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Value at Risk: Methodsintermediate

VaR With a Nonzero Mean (Drift)

Over short horizons the expected return, or drift μ\mu, is tiny and usually dropped. Over longer horizons it is not, and ignoring it overstates VaR. The mean-adjusted parametric VaR subtracts the expected gain before measuring the tail: VaRc=−(μ+zc σ) V\mathrm{VaR}_c = -(\mu + z_c\,\sigma)\,V. Because a positive μ\mu shifts the whole distribution rightward, it lowers the loss threshold. There are two conventions: absolute VaR is measured from zero (loss in dollars), while relative VaR is measured from the expected value μV\mu V and strips the drift out.

Why it matters

A drifting distribution is a bell curve sliding to the right at rate μ\mu. The loss tail is the same shape but starts from a higher expected level, so the dollar loss that marks your quantile is smaller. Over a day the slide is negligible and you can ignore it. Over a year it matters, and pretending μ=0\mu = 0 makes the position look riskier than it is. Relative VaR sidesteps the debate by measuring losses relative to where you expected to be, not relative to zero.

Formulas

Mean-adjusted (absolute) parametric VaR
VaRc=−(μ+zc σ) V\mathrm{VaR}_c = -(\mu + z_c\,\sigma)\,V
With zc<0z_c < 0, a positive drift μ\mu raises the bracket toward zero and so lowers VaR. Setting μ=0\mu = 0 recovers the short-horizon form −zc σ V-z_c\,\sigma\,V.
Relative VaR (loss from the mean)
VaRcrel=−zc σ V\mathrm{VaR}^{\text{rel}}_c = -z_c\,\sigma\,V
Relative VaR measures the shortfall below the expected value μV\mu V, so the drift cancels. Absolute and relative VaR differ by exactly μV\mu V.

Worked examples

Scenario

A US$10M position has an expected annual return μ=8%\mu = 8\% and annual volatility σ=20%\sigma = 20\%. Find the one-year 95% absolute VaR and compare it with the relative VaR.

Solution

Absolute VaR uses VaR=−(μ+zc σ) V\mathrm{VaR} = -(\mu + z_c\,\sigma)\,V. The bracket is $0.08 + (-1.65)(0.20) = -0.25,soabsoluteVaRis0.25timesUS, so absolute VaR is 0.25 times US10M, which is US$2.5M. Relative VaR uses VaRrel=−zc σ V\mathrm{VaR}^{\text{rel}} = -z_c\,\sigma\,V, giving 1.65 times 0.20 times US$10M, which is US$3.3M. The two differ by exactly the expected profit μV\mu V, here 0.08 times US$10M, which is US$0.8M, the gain the drift contributes over the year.

Common mistakes

  • ✗The mean is always safe to ignore. Over short horizons it is negligible, but over months or years a nonzero drift meaningfully lowers VaR; dropping it overstates risk.
  • ✗Absolute and relative VaR are the same number. They differ by the expected profit μV\mu V; relative VaR measures loss from the mean, absolute VaR measures loss from zero.
  • ✗A positive expected return raises VaR. A positive drift shifts the distribution toward gains, which lowers the loss threshold, so it reduces VaR.
  • ✗Including the mean changes which method you use. The drift adjustment applies within the same parametric framework; it shifts the location of the distribution, not its shape.

Revision bullets

  • •Drift μ\mu is negligible over short horizons, material over long ones
  • •Mean-adjusted absolute VaR: −(μ+zc σ) V-(\mu + z_c\,\sigma)\,V
  • •Positive drift lowers VaR by shifting the distribution right
  • •Relative VaR measures loss from the mean and cancels μ\mu
  • •Absolute and relative VaR differ by exactly μV\mu V

Quick check

Including a positive expected return (drift) in parametric VaR over a long horizon will

Absolute VaR and relative VaR for the same position differ by

Connected topics

VaR DefinitionParametric VaRVaR Time ScalingValuation Methods

Sources

  1. Jorion (2007), Ch. 5
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007. ISBN 978-0-07-146495-6.
    Distinguishes absolute VaR (from zero) and relative VaR (from the mean) and the role of drift over long horizons.
  2. Roncalli (2020), Ch. 2
    Roncalli, T. Handbook of Financial Risk Management. Chapman & Hall/CRC, 2020. ISBN 978-1-138-50187-4.
    Treats the Gaussian VaR with and without the expected-return term.
How to cite this page
Dr. Phil's Quant Lab. (2026). VaR With a Nonzero Mean (Drift). Derivatives Atlas. https://phucnguyenvan.com/concept/frm-var-with-mean
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