Skip to content
DP
Value at Risk: Methodsintermediate

Converting VaR Across Confidence Levels

Under the parametric (normal) model, two VaR figures for the same position and horizon differ only by their critical values. So you can convert one confidence level to another by rescaling with the ratio of z-values: VaRc2=VaRc1 zc2zc1\mathrm{VaR}_{c_2} = \mathrm{VaR}_{c_1}\,\dfrac{z_{c_2}}{z_{c_1}}. This works only under the normal assumption, where VaR is exactly proportional to zz. For historical-simulation or fat-tailed distributions the relationship between confidence levels is not a fixed z-ratio, and this conversion fails.

Why it matters

In the normal world VaR is just z×σ×Vz \times \sigma \times V, and σV\sigma V is the same regardless of confidence level. So moving from 95% to 99% only swaps one zz for another, and the conversion is a clean ratio. The trick is convenient but fragile: it is a property of the bell curve, not of VaR in general. Real loss distributions have heavier tails, so the jump from 95% to 99% is steeper than the normal z-ratio predicts.

Formulas

Confidence-level conversion (normal model)
VaRc2=VaRc1 zc2zc1\mathrm{VaR}_{c_2} = \mathrm{VaR}_{c_1}\,\frac{z_{c_2}}{z_{c_1}}
Valid only when VaR is proportional to zz, i.e. under normality. The common pair is z95%=−1.65z_{95\%} = -1.65 and z99%=−2.33z_{99\%} = -2.33.

Worked examples

Scenario

A one-day 95% VaR is US$3.3M under the normal model. Convert it to a 99% VaR for the same position and horizon.

Solution

Use the z-ratio VaR99%=VaR95% (z99%/z95%)\mathrm{VaR}_{99\%} = \mathrm{VaR}_{95\%}\,(z_{99\%}/z_{95\%}). The ratio is z99%/z95%=2.33/1.65≈1.412z_{99\%}/z_{95\%} = 2.33/1.65 \approx 1.412, so US$3.3M times 1.412 is about US$4.66M. The 99% figure is roughly 41 percent larger because the multiplier rises from 1.65 to 2.33. This matches computing US$4.66M directly from 2.33 times 0.02 times US$100M, confirming the conversion is internally consistent under normality.

Common mistakes

  • ✗The z-ratio conversion works for any VaR. It holds only under the normal model; with historical simulation or fat tails the confidence levels are not linked by a fixed z-ratio.
  • ✗Converting confidence levels needs the volatility. Under normality σV\sigma V cancels in the ratio, so you only need the two z-values and one VaR.
  • ✗Going from 95% to 99% scales VaR by a fixed percentage for all assets. The 41 percent uplift is specific to the normal z-ratio; heavier-tailed assets jump by more.
  • ✗You can convert confidence levels and horizons in one step with the same factor. They are separate transformations: z-ratio for confidence, square-root rule for horizon.

Revision bullets

  • •Normal-model VaR is exactly proportional to zz
  • •Convert with VaRc2=VaRc1 zc2/zc1\mathrm{VaR}_{c_2} = \mathrm{VaR}_{c_1}\,z_{c_2}/z_{c_1}
  • •σV\sigma V cancels, so only the two z-values are needed
  • •Fails under historical simulation or fat-tailed losses
  • •95% to 99% uplift is about 1.41 under normality

Quick check

A one-day 95% normal VaR is US$3.3M. The corresponding 99% VaR is closest to

The z-ratio method for converting between confidence levels is valid

Connected topics

VaR DefinitionParametric VaRHistorical SimFat Tails

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.
    Shows that under normality VaR scales with the standard-normal quantile, enabling confidence-level conversion.
  2. GARP (2023), FRM Part I: Valuation and Risk Models
    Global Association of Risk Professionals. FRM Part I: Valuation and Risk Models. GARP / Pearson, 2023.
    Covers parametric VaR and conversion across confidence levels under the normal assumption.
How to cite this page
Dr. Phil's Quant Lab. (2026). Converting VaR Across Confidence Levels. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-var-confidence-conversion
← Back to the atlasSee in the network →