Component VaR

A portfolio's total Value at Risk (VaR) comes back as a single number, say $2 million. A risk manager's next question is always: which positions are actually driving that number? You cannot answer this by computing each position's standalone VaR and adding them up, standalone VaR ignores diversification entirely and the individual numbers will sum to far more than the $2 million total. Component VaR solves this by splitting total portfolio VaR into pieces, one per position, that account for each position's diversification benefit and that add up exactly to the whole.

The analogy

Think of splitting a restaurant bill among friends who shared several dishes. You cannot just ask "how much would each person's meal have cost alone at a different restaurant," that number ignores that dishes were shared and discounts applied. You want to know each diner's true, fair share of this bill, in a way that the shares add up exactly to the total on the receipt. Component VaR is that fair-share calculation applied to portfolio risk: not "how risky is this position on its own," but "how much of the portfolio's actual, diversified risk belongs to this position."

Building the formula

Total portfolio VaR can be written as $\text{VaR}_p = z_\alpha \sigma_p$, where $\sigma_p$ is portfolio volatility and $z_\alpha$ is the confidence-level multiplier (e.g. 1.65 for 95%). The key building block is each asset's marginal contribution to portfolio volatility,

$\frac{\partial \sigma_p}{\partial w_i} = \frac{(\Sigma w)_i}{\sigma_p},$

where $w$ is the weight vector and $\Sigma$ the covariance matrix. In words: this is "how much would total portfolio volatility change if you nudged position $i$'s weight up a tiny bit," and it depends not just on asset $i$'s own volatility but on how it covaries with everything else in the book, $(\Sigma w)_i$ is asset $i$'s covariance with the whole portfolio, not with itself alone.

Component VaR multiplies that marginal sensitivity by the position's actual size, and it uses a mathematical identity (Euler's theorem, since VaR here is a function that scales linearly with the whole portfolio's size) to guarantee the pieces sum exactly to the total:

$\text{CVaR}_i = w_i \cdot \frac{\partial \text{VaR}_p}{\partial w_i} = \text{VaR}_p \times \frac{w_i (\Sigma w)_i}{\sigma_p^2}, \qquad \sum_i \text{CVaR}_i = \text{VaR}_p.$

In words: each position's component is its weight times its covariance with the whole portfolio, rescaled so all the pieces add back up to the total VaR dollar figure exactly. A position can even have a negative component VaR, meaning it is a genuine hedge that reduces total portfolio risk below what the other positions alone would produce.

Drag the correlation slider here. As correlation falls, the frontier bows further left, meaning total portfolio risk for a given mix drops below what either asset's standalone risk would suggest. Component VaR is the accounting exercise that assigns each asset its fair share of that leftward bow, not its share of the two standalone risk numbers you started with.

Worked example 1: two positions

A portfolio holds $60 in Asset 1 and $40 in Asset 2 (weights $w = (0.6, 0.4)$). Their covariance matrix (annualized) is $\Sigma = \begin{pmatrix} 0.04 & 0.01 \\ 0.01 & 0.09 \end{pmatrix}$.

$\Sigma w$: first row $0.04 \times 0.6 + 0.01 \times 0.4 = 0.024 + 0.004 = 0.028$; second row $0.01 \times 0.6 + 0.09 \times 0.4 = 0.006 + 0.036 = 0.042$.

Portfolio variance: $w^\top \Sigma w = 0.6 \times 0.028 + 0.4 \times 0.042 = 0.0168 + 0.0168 = 0.0336$, so $\sigma_p = \sqrt{0.0336} \approx 0.1833$ (18.33% annual vol).

At 95% VaR ($z=1.65$) on a $100 portfolio: $\text{VaR}_p = 1.65 \times 0.1833 \approx 0.3025$, i.e. $30.25.

Component shares: Asset 1's share of variance is $w_1(\Sigma w)_1/\sigma_p^2 = 0.6 \times 0.028/0.0336 = 0.5$. Asset 2's is $w_2(\Sigma w)_2/\sigma_p^2 = 0.4 \times 0.042/0.0336 = 0.5$ as well. So \text{CVaR}_1 = 0.5 \times \30.25 = $15.13$and$\text{CVaR}_2 = $15.13, summing exactly to \30.25, even though the positions are unequal in size ($60 vs $40), because Asset 2's higher own-volatility (9% variance vs 4%) offsets its smaller weight.

Worked example 2: a hedge with negative component VaR

Add a third, small short position in an asset negatively correlated with Asset 1, with weight $w_3 = -0.10$ and $(\Sigma w)_3 = -0.02$ once its covariance with the rest of the book is included. Its component is proportional to $w_3(\Sigma w)_3 = (-0.10)(-0.02) = 0.002$, positive, since two negatives multiply to a positive. Counter-intuitively, this short "hedge" adds a small slice to total VaR here rather than reducing it, because component VaR depends on the product of a position's own weight and its covariance with the whole book, not on whether it looks like a hedge in isolation.

What this means in practice

Component VaR is the standard tool for risk budgeting across a multi-strategy book: a risk manager sets a dollar VaR limit per desk and monitors component VaR against it, catching a position whose contribution has grown even if its own size hasn't changed, because its correlation with the rest of the book shifted. It also feeds Incremental VaR, which asks what happens if you add or remove a position entirely.