Incremental VaR

A trader wants to add a new position. The compliance question is not "how risky is this trade in isolation," it is "how much does the portfolio's total VaR actually go up if I add it?" These can be very different numbers: a wildly volatile trade that happens to hedge the existing book can lower total VaR, while a seemingly mild trade that is highly correlated with existing exposure can push total VaR up by far more than its standalone size would suggest. Incremental VaR (IVaR) is defined precisely to answer this before-and-after question, exactly, for any proposed trade.

The analogy

Think of a bus that is already full to a certain weight limit. Asking "how heavy is this new passenger" tells you nothing about whether the bus will now be overweight, what matters is how much the bus's total weight changes once they board, and that depends on who is already on the bus. A 200-pound passenger who happens to be carried on their own personal float (offsetting some other cargo) might add less net weight than a 100-pound passenger sitting on top of already-strained axles. Incremental VaR is the "how much does the total actually change" measurement, not the "how heavy is this one item alone" measurement.

Building the idea

The exact definition is simply the before-and-after difference:

$\text{IVaR} = \text{VaR}(P + \text{new trade}) - \text{VaR}(P).$

In words: recompute total portfolio VaR with the new trade included, subtract the VaR you had before, and whatever is left over is the true marginal risk cost (or benefit) of the trade. This exact calculation requires rebuilding the full portfolio VaR twice, which is expensive for a large book, so in practice IVaR is usually approximated using the same marginal-sensitivity machinery as Component VaR:

$\text{IVaR}_i \approx \Delta w_i \cdot \frac{\partial \text{VaR}_p}{\partial w_i} = \Delta w_i \times z_\alpha \times \frac{(\Sigma w)_i}{\sigma_p},$

where $\Delta w_i$ is the size of the new (or changed) position. In words: take the trade's size, multiply by how sensitive total portfolio VaR is to a small change in that position (its covariance with the rest of the book, rescaled), and that product approximates the true incremental effect. This approximation is a linearization, it is exact only for an infinitesimally small trade; for a large trade relative to the existing book, the true before-and-after difference can deviate noticeably from this linear estimate, because $(\Sigma w)_i$ itself shifts once the trade materially changes the weight vector.

Drag the weight slider here. The change in the portfolio's total risk as you move the slider a small step is exactly the incremental-VaR idea applied to a two-asset book: not the new asset's standalone risk, but how much the curve's height actually shifts as you nudge the mix.

Worked example 1: a diversifying trade

Reuse the two-asset portfolio from Component VaR: $60 in Asset 1, $40 in Asset 2, $\Sigma = \begin{pmatrix} 0.04 & 0.01 \\ 0.01 & 0.09 \end{pmatrix}$, giving $\sigma_p \approx 0.1833$ and \text{VaR}_p \approx \30.25$at 95$100.

A trader proposes adding $5 of a third asset with variance 0.05 and covariance $-0.015$ with Asset 1 and $-0.005$ with Asset 2 (a genuine diversifier). Its marginal covariance-with-portfolio term (using the existing weights as an approximation) is roughly $(\Sigma w)_3 \approx 0.6\times(-0.015) + 0.4\times(-0.005) = -0.009 - 0.002 = -0.011$.

Approximate incremental VaR: $\Delta w_3 \times z_\alpha \times (\Sigma w)_3 / \sigma_p = 0.05 \times 1.65 \times (-0.011)/0.1833 \approx -0.00496$, i.e. roughly -\0.50. Adding this \5 position is estimated to reduce total portfolio VaR by about half a dollar, even though the position itself is not tiny, because it is negatively correlated with the existing book.

Worked example 2: a concentrating trade

Now consider adding $5 more to Asset 2 itself instead (increasing its weight, not a new asset). Using $(\Sigma w)_2 = 0.042$ from the earlier calculation: incremental VaR $\approx \Delta w_2 \times z_\alpha \times (\Sigma w)_2/\sigma_p = 0.05 \times 1.65 \times 0.042/0.1833 \approx 0.0189$, about +\1.89$.

Compare the two trades: both are the same dollar size ($5), but one reduces total VaR by roughly $0.50 and the other increases it by roughly $1.89, a swing of well over $2 between two trades of identical size. Sizing limits based on notional alone would treat them identically; incremental VaR reveals they have essentially opposite effects on the book's real risk.

What this means in practice

Pre-trade risk checks at banks and multi-strategy funds run this calculation before a large order is approved: does this trade push the book past its VaR limit, net of everything already on it. It also prices internal capital charges, a desk adding a diversifying trade can face a lower internal cost of risk capital than its gross notional would suggest, precisely the incentive risk management wants to create.