Aggregating Risk Across Books and Desks

A bank has an equities desk, a rates desk, and a credit desk, each with its own VaR limit, each managed by a different head who has never met the others' books. The chief risk officer needs one number: total firm-wide VaR. Simply adding the three desks' individual VaR figures overstates the truth, because the desks are not perfectly correlated and some of their risks cancel. Simply pooling every position into one giant covariance matrix and computing a single VaR gets a more accurate number, but destroys the ability to say which desk is responsible for how much of it, and desks are run, limited, and compensated separately for good reasons. Real risk aggregation has to solve both problems at once.

The analogy

Picture three separate households each keeping their own grocery budget, and a landlord who wants to know the building's total food spending risk (how much it could swing month to month). Simply summing each household's own volatility overstates the building's swing, because one household stocking up before a storm and another cutting back barely move the building total; some of that variation cancels across households. But the landlord still needs to know which household is actually driving the building-level swings, they cannot manage risk by managing an anonymous total. Aggregation has to produce one honest number for the building and a fair breakdown by household.

Building the two-step approach

Step one, the honest total. Build a single covariance matrix $\Sigma$ across all positions firm-wide, spanning every desk, and compute one enterprise-level VaR the normal way:

$\text{VaR}_{\text{firm}} = z_\alpha \sqrt{w^\top \Sigma w}.$

This is strictly less than or equal to the naive sum of each desk's standalone VaR, $\sum_k \text{VaR}_k$, with equality only if every desk's returns are perfectly correlated with every other desk's, which essentially never happens. The gap between the naive sum and the true firm-wide figure is the firm's diversification benefit:

$\text{Diversification Benefit} = \sum_k \text{VaR}_k - \text{VaR}_{\text{firm}}.$

In words: how much risk cancels out across desks that would otherwise be double-counted if you just added up each desk's standalone number.

Step two, fair attribution. Use Component VaR logic at the desk level rather than the position level: each desk's component VaR is its weighted covariance with the whole firm, rescaled so the desk-level components sum exactly back to $\text{VaR}_{\text{firm}}$. In words: instead of "how risky is this desk on its own," ask "how much of the firm's actual, diversified risk belongs to this desk," using the same fair-share machinery used for individual positions.

Picture each point here as a day's joint P&L outcome for two desks instead of two assets. A low correlation means the desks' good and bad days do not line up, so a naive sum of their standalone VaRs overstates the risk of holding both desks together.

Worked example 1: three desks, naive sum versus true total

Equities desk standalone VaR: $5m. Rates desk: $4m. Credit desk: $3m. Naive sum: $12m.

Suppose pairwise correlations between the desks' daily P&L are: equities–rates $0.3$, equities–credit $0.4$, rates–credit $0.2$. Using the portfolio-variance formula for three components (variance is the sum of each squared term plus twice each pairwise covariance term):

$\text{Var}_{\text{firm}} = 5^2+4^2+3^2 + 2(0.3)(5)(4) + 2(0.4)(5)(3) + 2(0.2)(4)(3) = 25+16+9+12+12+4.8 = 78.8.$

\text{VaR}_{\text{firm}} = \sqrt{78.8} \approx \8.88\text{m}$.

Naive sum was $12m; the true firm-wide figure is $8.88m, a diversification benefit of $3.12m, over a quarter of the naive total simply vanishes once cross-desk correlation is properly accounted for.

Worked example 2: attributing the $8.88m back to desks

Component VaR for a desk is its own standalone VaR times its correlation-weighted exposure to the whole firm, divided by firm volatility. Concretely, first find each desk's total correlation-weighted link to the firm: equities' is $1(5) + 0.3(4) + 0.4(3) = 5 + 1.2 + 1.2 = 7.4$; rates' is $0.3(5) + 1(4) + 0.2(3) = 1.5+4+0.6=6.1$; credit's is $0.4(5)+0.2(4)+1(3)=2+0.8+3=5.8$.

Multiply each by its own standalone VaR and divide by firm VaR ($8.88m): equities = 5 \times 7.4 / 8.88 \approx \4.17\text{m}$; rates$= 4 \times 6.1/8.88 \approx $2.75\text{m}$; credit$= 3 \times 5.8/8.88 \approx $1.96\text{m}$. Check:$4.17+2.75+1.96 = 8.88$, exactly the firm total, as required.

So of the $8.88m firm-wide VaR, equities carries $4.17m (47%), rates $2.75m (31%), and credit $1.96m (22%), not identical to their standalone-VaR shares (42% / 33% / 25%), because equities' higher average correlation to the other two desks means it absorbs a slightly larger share of the diversified total than its raw size alone would suggest.

What this means in practice

Firm-wide risk limits, regulatory capital, and desk-level compensation all depend on getting this decomposition right. Getting it wrong in the naive-sum direction over-reserves capital and penalizes desks that genuinely diversify the firm; pooling everything with no attribution leaves nobody accountable for a risk buildup until it is too large to unwind quietly.