What is incremental VaR, and how does it differ from marginal VaR when sizing a discrete new trade?

Question

AcadiFi · Accepted Answer

Incremental VaR (IVaR) measures the actual change in portfolio VaR when a specific, discrete trade is added. Unlike marginal VaR (which is a derivative and works only for infinitesimally small changes), incremental VaR captures the full nonlinear impact of a realistic position change.

**Key Distinction:**

- Marginal VaR: dVaR/dw (slope of VaR at current weights, local approximation)
- Incremental VaR: VaR(portfolio + trade) - VaR(portfolio) (exact difference, global measure)

For small trades, MVaR and IVaR converge. For large allocations, they diverge because the correlation structure changes as weights shift.

**Formula:**

IVaR = VaR(P + a) - VaR(P)

where P is the current portfolio and a is the proposed trade.

For parametric normal VaR, the new portfolio variance is:

sigma_new^2 = w'^T x Sigma x w'

where w' is the new weight vector after the trade.

**Worked Example:**

Fairhaven Partners has a two-asset portfolio worth $80M:

| Asset | Weight | sigma (annual) |
|---|---|---|
| Whitmore Equities | 60% | 20% |
| Dunmore Bonds | 40% | 6% |

Correlation: rho = 0.25. Confidence: 99% (z = 2.326).

Current portfolio sigma:
sigma_P = sqrt(0.6^2 x 0.20^2 + 0.4^2 x 0.06^2 + 2 x 0.6 x 0.4 x 0.25 x 0.20 x 0.06)
sigma_P = sqrt(0.0144 + 0.000576 + 0.001440) = sqrt(0.016416) = **12.81%**

Current VaR = 2.326 x 0.1281 x $80M = **$23.84M**

Propose adding $10M of Larkfield Real Estate (sigma = 15%, rho with equities = 0.40, rho with bonds = 0.10). New portfolio = $90M.

New weights: Whitmore = 53.3%, Dunmore = 35.6%, Larkfield = 11.1%

New portfolio sigma = sqrt(0.533^2 x 0.04 + 0.356^2 x 0.0036 + 0.111^2 x 0.0225 + 2 x 0.533 x 0.356 x 0.25 x 0.20 x 0.06 + 2 x 0.533 x 0.111 x 0.40 x 0.20 x 0.15 + 2 x 0.356 x 0.111 x 0.10 x 0.06 x 0.15)

sigma_new = sqrt(0.01136 + 0.000456 + 0.000277 + 0.001138 + 0.001421 + 0.0000711) = sqrt(0.014723) = **12.13%**

New VaR = 2.326 x 0.1213 x $90M = **$25.39M**

Incremental VaR = $25.39M - $23.84M = **$1.55M**

Marginal VaR approximation would have predicted: MVaR x $10M, which could differ meaningfully from $1.55M for this 12.5% portfolio weight change.

**When to Use Each:**
- Marginal VaR: quick screening of many candidate trades
- Incremental VaR: final decision on a specific trade with a known notional
- Component VaR: decomposing existing portfolio risk for reporting

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What is incremental VaR, and how does it differ from marginal VaR when sizing a discrete new trade?

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