Skip to content

FRM Exam Part II · Portfolio Risk: Analytical Methods

Portfolio VaR Formula and Parametric VaR Explained

Updated 11 October 2026 · Fact-checked

Value at Risk is the loss that will not be exceeded over a set horizon with a set confidence level. Under normality, compute portfolio standard deviation from weights, volatilities and correlations, multiply by the z-value (1.645 at 95%, 2.326 at 99%) and the portfolio value, then scale by √T for longer horizons.

Understand Value at Risk (VaR) Fundamentals for Portfolios

Value at Risk (VaR) answers one question: how much can I lose over a given horizon, at a given confidence level, under normal market conditions? A one-day 99% VaR of $2 million means you expect to lose more than $2 million on only about 1 day in 100. It does not say how bad the loss is on that day.

Every VaR number needs two parameters: the confidence level (commonly 95% or 99%) and the horizon (one day for trading desks, ten days in older Basel rules, longer for asset managers). A higher confidence level or a longer horizon gives a larger VaR.

Parametric VaR assumes returns follow a normal distribution. Then the whole distribution is fixed by the mean and the standard deviation, and VaR is just a multiple of the standard deviation. For a portfolio, the key input is the portfolio standard deviation, which depends on each asset's volatility, its weight, and the correlations between assets. Lower correlation means more diversification and a lower portfolio VaR.

This is also called delta-normal VaR when the portfolio holds instruments such as options. Each position is replaced by its linear (delta) exposure to risk factors, and the risk factors are assumed jointly normal. It works well for linear positions and poorly for strongly nonlinear ones.

Absolute VaR measures the loss in value relative to zero, so it is reduced by the expected return. Relative VaR measures the loss relative to the expected (mean) value, so it ignores the mean. Over short horizons the mean is tiny and the two are almost equal. Over a year they can differ a lot.

Key formulas to remember

Portfolio variance (two assets)
σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2
Use weights with return volatilities, or dollar positions with dollar volatilities. Do not mix them.
Portfolio variance (matrix form)
σp² = wᵀ Σ w
Σ is the covariance matrix. Use this form for three or more assets.
Relative VaR (to the mean)
VaR(relative) = z × σp × V
z is the one-tailed standard normal value. V is portfolio value.
Absolute VaR (to zero)
VaR(absolute) = (z × σp − μp) × V
μp is the expected portfolio return over the same horizon. A positive mean lowers absolute VaR.
Common z-values (one-tailed)
90%: 1.282; 95%: 1.645; 97.5%: 1.960; 99%: 2.326
Use one-tailed values. 1.96 is the 95% two-tailed value and is the 97.5% one-tailed value.
Horizon scaling
σ(T) = σ(1) × √T and μ(T) = μ(1) × T
The √T rule assumes independent, identically distributed returns.
Undiversified VaR bound
VaR(p) ≤ VaR1 + VaR2 when ρ ≤ 1; equality when ρ = 1
For normal returns, portfolio VaR equals the sum of stand-alone VaRs only if all correlations are 1 (with a zero mean).

How to solve Value at Risk (VaR) Fundamentals for Portfolios questions

Use this sequence for any parametric portfolio VaR question.

  1. 1Read the confidence level, horizon, and whether the question asks for absolute or relative VaR.
  2. 2Convert every position to dollars (or rupees) and make sure each volatility is for the same period as your base data (daily, annual).
  3. 3Compute each position's dollar volatility: position value × return volatility.
  4. 4Combine them using the portfolio variance formula with the correlations. Take the square root to get portfolio dollar standard deviation.
  5. 5Pick the one-tailed z-value for the confidence level.
  6. 6Multiply z by the portfolio dollar standard deviation. This is relative VaR for that period.
  7. 7If the horizon is longer than the data period, multiply σ by √T first. If absolute VaR is requested, subtract the mean return scaled to the horizon times V.
  8. 8Check the answer: it should be below the sum of stand-alone VaRs unless correlation is 1.

Quickest way: Dollar-volatility shortcut

When to use it: Use for two-asset multiple-choice questions where time is short.

  1. Work in dollar volatilities: σA$ = weight × value × σA, and the same for B.
  2. Compute σp$ = √(σA$² + σB$² + 2ρ·σA$·σB$).
  3. Multiply by z. Then multiply by √T if needed.
  4. Sanity check: σp$ must lie between |σA$ − σB$| and σA$ + σB$. If it does not, you made an arithmetic slip.
  5. If an option equals the plain sum of the stand-alone VaRs, it is almost always a distractor unless ρ = 1.

Common mistakes in Value at Risk (VaR) Fundamentals for Portfolios

  • Adding the stand-alone VaRs of the assets to get portfolio VaR.

    Adding looks natural, and it is correct for ρ = 1, so students forget it is only an upper bound.

    Fix: Always combine standard deviations with the correlation term, then apply z once.

  • Using weights and volatilities in the formula and then forgetting to multiply by portfolio value.

    The variance formula gives a percentage, so the last step gets dropped.

    Fix: Finish every answer with × V and write the currency unit.

  • Using the two-tailed z-value, for example 1.96 for 95% VaR.

    Students remember 1.96 from confidence intervals.

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

  • Scaling the variance or the VaR by T instead of by √T.

    Variance scales with T, and this gets mixed up with standard deviation.

    Fix: Scale σ by √T. Equivalently, scale VaR by √T. Scale the mean by T.

  • Ignoring the mean when the question asks for absolute VaR over a long horizon.

    Daily examples set the mean to zero, so the habit sticks.

    Fix: Check the wording. Relative VaR ignores the mean. Absolute VaR subtracts the expected return.

  • Mixing daily volatility with annual returns, or 250 and 252 trading days, without checking the question.

    Inputs are given in different periods to test attention.

    Fix: Convert everything to one period first, using the day count the question states.

Worked examples

Example 1

A portfolio holds $6 million in Asset A and $4 million in Asset B. Daily return volatilities are 1.5% for A and 2.0% for B, and the correlation is 0.30. Assuming normal returns and zero mean, find the 1-day 99% VaR and the 10-day 99% VaR.

Show the solution
  1. Dollar volatilities: A = 6,000,000 × 0.015 = $90,000. B = 4,000,000 × 0.02 = $80,000.
  2. Portfolio variance = 90,000² + 80,000² + 2 × 0.30 × 90,000 × 80,000 = 8.1 × 10⁹ + 6.4 × 10⁹ + 4.32 × 10⁹ = 18.82 × 10⁹.
  3. Portfolio standard deviation = √(18.82 × 10⁹) ≈ $137,186.
  4. 1-day 99% VaR = 2.326 × 137,186 ≈ $319,100.
  5. 10-day VaR = 319,095 × √10 ≈ 319,095 × 3.1623 ≈ $1,009,100.
  6. Check: the sum of stand-alone VaRs is 2.326 × (90,000 + 80,000) ≈ $395,400. Portfolio VaR is lower, as expected.

Answer: 1-day 99% VaR ≈ $319,100; 10-day 99% VaR ≈ $1.01 million.

Example 2

A $50 million portfolio has an expected annual return of 8% and an annual volatility of 12%. Assuming normality, compute the one-year 95% relative VaR and absolute VaR.

Show the solution
  1. z at 95% one-tailed = 1.645.
  2. Relative VaR = 1.645 × 0.12 × 50,000,000 = 0.1974 × 50,000,000 = $9.87 million.
  3. Absolute VaR = (1.645 × 0.12 − 0.08) × 50,000,000 = (0.1974 − 0.08) × 50,000,000 = 0.1174 × 50,000,000 = $5.87 million.
  4. The difference of $4 million is the expected gain: 0.08 × 50 million.

Answer: Relative VaR = $9.87 million; absolute VaR = $5.87 million.

Exam tips

  • Look at the last line of the question first. It tells you whether you need relative or absolute VaR and which horizon.
  • Memorise 1.645 and 2.326 and the √T rule. Most numerical items need only these.
  • Expect interpretation items: a 99% one-day VaR is exceeded about once in 100 days, and VaR says nothing about the size of losses beyond it.
  • Know the assumptions: normal returns, stable correlations, linear exposures. Questions often ask which one fails for options or in a crisis.
  • When a question lowers correlation or adds an asset, reason first: lower correlation lowers portfolio VaR, and that can save you a calculation.

Practice questions from Portfolio Risk: Analytical Methods

Value at Risk (VaR) Fundamentals for Portfolios in other exams

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

Value at Risk (VaR) Fundamentals for Portfolios: frequently asked questions

What is the portfolio VaR formula for FRM Part II?

Under normality, VaR = z × σp × V, where σp is portfolio standard deviation built from weights, volatilities and correlations. For two assets, σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2. Scale by √T for longer horizons.

How do I calculate parametric VaR for a portfolio?

Convert positions to dollar volatilities, combine them with the correlation term, take the square root, and multiply by the one-tailed z-value. Adjust for horizon with √T. Subtract the expected return if the question asks for absolute VaR.

What is the difference between absolute VaR and relative VaR?

Relative VaR measures the potential loss from the expected value, so it equals z × σ × V. Absolute VaR measures the potential loss from the current value, so it is z × σ × V minus the expected gain. They are nearly equal over short horizons.

What is delta-normal VaR?

It is parametric VaR applied to portfolios with derivatives. Each position is replaced by its linear delta exposure to normally distributed risk factors. It becomes inaccurate for options with large gamma or for large moves.