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Show how VaR may be computed and used when measured against a performance benchmark.

FRM® Part II · Market Risk Measurement and Management · VaR Mapping · FRM2.MR.5.f

The idea

Relative VaR measures the risk of deviating from a performance benchmark rather than the risk of losing money. With portfolio weights wP and benchmark weights wB, the active exposure vector is x = wP − wB. Tracking error volatility uses the covariance matrix Σ of the risk factors, written TEV = √(x′Σx). Let z be the confidence multiplier, V the portfolio value and T the horizon in units of the TEV period. Relative VaR is scaled to the horizon by the square root of time, RelVaR = z × TEV × V × √T. A portfolio that replicates the benchmark has zero relative VaR even though its absolute VaR is large, while a cash portfolio held against an equity benchmark has low absolute VaR but a large relative VaR. For a portfolio and benchmark of equal value, with return correlation ρ, relative VaR is RelVaR = √(VaRP² + VaRB² − 2 × ρ × VaRP × VaRB). Common errors are subtracting the two VaRs or ignoring correlation. Exam point: relative VaR is the natural basis for active-risk limits.

Formulas

Tracking error volatility

TEV = √(x′ × Σ × x)

x
the active weight vector, portfolio weights minus benchmark weights, x′ its transpose
Σ
the covariance matrix of the risk factors

Uses active weights, not portfolio weights. For a +10% equity and −10% bond tilt the covariance term is subtracted.

Relative VaR against a benchmark

RelVaR = z × TEV × V × √T

z
the standard normal quantile
TEV
the tracking error volatility per period
V
the portfolio value
T
the horizon in units of the TEV period

Square-root-of-time scaling: a quarter is √0.25 of an annual TEV. Zero for a portfolio that replicates its benchmark.

Relative VaR from two absolute VaRs

RelVaR = √(VaRP² + VaRB² − 2 × ρ × VaRP × VaRB)

VaRP
the portfolio's absolute VaR
VaRB
the benchmark's absolute VaR
ρ
the correlation between portfolio and benchmark returns

Portfolio and benchmark of equal value. The cross term is subtracted; simply subtracting the VaRs holds only at ρ = 1.

A practice question

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

Relative VaR measured against a performance benchmark is the VaR of:

  1. the benchmark's return alone.Why not: Measures the benchmark's own risk rather than the deviation from it.
  2. the portfolio's return in excess of the risk-free rate.Why not: Describes the risk of an excess return over cash, not over the benchmark.
  3. the difference between the portfolio's return and the benchmark's return. - correct
  4. the portfolio's return after subtracting its own expected return.Why not: Describes a mean-adjusted absolute VaR; no benchmark is involved.

Relative (tracking-error) VaR is computed on the active position, portfolio minus benchmark. It is the loss relative to the benchmark that will not be exceeded with the stated confidence over the horizon.

A harder one

A USD 500 million portfolio is invested 70 percent in equities and 30 percent in bonds against a 60/40 benchmark. Annual volatilities are 16 percent for equities and 6 percent for bonds, with a correlation of 0.2. The one-month 95 percent relative VaR, using the square root of 12, is closest to:

  1. USD 4.06 millionWhy not: Sets the equity-bond correlation to zero.
  2. USD 4.32 millionWhy not: Adds the covariance term, +2 x rho x sigma(E) x sigma(B), instead of subtracting it for the +10/-10 active positions.
  3. USD 27.76 millionWhy not: Computes the absolute VaR of the portfolio rather than the VaR of the active position.
  4. USD 3.78 million - correct

Active weights are +10% equities and -10% bonds. Tracking-error variance = 0.10² x (0.16² + 0.06² - 2 x 0.2 x 0.16 x 0.06) = 0.000254, so tracking error is 1.59% a year. Relative VaR = 1.645 x 1.59%/sqrt(12) x 500m = 3.78m.

The hardest tier

A USD 400 million portfolio has a beta of 1.2 to its benchmark index, whose annual volatility is 15 percent, plus an uncorrelated residual tracking risk of 2 percent a year. The one-month 95 percent relative VaR, using the square root of 12, is closest to:

  1. USD 5.70 millionWhy not: Counts only the beta mismatch and omits the residual tracking risk.
  2. USD 3.80 millionWhy not: Counts only the residual tracking risk and omits the beta mismatch.
  3. USD 6.85 million - correct
  4. USD 9.50 millionWhy not: Adds the beta-mismatch risk and the residual risk linearly rather than as a square root of summed squares.

Active beta = 1.2 - 1.0 = 0.2, so the beta-driven tracking volatility is 0.2 x 15% = 3%. Combine with the 2% residual: sqrt(3%² + 2%²) = 3.61% a year. Relative VaR = 1.645 x 3.61%/sqrt(12) x 400m = 6.85m.

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