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FRM Exam Part II · Estimating Market Risk Measures: An Introduction and Overview

Parametric VaR: Normal and Lognormal Methods

Updated 11 October 2026 · Fact-checked

Parametric VaR assumes returns follow a known distribution and reads the loss cut-off from its mean, standard deviation and a z-score. Normal VaR = −(μ − zσ) × portfolio value in return terms. Lognormal VaR uses geometric returns, so the price cannot fall below zero. Pick the confidence level, find z, plug in, and scale to a dollar loss.

Understand Parametric VaR: Normal and Lognormal

Value at Risk (VaR) is the loss you expect not to exceed at a chosen confidence level over a set horizon. Parametric VaR gets it from a formula instead of from past data. You assume a distribution, estimate its parameters, and read the quantile.

With the normal approach, you assume the periodic P/L or return is normal with mean μ and standard deviation σ. The loss quantile at confidence level α is μ − zα·σ, where zα is the standard normal value that leaves α in the left of the distribution. Common values are 1.645 for 95% and 2.326 for 99% (one-tailed). VaR is quoted as a positive loss, so VaR = zα·σ − μ.

With the lognormal approach, you assume the log (continuously compounded) return is normal. Then the price itself is lognormal and can never go below zero. This fixes a flaw of the normal model, which allows losses larger than the whole portfolio. The lognormal VaR is the portfolio value minus the lowest price at the cut-off: VaR = P0 × (1 − exp(μ − zα·σ)), with μ and σ being the mean and standard deviation of the log return.

For short horizons and small σ, the two give almost the same number. The gap grows with long horizons and high volatility. Lognormal VaR is slightly smaller than normal VaR for the same μ and σ, because 1 − exp(−x) is less than x for positive x.

To change horizon, assume returns are independent and identically distributed. Mean scales with time (μ × T) and volatility scales with the square root of time (σ × √T). Always convert annual figures to the holding period first, such as 10 days out of 250 trading days.

Key formulas to remember

Normal VaR (return terms)
VaR% = zα × σ − μ
Use one-tailed z: 1.645 at 95%, 2.326 at 99%. Result is a positive loss rate.
Normal VaR (value terms)
VaR = P0 × (zα × σ − μ)
P0 is the current portfolio value. If the mean is assumed zero, VaR = P0 × zα × σ.
Lognormal VaR
VaR = P0 × (1 − exp(μ − zα × σ))
μ and σ are the mean and standard deviation of the log return over the holding period.
Lowest price at the cut-off
P* = P0 × exp(μ − zα × σ)
Lognormal VaR = P0 − P*.
Time scaling
μT = μ × T; σT = σ × √T
Valid under i.i.d. returns. Convert to the holding period before applying z.
Common one-tailed z-scores
90%: 1.282; 95%: 1.645; 97.5%: 1.960; 99%: 2.326
Do not use two-tailed values such as 1.96 for 95% VaR.

How to solve Parametric VaR: Normal and Lognormal questions

Use this order for any parametric VaR question. It keeps units and tails straight.

  1. 1Identify the distribution: normal P/L or return, or lognormal price (normal log return).
  2. 2Write down μ, σ, portfolio value, confidence level and the horizon.
  3. 3Convert μ and σ to the holding period: μ × T and σ × √T, with T in the same units as the inputs.
  4. 4Find the one-tailed z for the confidence level.
  5. 5Compute the cut-off return: normal gives μ − zσ; lognormal gives exp(μ − zσ) − 1 or the price P0 × exp(μ − zσ).
  6. 6Convert to a positive loss in currency: P0 × (−cut-off return), or P0 minus P*.
  7. 7Check the sign and size. VaR should be positive, and lognormal VaR should be below P0.
  8. 8State the interpretation: with α confidence, the loss over the horizon will not exceed this amount.

Quickest way: Zero-mean shortcut with square-root scaling

When to use it: When the question gives daily volatility and a short horizon, or says to assume zero mean. Most normal VaR questions fit this.

  1. Daily dollar VaR = z × σ(daily) × P0.
  2. Multiply by √(days) for a multi-day VaR.
  3. Memorise z of 1.645 and 2.326. Use them without a table.
  4. If a mean is given and not small, subtract it: z × σ − μ.
  5. For lognormal, compute exp(μ − zσ) first, then 1 minus that, times P0. Only switch to this when the question says lognormal or log returns.

Common mistakes in Parametric VaR: Normal and Lognormal

  • Using 1.96 for 95% VaR

    1.96 is the familiar two-tailed value for a 95% interval.

    Fix: VaR is a one-tailed measure. Use 1.645 at 95% and 2.326 at 99%.

  • Scaling volatility by T instead of √T

    Students scale the mean and volatility the same way.

    Fix: Mean scales with T, standard deviation with √T. Variance scales with T.

  • Forgetting to subtract the mean, or subtracting it with the wrong sign

    Memorising VaR = zσ and ignoring μ.

    Fix: Use VaR = zσ − μ in return terms. A positive mean reduces VaR.

  • Applying the normal formula to a lognormal question

    Both use μ, σ and z, so they look alike.

    Fix: If the inputs are log-return parameters, use P0 × (1 − exp(μ − zσ)).

  • Mixing annual and daily parameters

    Volatility is quoted annually but the horizon is 1 or 10 days.

    Fix: Convert first. Daily σ = annual σ ÷ √252 (or the trading-day count given).

  • Reporting VaR as a negative number without comment

    The cut-off return is negative, so the sign carries over.

    Fix: Report VaR as a positive loss, unless the question asks for the quantile of P/L.

Worked examples

Example 1

A portfolio is worth $50 million. Daily returns are normal with mean 0.05% and standard deviation 1.2%. Compute the 1-day 99% parametric VaR.

Show the solution
  1. z at 99% one-tailed = 2.326.
  2. VaR% = zσ − μ = 2.326 × 1.2% − 0.05% = 2.7912% − 0.05% = 2.7412%.
  3. VaR = $50,000,000 × 0.027412 = $1,370,600.

Answer: About $1.37 million. With 99% confidence, the 1-day loss should not exceed this amount.

Example 2

A $20 million position has annual log-return mean 8% and annual volatility 20%. Assume lognormal prices. Compute the 1-year 95% VaR.

Show the solution
  1. z at 95% = 1.645.
  2. Log-return cut-off = μ − zσ = 0.08 − 1.645 × 0.20 = 0.08 − 0.329 = −0.249.
  3. exp(−0.249) ≈ 0.7796.
  4. Lowest price P* = $20,000,000 × 0.7796 = $15,592,000.
  5. VaR = P0 − P* = $20,000,000 − $15,592,000 = $4,408,000.

Answer: About $4.41 million. For comparison, normal VaR would be 20 × (0.329 − 0.08) = $4.98 million, so lognormal VaR is lower.

Exam tips

  • Check the first line for the word lognormal, log returns or geometric returns. It decides the formula.
  • Read whether the mean is given or to be assumed zero. Short-horizon questions often assume zero.
  • Watch the horizon. A 10-day VaR needs √10, and an annual σ needs conversion to daily first.
  • Expect conceptual items: lognormal VaR is below normal VaR for the same μ and σ, and normal VaR can imply losses beyond the portfolio value.
  • Remember that fat tails make normal VaR understate extreme losses, especially at high confidence.

Practice questions from Estimating Market Risk Measures: An Introduction and Overview

Parametric VaR: Normal and Lognormal in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Parametric VaR: Normal and Lognormal: frequently asked questions

What is the formula for parametric VaR under a normal distribution?

VaR = P0 × (zα × σ − μ), where σ and μ are for the holding period. With zero mean this becomes P0 × zα × σ. Use one-tailed z values such as 1.645 at 95% and 2.326 at 99%.

What is the difference between normal VaR and lognormal VaR?

Normal VaR assumes returns are normal, so prices can in theory fall below zero. Lognormal VaR assumes log returns are normal, so the price stays positive. For the same μ and σ, lognormal VaR is slightly smaller, and the two converge for short horizons.

How do I calculate normal VaR using a z-score?

Find the one-tailed z for your confidence level, multiply by the holding-period volatility, subtract the mean if given, and multiply by the portfolio value. Scale daily volatility by the square root of the number of days for longer horizons.

Why does parametric VaR scale with the square root of time?

If returns are independent with equal variance, variances add across periods. Standard deviation is the square root of variance, so it grows with √T. This fails when returns are autocorrelated.