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Specify, compute and set against each other six portfolio VaR figures: individual, undiversified, diversified, incremental, marginal and component.

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

The idea

Six VaR figures describe one portfolio, and each answers a different question. Individual VaR is a position's own VaR, z × σ × position value, ignoring everything else held. Undiversified VaR is the sum of the individual VaRs and assumes every correlation is +1. Diversified VaR uses the actual correlations, under normal returns, and is never above the undiversified figure. Let x be the vector of position values and Σ the covariance matrix of returns. Then the diversified figure is DVaR = z × √(x′Σx). Incremental VaR is the change in diversified VaR from adding or dropping a whole position. Marginal VaR is the change in portfolio VaR per extra unit of one position. With σp the portfolio volatility, the marginal VaR of position i is MVaR = z × cov(Ri, Rp) ÷ σp. It also equals portfolio VaR per unit of value times the position's beta. Component VaR is marginal VaR × position size, and the components add up exactly to diversified VaR. A hedge can have a negative component. The usual mistake is expecting incremental VaRs to sum to the total, which they generally do not.

Formulas

Parametric VaR of a normal portfolio

VaR = z × σ × V

z
the standard normal quantile for the confidence level, 2.326 at 99%
σ
the standard deviation of returns over the horizon
V
the portfolio value

This is individual VaR when applied to one position on its own; the sum of individual VaRs is the undiversified VaR, which assumes every correlation is +1.

Marginal VaR of a position

marginal VaR = z × cov(Ri, Rp) ÷ σp

z
the standard normal quantile
cov(Ri, Rp)
the covariance of the position's return with the portfolio's
σp
the portfolio's volatility

A per-unit, local measure under normal returns. Negative for a hedge.

Component VaR

component VaR = marginal VaR × position

marginal VaR
the change in portfolio VaR per unit of the position
position
the money held in the position

Components add up exactly to diversified VaR; incremental VaRs generally do not.

Incremental VaR of a position

incremental VaR = VaR with − VaR without

VaR with
diversified VaR of the portfolio including the whole position
VaR without
diversified VaR of the portfolio without it

A full revaluation for adding or dropping a whole position, not a per-unit sensitivity.

A practice question

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

A risk manager adds up the stand-alone VaRs of every position in a portfolio. Which VaR measure does the total represent?

  1. Undiversified VaR, which assumes all correlations are +1 - correct
  2. Diversified VaR, which reflects the actual correlationsWhy not: Diversified VaR uses the actual correlations and is normally lower than the sum of the stand-alone figures.
  3. Incremental VaR, the change from adding the last positionWhy not: Incremental VaR measures the change in portfolio VaR from adding or removing a position, not a sum of stand-alone VaRs.
  4. Component VaR, the sum of which is the portfolio VaRWhy not: Component VaRs add up to the diversified VaR, not to the sum of the stand-alone VaRs.

The sum of individual (stand-alone) VaRs is the undiversified VaR. It is only right if all correlations are +1, so it is an upper bound on diversified VaR.

A harder one

A portfolio holds USD 6 million in asset A (daily volatility 1.5%) and USD 4 million in asset B (daily volatility 2.0%). The correlation of daily returns is 0.3. Use z = 1.645 for the 95% one-day VaR. The diversified VaR is closest to:

  1. USD 0.280 millionWhy not: Sum of the individual VaRs (0.148 + 0.132), the undiversified VaR, which assumes correlation +1.
  2. USD 0.198 millionWhy not: Sets the correlation to zero: sqrt(0.148² + 0.132²). The stated correlation of 0.3 adds to risk.
  3. USD 0.226 million - correct
  4. USD 0.166 millionWhy not: Subtracts the correlation term (-2 x 0.3 x ...) instead of adding it, which treats the assets as negatively related.

Individual VaRs: 1.645 x 0.015 x 6 = 0.148 and 1.645 x 0.020 x 4 = 0.132. Diversified VaR = sqrt(0.148² + 0.132² + 2 x 0.3 x 0.148 x 0.132) = sqrt(0.0509) = 0.226 million.

The hardest tier

A fund has individual VaRs of USD 4.0 million for position A and USD 3.0 million for position B. The correlation between the two positions' returns is 0.5. The component VaR of A is closest to:

  1. USD 3.48 millionWhy not: Allocates diversified VaR in proportion to individual VaRs (6.08 x 4/7), ignoring that A's correlation with B changes its contribution.
  2. USD 3.62 million - correct
  3. USD 3.08 millionWhy not: Incremental VaR of A: portfolio VaR 6.08 minus B's individual VaR of 3.0. This is not the component VaR.
  4. USD 4.00 millionWhy not: A's individual VaR, which is the stand-alone figure with no diversification.

Portfolio VaR = sqrt(4² + 3² + 2 x 0.5 x 4 x 3) = sqrt(37) = 6.083. Component VaR of A = (4² + 0.5 x 4 x 3) / 6.083 = 22 / 6.083 = 3.62. B's is 2.47 and they sum to 6.08.

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