Use marginal VaR when making decisions about managing a portfolio.

FRM® Part II · Risk Management and Investment Management · Portfolio Risk: Analytical Methods · FRM2.IM.5.c

The idea

Marginal VaR tells a manager how portfolio VaR responds to a small change in one position. Let σp be the portfolio's volatility. Then the marginal VaR per unit of position i is MVaR = z × cov(Ri, Rp) ÷ σp. This also equals portfolio VaR divided by portfolio value, times the position's beta to the portfolio. The estimated change in VaR from a trade is marginal VaR × the amount traded. It is a linear approximation, good for small trades only, because marginal VaRs change once the mix changes. Rank positions by marginal VaR, not by individual or component VaR. The position with the highest marginal VaR cuts risk most per unit sold, and a negative marginal VaR marks a hedge, whose increase lowers VaR. A large position with a low marginal VaR can be a poor place to cut. Exam point: use marginal VaR for the direction and rough size of a trade, and recompute for big ones.

Formulas

Marginal VaR from beta

marginal VaR = (VaRp ÷ Wp) × βi

VaRp: the portfolio's VaR
Wp: the portfolio's value
βi: the position's beta to the portfolio

Per unit of position. Rank positions by it to see which cut reduces VaR most per unit sold.

Change in VaR from a trade

ΔVaR = marginal VaR × Δposition

marginal VaR: the position's marginal VaR per unit
Δposition: the amount bought (positive) or sold (negative)

A linear approximation for small trades; recompute VaR for large ones because marginal VaRs change with the mix.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which description best fits the marginal VaR of a position?

The change in portfolio VaR if the entire position is removedWhy not: This is incremental VaR, which measures a full add or removal rather than a small step.
The VaR of the position taken on its ownWhy not: This is individual VaR, which ignores the rest of the portfolio.
The change in portfolio VaR for a small change in the position, per unit of the position - correct
The share of portfolio VaR attributed to the positionWhy not: This is component VaR, which equals marginal VaR times the position size.

Marginal VaR is the derivative of portfolio VaR with respect to the position: z x cov(R_i, R_p) / sigma_p. It is a local, per-unit measure.

A harder one

A portfolio of USD 90 million has a VaR of USD 1.86 million. Marginal VaRs per USD of position are 0.030 for equity (USD 40 million held), 0.018 for credit (USD 30 million) and 0.006 for rates (USD 20 million). Selling USD 5 million of equity is estimated to change VaR, in USD million, by about:

-0.150 - correct
-0.090Why not: Uses the credit marginal VaR (0.018), not the equity position's 0.030.
-0.103Why not: Uses average portfolio VaR per USD (1.86 / 90) instead of the equity position's own marginal VaR.
-1.200Why not: Uses the whole equity component VaR (40 x 0.030), which is the contribution of the entire position, not of a USD 5 million sale.

Change in VaR is about marginal VaR x change in position = 0.030 x (-5) = -0.150 million. Component VaR (1.20 for equity) is the total contribution of the position, not a sensitivity to a trade.

The hardest tier

A portfolio of USD 120 million has a 1-day VaR of USD 4.2 million. A position has a beta of 1.4 to the portfolio. Increasing that position by USD 3 million is expected to change portfolio VaR by about:

+USD 0.105 millionWhy not: Uses the portfolio's average VaR per USD (0.035) without scaling by the beta.
+USD 0.206 millionWhy not: Multiplies by beta² (1.96) instead of beta.
+USD 0.075 millionWhy not: Divides by the beta (0.035 / 1.4) rather than multiplying.
+USD 0.147 million - correct

Marginal VaR = (VaR_p / W_p) x beta = (4.2 / 120) x 1.4 = 0.049 per USD. Change = 0.049 x 3 = +0.147 million.

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