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FRM Exam Part I · Calculating and Applying VaR

Parametric (Delta-Normal) VaR for FRM Part I

Updated 11 October 2026 · Fact-checked

Parametric VaR assumes returns are normally distributed, so VaR is a z-score times the standard deviation, adjusted for the mean and scaled by portfolio value. For one asset: VaR = (z × σ − μ) × V. For a portfolio, first compute portfolio σ using weights and correlations, then apply the same formula.

Understand Parametric (Delta-Normal) VaR

Value at Risk (VaR) is the loss that will not be exceeded with a stated confidence over a stated horizon. If you assume returns follow a normal distribution, you do not need to look at past data point by point. You only need two numbers: the mean (μ) and the standard deviation (σ).

The normal distribution tells you how far the loss quantile sits from the mean. At 95% confidence the one-tailed z-score is 1.645. At 99% it is 2.326. VaR is the distance of that quantile below the mean, multiplied by the position value. That is why it is called parametric: the answer comes from distribution parameters, not from simulation.

Delta-normal means two things together. Risk factors are normally distributed, and each position is linear in those factors through its delta (a first-order approximation). This works well for stocks, bonds with duration, and futures. It works poorly for options with large gamma, where the linear approximation misses curvature.

For a portfolio, VaR depends on correlation. Portfolio σ is not the sum of individual σ values. It uses the covariance matrix. Lower correlation means more diversification and a lower portfolio VaR. The individual VaRs can also be combined directly, using the same correlation structure, because VaR scales linearly with σ when the mean is ignored.

Two practical features matter. First, VaR scales with the square root of time if returns are independent and identically distributed and the mean is zero: VaR(T days) = VaR(1 day) × √T. Second, the normal assumption understates fat tails, so parametric VaR can underestimate extreme losses.

Key formulas to remember

Single-asset parametric VaR (return form)
VaR = (z × σ − μ) × V
Use z = 1.645 for 95% and 2.326 for 99% (one-tailed). μ and σ must be for the same horizon as the VaR.
Zero-mean shortcut
VaR = z × σ × V
Use when the question says to ignore the mean or assumes μ = 0, as is common for daily VaR.
Square-root-of-time scaling
σ(T) = σ(1) × √T; VaR(T) = VaR(1) × √T
Valid for i.i.d. returns with zero mean. If the mean is not zero, scale μ by T and σ by √T separately.
Two-asset portfolio variance
σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2
Weights are fractions of portfolio value. Take the square root to get σp.
Two-asset portfolio VaR from individual VaRs
VaRp = √(VaR1² + VaR2² + 2ρ × VaR1 × VaR2)
Holds for zero-mean normal returns with the same confidence level and horizon. If ρ = 1, VaRp = VaR1 + VaR2.
Annual to daily volatility
σ(daily) = σ(annual) ÷ √252
Use the trading-day count given in the question; 252 is the common convention.
Delta-normal VaR of a linear position
VaR ≈ |Δ| × S × z × σ(return)
Δ is the delta of the position and S the underlying price. Ignores gamma.

How to solve Parametric (Delta-Normal) VaR questions

Use this sequence for any parametric VaR question, single asset or portfolio.

  1. 1Read the confidence level, horizon and position value. Pick the one-tailed z-score: 1.645 for 95%, 2.326 for 99%.
  2. 2Check whether the mean is given or should be ignored. Note whether inputs are daily, monthly or annual.
  3. 3Convert σ and μ to the VaR horizon. Multiply σ by √T and μ by T, or divide annual σ by √252 for daily.
  4. 4For a portfolio, compute portfolio σ using weights, individual σ values and correlation. Or compute each VaR first and combine with the correlation formula.
  5. 5Apply VaR = (z × σ − μ) × V. Keep the sign convention: VaR is reported as a positive loss.
  6. 6Sanity check: portfolio VaR must lie between the diversified value (ρ = −1 or low) and the simple sum of individual VaRs (ρ = 1).
  7. 7State the answer in currency units with the confidence level and horizon.

Quickest way: Combine VaRs directly with correlation

When to use it: Use when the question gives or lets you compute each asset's VaR and asks for the two-asset portfolio VaR with zero mean.

  1. Compute each asset VaR as z × σ × position value.
  2. Plug into √(VaR1² + VaR2² + 2ρ VaR1 VaR2).
  3. Check the bounds: the result must be below VaR1 + VaR2 when ρ < 1.
  4. Remember the z-scores 1.645 and 2.326 so you never search for them.

Common mistakes in Parametric (Delta-Normal) VaR

  • Using the two-tailed z-score (1.96) for 95% VaR.

    1.96 is familiar from confidence intervals and hypothesis tests.

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

  • Adding individual VaRs to get portfolio VaR.

    It feels natural to add risks, like adding values.

    Fix: Add only if ρ = 1. Otherwise use the correlation formula, which gives a smaller number.

  • Scaling σ to the wrong horizon, for example multiplying daily σ by 10 instead of √10.

    Mixing up how mean and volatility scale.

    Fix: σ scales with √T and the mean scales with T.

  • Mishandling the sign of the mean.

    A positive expected return reduces VaR, but students add it.

    Fix: Use VaR = (zσ − μ) × V. A positive μ lowers VaR.

  • Using weights that do not add to 1 or forgetting to multiply by portfolio value at the end.

    Rushing between the σp step and the currency answer.

    Fix: Write weights as fractions first and finish with × V.

  • Applying delta-normal VaR to option positions without noting its limits.

    The formula looks general.

    Fix: Delta-normal is a linear approximation. It is poor for options with high gamma. Use full revaluation there.

Worked examples

Example 1

A portfolio is worth $10 million. Its daily return has mean 0 and standard deviation 1.2%. Assuming normal returns, calculate the 1-day 99% VaR and the 10-day 99% VaR.

Show the solution
  1. z at 99% (one-tailed) = 2.326.
  2. 1-day VaR = 2.326 × 0.012 × 10,000,000 = 2.326 × 120,000 = $279,120.
  3. 10-day VaR = 279,120 × √10 = 279,120 × 3.1623 ≈ $882,666.
  4. Zero mean means square-root-of-time scaling is valid.

Answer: 1-day 99% VaR ≈ $279,120; 10-day 99% VaR ≈ $882,700.

Example 2

A portfolio holds $6 million in Asset A and $4 million in Asset B. The daily σ is 2% for A and 3% for B. The correlation is 0.30. Assume zero mean. Calculate the 1-day 95% VaR.

Show the solution
  1. z at 95% = 1.645.
  2. VaR_A = 1.645 × 0.02 × 6,000,000 = 1.645 × 120,000 = $197,400.
  3. VaR_B = 1.645 × 0.03 × 4,000,000 = 1.645 × 120,000 = $197,400.
  4. VaRp² = 197,400² + 197,400² + 2 × 0.30 × 197,400 × 197,400 = 197,400² × (1 + 1 + 0.6) = 197,400² × 2.6.
  5. VaRp = 197,400 × √2.6 = 197,400 × 1.61245 ≈ $318,300.
  6. Check: this is below the undiversified sum of $394,800, as expected for ρ < 1.

Answer: 1-day 95% portfolio VaR ≈ $318,300.

Exam tips

  • Memorise 1.645 and 2.326. Questions rarely give a z-table.
  • Read whether the question says to ignore the mean. If not and μ is given, subtract it.
  • Check the time units before calculating. Annual to daily conversions are a favourite trap.
  • For the portfolio, the combined-VaR formula is faster than building σp from weights when individual VaRs are easy to compute.
  • Expect conceptual questions on why delta-normal fails for options and fat tails.

Practice questions from Calculating and Applying VaR

Parametric (Delta-Normal) VaR in other exams

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

Parametric (Delta-Normal) VaR: frequently asked questions

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

VaR = (z × σ − μ) × V, where z is the one-tailed z-score, σ and μ are for the VaR horizon, and V is the position value. With zero mean it reduces to z × σ × V.

How do I calculate portfolio VaR with correlation?

Compute each position's VaR, then combine with √(VaR1² + VaR2² + 2ρ VaR1 VaR2). Alternatively compute portfolio σ from weights and correlation, then multiply by z and portfolio value.

Which z-scores do I need for FRM Part I?

Use 1.645 for 95% and 2.326 for 99% one-tailed. Use 1.96 only for two-tailed intervals, not for VaR.

Why is delta-normal VaR criticised?

It assumes normal returns, which understates fat tails, and it uses a linear approximation, which misses option convexity. It can therefore underestimate extreme losses.