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

Value at Risk (VaR) Basics: Confidence Level and Holding Period

Updated 11 October 2026 · Fact-checked

Value at Risk (VaR) is the loss that a portfolio should not exceed over a set holding period, with a stated confidence level. To solve a question, pick the tail probability (1 − confidence), find that quantile of the P/L or return distribution, and report the loss as a positive number.

Understand Value at Risk (VaR) Basics and Parameters

Value at Risk (VaR) answers one question: how much could I lose over a given period, if things are not extremely bad? It gives a single loss figure. A 1-day 99% VaR of $2 million means that, on 99% of days, the loss should be $2 million or less. On the other 1% of days, the loss should be larger.

Two parameters define VaR. The confidence level (for example 95% or 99%) sets how far into the tail you look. The holding period (for example 1 day or 10 days) sets how long the portfolio is held with no change. Change either one and the VaR changes. Always state both with the number.

A higher confidence level gives a larger VaR. The 99% VaR sits further out in the loss tail than the 95% VaR. A longer holding period also gives a larger VaR, because more time means more risk. Under the usual assumption of independent, identically distributed returns with zero mean, VaR scales with the square root of time.

VaR can be measured on profit/loss (P/L) data, in currency units, or on return data, in percentages. With P/L data, VaR is the negative of the loss-tail quantile. With returns, you find the return quantile first and multiply by the portfolio value. VaR is a quantile, so it says nothing about how bad losses are beyond that point. That is why Expected Shortfall is studied next.

Key formulas to remember

Definition of VaR
P(Loss > VaR) = 1 − confidence level = α
VaR is the loss quantile at the (1 − α) level. At 99% confidence, α = 1%.
VaR from P/L data
VaR = −(P/L quantile at α)
The P/L quantile at 5% or 1% is usually negative. Flip the sign to quote VaR as a positive loss.
VaR from return data
VaR = −(return quantile at α) × portfolio value
Use this when you are given returns in percent and a portfolio value.
Order-statistic rule for historical data
With n observations, the α quantile is roughly the (n × α)th worst outcome
For 500 observations at 99%, VaR is about the 5th worst loss. Conventions on interpolation vary, so follow the question.
Square-root-of-time scaling
VaR(h days) = VaR(1 day) × √h
Valid only for i.i.d. returns with zero mean (or a negligible mean). Not exact otherwise.
Normal VaR (reference)
VaR = −μ + z × σ, with z = 1.645 at 95% and 2.326 at 99%
Used when P/L or returns are assumed normal. μ and σ must match the holding period.

How to solve Value at Risk (VaR) Basics and Parameters questions

Use this method for any basic VaR question on P/L or return data.

  1. 1Read the confidence level and holding period. Write both down.
  2. 2Compute the tail probability α = 1 − confidence level.
  3. 3Check whether the data are P/L (currency) or returns (percent). Note the portfolio value if you need it.
  4. 4Find the α quantile. For sorted data, count in from the worst outcome. For a normal assumption, use the z-value.
  5. 5Convert to VaR: flip the sign so the loss is positive. For returns, multiply by portfolio value.
  6. 6If the holding period differs from the data period, scale by √h only if the question allows the i.i.d. zero-mean assumption.
  7. 7State the answer with its meaning: 'With X% confidence, the loss over h days will not exceed VaR.'

Quickest way: Count-from-the-tail shortcut

When to use it: Use when you are given a list of historical P/L or returns and asked for VaR at a given confidence level.

  1. Multiply the number of observations by α to get the tail count k.
  2. If k is a whole number, the VaR is the loss at that worst-rank position (or as the question specifies). Otherwise follow the question's convention.
  3. Quoted as a positive loss; for returns, multiply by portfolio value.
  4. For horizon scaling, multiply by √h. For a switch from 95% to 99% under normality, multiply by 2.326 ÷ 1.645 (about 1.41).

Common mistakes in Value at Risk (VaR) Basics and Parameters

  • Reporting VaR as a negative number.

    The quantile of P/L is negative, so students copy it directly.

    Fix: VaR is quoted as a positive loss. Change the sign of the P/L or return quantile.

  • Using the confidence level as the tail probability.

    The number 99% is in the question, so students look up 99% in the data tail.

    Fix: Compute α = 1 − confidence first. A 99% VaR looks at the worst 1%.

  • Applying the square-root-of-time rule without checking the assumptions.

    It is a familiar shortcut.

    Fix: Use it only for i.i.d. returns with zero mean, or when the question says so. Otherwise scale the mean and the volatility separately.

  • Thinking VaR is the maximum possible loss.

    The word 'risk' suggests a worst case.

    Fix: VaR is a quantile. Losses beyond it can happen, with probability α, and VaR gives no size for them.

  • Forgetting to multiply a return-based VaR by portfolio value.

    The percent number looks like a final answer.

    Fix: If the question asks for a currency figure, multiply the percent loss by the portfolio value.

  • Assuming VaR at 95% and 99% differ only by a small amount.

    Both are 'high' confidence levels.

    Fix: Under normality, 99% VaR is about 1.41 times the 95% VaR. With fat tails, the gap can be larger.

Worked examples

Example 1

A trading desk has 200 daily P/L observations. The five worst outcomes, in USD million, are −4.8, −4.1, −3.6, −3.3 and −3.0. Using the historical approach, with the tail count equal to n × α, what is the 1-day 97.5% VaR?

Show the solution
  1. Confidence level 97.5%, so α = 1 − 0.975 = 2.5%.
  2. Tail count k = 200 × 0.025 = 5.
  3. Under the stated convention, VaR is the 5th worst outcome: −3.0 million.
  4. Flip the sign to quote VaR as a loss: 3.0 million.

Answer: 1-day 97.5% VaR = USD 3.0 million.

Example 2

A portfolio is worth EUR 50 million. Daily returns are assumed normal with mean 0 and standard deviation 1.2%. Compute the 1-day 99% VaR, then the 10-day 99% VaR using square-root-of-time scaling. Use z = 2.326.

Show the solution
  1. 1-day VaR in percent = 2.326 × 1.2% = 2.7912%.
  2. 1-day VaR in currency = 2.7912% × 50 million = EUR 1.3956 million.
  3. 10-day VaR = 1.3956 × √10. √10 ≈ 3.1623.
  4. 1.3956 × 3.1623 ≈ 4.413 million.
  5. Scaling is allowed here because returns are i.i.d. with zero mean.

Answer: 1-day 99% VaR ≈ EUR 1.40 million; 10-day 99% VaR ≈ EUR 4.41 million.

Exam tips

  • Write confidence level and holding period at the top of your rough work before you compute anything.
  • Check the sign and units of the answer: positive loss, in currency or in percent as asked.
  • If options show a number and its negative, VaR is the positive one.
  • For 95% versus 99% questions, expect the larger confidence level to give the larger VaR. Eliminate options that go the other way.
  • Watch for traps where the data are weekly or monthly and the question asks for a different horizon.

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

Value at Risk (VaR) Basics and Parameters 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) Basics and Parameters: frequently asked questions

What is value at risk in simple words?

It is the loss you do not expect to exceed over a set period, at a stated confidence level. For example, a 1-day 99% VaR of $1 million means a loss above $1 million is expected on about 1 day in 100.

What is the difference between VaR at 95% and 99% confidence?

The 99% VaR looks further into the tail, so it is larger. Under a normal distribution it is about 1.41 times the 95% VaR. The 99% figure is exceeded less often, about 1% of the time against 5%.

How do I calculate VaR from P/L data?

Sort the P/L values and find the α quantile, where α = 1 − confidence. Then flip the sign so VaR is a positive loss. With n observations, the tail count is about n × α.

Why does a longer holding period increase VaR?

There is more time for the portfolio value to move. Under i.i.d. returns with zero mean, VaR grows with the square root of the number of days.