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FRM Exam Part I · Calculating and Applying VaR

Value at Risk (VaR): Definition, Confidence Level and Horizon

Updated 11 October 2026 · Fact-checked

Value at Risk (VaR) is the loss that will not be exceeded over a set time horizon with a stated confidence level. For example, a 1-day 99% VaR of $2 million means a 1% chance of losing more than $2 million in a day. To scale, multiply by √(days) when returns are i.i.d. with zero mean.

Understand Value at Risk (VaR) Definition and Parameters

Value at Risk (VaR) answers one question: how much could I lose over a given period, in normal conditions, at a given level of confidence? It is a single number in currency units (or sometimes a percentage of portfolio value).

VaR has two parameters you must always state. The confidence level (for example 95% or 99%) sets how rare the loss is. The holding period or time horizon (for example 1 day or 10 days) sets how long the position is exposed. A 99% 1-day VaR of $5 million says: on 99 days out of 100 you expect to lose less than $5 million. On about 1 day in 100 you expect to lose more.

VaR is a quantile of the loss distribution. A higher confidence level pushes you further into the tail, so VaR rises. A longer horizon gives more time for prices to move, so VaR rises. Notice what VaR does not say: it gives no information about how large the loss is once it exceeds VaR. That is the job of expected shortfall.

If daily returns are independent and identically distributed (i.i.d.) with a mean of zero, the standard deviation grows with the square root of time. So the 10-day VaR equals the 1-day VaR times √10. This is the square-root-of-time rule. It also holds under normality when the mean is ignored. If the mean is not zero, the mean scales with time (not √time), so the shortcut changes.

Changing the confidence level under normality works through z-values. The one-tailed z for 95% is 1.645, for 97.5% is 1.96, and for 99% is 2.326. VaR = z × σ × portfolio value (with a zero mean). Regulators and banks choose parameters by purpose: short horizons such as 1 day for trading books, and longer horizons to reflect the time needed to liquidate or hedge.

Key formulas to remember

VaR definition
P(Loss > VaR) = 1 − confidence level
VaR is the loss quantile. At 99%, there is a 1% chance of a larger loss over the horizon.
Normal VaR (percentage)
VaR% = z × σ − μ
σ and μ are for the chosen horizon. Use z = 1.645 (95%), 2.326 (99%). With μ = 0, VaR% = z × σ.
Normal VaR (currency)
VaR = Portfolio value × (z × σ − μ)
State it as a positive loss number.
Square-root-of-time rule
VaR(T days) = VaR(1 day) × √T
Valid for i.i.d. returns with zero mean (or normal with mean ignored). Volatility scales the same way: σ(T) = σ(1) × √T.
Scaling volatility to annual
σ(annual) = σ(daily) × √252
252 trading days is a common convention; use the number the question gives.
Change of confidence level (normal)
VaR(new) = VaR(old) × z(new) ÷ z(old)
Only valid under normality and zero mean.

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

Use this routine for any VaR definition or scaling question.

  1. 1Identify the confidence level and horizon you are given, and the ones you are asked for.
  2. 2Decide whether the question states a loss distribution, a standard deviation, or an existing VaR figure.
  3. 3Check the assumptions: normal returns, i.i.d., zero mean. If the question says so, you may use the formulas directly.
  4. 4If you have σ, compute VaR = z × σ × portfolio value, using σ for the right horizon.
  5. 5If you have a VaR already, rescale: multiply by √T for the horizon and by z(new) ÷ z(old) for the confidence level.
  6. 6Handle the mean separately if it is non-zero: subtract the horizon mean from z × σ.
  7. 7State the answer as a positive loss and read it back in words: 'with X% confidence, losses over T days will not exceed this amount'.

Quickest way: Ratio scaling from a known VaR

When to use it: When the question gives a VaR for one confidence level or horizon and asks for another, under normality with zero mean.

  1. Write the new VaR as old VaR × √(new days ÷ old days).
  2. Multiply by z(new) ÷ z(old) if the confidence level changes.
  3. Memorise z: 1.645 for 95%, 1.96 for 97.5%, 2.326 for 99%.
  4. Sanity check: longer horizon and higher confidence must both give a larger VaR.

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

  • Scaling VaR by the number of days instead of its square root.

    Students think loss simply adds up day by day.

    Fix: Independent daily shocks partly offset, so volatility grows with √T. Multiply by √10, about 3.162, for 10 days.

  • Reading 99% VaR as the maximum possible loss.

    The word 'value at risk' sounds like a worst case.

    Fix: VaR is a threshold. Losses exceed it with probability 1 − confidence. It says nothing about the size of those larger losses.

  • Using the wrong tail probability for z.

    Mixing up one-tailed and two-tailed z-values (1.96 vs 1.645 for 95%).

    Fix: VaR is one-tailed. 95% uses 1.645 and 99% uses 2.326.

  • Applying the square-root rule with a non-zero mean or non-i.i.d. returns without noting it.

    Students treat the rule as universal.

    Fix: State the conditions. The mean scales with T, not √T. Autocorrelation or changing volatility breaks the rule.

  • Mixing horizons: using daily σ with a 10-day VaR request.

    The horizon is easy to overlook in the final line of the question.

    Fix: Convert σ to the target horizon first, then multiply by z.

  • Treating an increase in confidence level as unrelated to horizon scaling.

    Two adjustments in one question cause confusion.

    Fix: Apply them as two independent multipliers: √T for time and z-ratio for confidence.

Worked examples

Example 1

A portfolio has a 1-day 99% VaR of $3 million. Assuming i.i.d. normal returns with zero mean, what is the 10-day 99% VaR?

Show the solution
  1. Formula: VaR(10) = VaR(1) × √10.
  2. √10 ≈ 3.1623.
  3. VaR(10) = 3 × 3.1623 = 9.487 million.

Answer: About $9.49 million.

Example 2

A portfolio worth $50 million has daily return volatility of 1.2% and zero mean returns. Assuming normality, what is the 5-day 95% VaR?

Show the solution
  1. 5-day volatility = 1.2% × √5 = 1.2% × 2.2361 = 2.683%.
  2. z for 95% one-tailed = 1.645.
  3. VaR% = 1.645 × 2.683% = 4.414%.
  4. VaR = $50 million × 0.04414 = $2.207 million.

Answer: About $2.21 million.

Exam tips

  • Always check the stated assumptions (normal, i.i.d., zero mean) before using √T scaling.
  • Memorise 1.645 and 2.326, and be ready to get the z-ratio between 95% and 99% (2.326 ÷ 1.645 ≈ 1.414).
  • Read the final sentence of the question for the target horizon and confidence level before calculating.
  • Conceptual questions often test what VaR does not tell you: loss beyond the threshold.
  • Work in the same unit throughout: percent or currency, daily or annual.

Practice questions from Calculating and Applying VaR

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

What is the difference between confidence level and holding period in VaR?

The confidence level sets how rare the loss threshold is, such as 99%. The holding period is the time the position is exposed, such as 1 or 10 days. Both must be stated for a VaR number to be meaningful.

How do I scale VaR from 1 day to 10 days?

Multiply the 1-day VaR by √10, about 3.162. This works when daily returns are i.i.d. with zero mean, or normal with the mean ignored. Check the question states or allows these assumptions.

Does a higher confidence level always increase VaR?

Yes. A higher confidence level moves the quantile further into the loss tail, so the VaR figure is larger. Under normality, the VaR is proportional to the z-value.

Is VaR the worst loss I can suffer?

No. A 99% VaR is exceeded on about 1% of periods, and VaR gives no information on how big the loss is when it is exceeded. Expected shortfall addresses that.