size or standalone VaR, is the standard way trading desks find the position actually driving the number that matters.
• Works for any homogeneous risk measure, not just normal-VaR: expected shortfall admits an analogous Euler decomposition (the average of the worst-case marginal contributions), which is one more reason ES is preferred to VaR in modern risk frameworks; see Coherent Risk Measures.
• Requires the full covariance matrix, so its quality is only as good as the covariance estimate; see Covariance Matrix Estimation and Estimating the Factor Covariance Matrix for how that estimate is actually built at scale.

Practice

1. Three positions have standalone VaRs of $2m, $3m, and $1m. Explain why their Euler contributions cannot be computed from these three numbers alone.
2. In the two-position worked example, what happens to A's Euler contribution as $\rho \to -1$? Sketch the intuition without recomputing the full formula.
3. A trader argues a position with negative Euler contribution should have no risk limit at all. What is wrong with that reasoning?