FRM Exam Part I · Measures of Financial Risk

Value at Risk (VaR) Basics for FRM Part I

Updated 11 October 2026 · Fact-checked

Value at Risk (VaR) is the loss level that a portfolio is not expected to exceed over a set horizon at a stated confidence level. Under the normal model, VaR = z × σ × √t × portfolio value (minus any expected return). Historical simulation and Monte Carlo instead read the quantile from simulated or past P&L.

Understand Value at Risk (VaR) Basics

Value at Risk (VaR) answers one question: how much can I lose over a given period, with a given level of confidence? A 1-day 99% VaR of $2 million means that on 99 days out of 100 you expect to lose less than $2 million. On the remaining 1 day in 100, the loss is expected to be at least $2 million. VaR says nothing about how much worse that loss could be.

Every VaR figure has two parameters. The confidence level (for example 95% or 99%) sets how far into the tail you look. The horizon (1 day, 10 days, 1 year) sets the period over which the loss is measured. Change either and the number changes. A higher confidence level or a longer horizon gives a larger VaR.

There are three standard ways to calculate it. Delta-normal (parametric) VaR assumes returns are normally distributed and the position is linear in the risk factors. You multiply the standard deviation by a z-value. Historical simulation takes past returns, applies them to today's portfolio, and reads off the loss at the chosen percentile. It makes no distribution assumption. Monte Carlo simulates many random scenarios from a chosen model and reads the percentile from the results. It handles non-linear positions such as options best, but it needs a model and more computing.

To move between horizons, the usual shortcut is the square-root-of-time rule: VaR over t days = 1-day VaR × √t. It holds when daily returns are independent and identically distributed with zero mean (or when the mean is ignored). It is an approximation, and it weakens when returns are autocorrelated or volatility changes.

VaR is a quantile, not an average. It is not a coherent risk measure because it can fail subadditivity, and it ignores the size of losses beyond the cutoff. That is why Expected Shortfall is studied alongside it.

Key formulas to remember

VaR definition: P(Loss > VaR) = 1 − confidence level; VaR is the quantile of the loss distribution at the chosen confidence level.
Normal VaR (return form): VaR = (z × σ − μ) × Portfolio value; σ and μ must be for the same horizon as the VaR. Many exam questions set μ = 0.
Common z-values (one-tailed): 90%: 1.28 | 95%: 1.645 | 97.5%: 1.96 | 99%: 2.326; Use one-tailed values for VaR. Using 1.96 for 95% is a classic error.
Square-root-of-time rule: VaR(t days) = VaR(1 day) × √t; Valid for i.i.d. returns with zero mean. Scale σ, not the mean.
Annual to daily volatility: σ(daily) = σ(annual) ÷ √252; Use the trading-day count given in the question; 250 or 252 are common.
Historical simulation quantile: VaR = loss at rank (1 − c) × N in the sorted loss list; With 500 scenarios at 99%, VaR is around the 5th worst loss. Conventions on interpolation vary, so follow the question.
Linear (delta) approximation: ΔP ≈ Δ × ΔS, so VaR(position) ≈ |Δ| × VaR(underlying); Delta-normal ignores gamma, so it can misstate option VaR.

How to solve Value at Risk (VaR) Basics questions

Use this sequence for any VaR question, whether it asks for a number or for a comparison of methods.

1Read the confidence level and horizon. Note whether VaR is wanted in percent or in currency.
2Decide the method. If the question says normal, parametric or delta-normal, use z × σ. If it gives a list of past returns or P&Ls, use historical simulation. If it describes random scenarios, think Monte Carlo.
3Put σ and μ on the VaR horizon. Convert annual volatility by dividing by √252 (or the stated count), or scale a daily σ up by √t.
4Pick the correct one-tailed z-value for the confidence level.
5Compute VaR = (z × σ − μ) × position value. If μ is not mentioned, or the question says to ignore it, use zero.
6For historical simulation, sort the losses from worst to best and find the observation at the tail position (1 − c) × N.
7Sanity-check: a higher confidence level or longer horizon must give a larger VaR. Check units and that the answer is a positive loss figure.

Quickest way: Memorize z, scale by √t, multiply

When to use it: Use for any delta-normal question with zero mean and independent returns, which covers most numeric VaR questions.

Memorize 1.645 (95%) and 2.326 (99%).
Compute daily σ in currency: position × daily volatility.
Multiply by z, then by √t for the horizon. Common roots: √10 ≈ 3.162, √5 ≈ 2.236, √252 ≈ 15.875.
To move from 95% to 99% at the same horizon, multiply by 2.326 ÷ 1.645 ≈ 1.414.
Eliminate options that use the wrong tail, scale the mean by √t, or forget to scale at all.

Common mistakes in Value at Risk (VaR) Basics

Using a two-tailed z-value such as 1.96 for 95% VaR.
1.96 is the best-known z-value from confidence intervals.
Fix: VaR looks at one tail only. Use 1.645 for 95% and 2.326 for 99%.
Forgetting to convert volatility to the VaR horizon.
Annual volatility is quoted, but the question asks for 1-day or 10-day VaR.
Fix: Divide annual σ by √252 for daily, then multiply by √t for longer horizons.
Scaling the mean by √t as well as the volatility.
Students apply √t to the whole expression.
Fix: The mean scales with t, volatility with √t. Most questions set the mean to zero, so just scale σ.
Reading VaR as the maximum possible loss.
The phrase 'worst case' is used loosely.
Fix: VaR is a threshold. Losses beyond it occur with probability 1 − c, and VaR says nothing about their size.
Choosing the wrong tail position in historical simulation.
Students confuse 99% confidence with the 99th worst observation.
Fix: Find (1 − c) × N. For 1,000 scenarios at 99%, that is around the 10th worst loss.
Assuming delta-normal VaR is accurate for options and fat-tailed returns.
The formula is simple and looks general.
Fix: Delta-normal assumes normality and linearity. For non-linear positions, use full revaluation or Monte Carlo.

Worked examples

Example 1

A portfolio is worth $50 million. Daily return volatility is 1.2%. Assume normal returns with zero mean. Compute the 10-day 99% VaR.

Show the solution

Daily σ in dollars = 50,000,000 × 0.012 = $600,000.
One-tailed z at 99% = 2.326.
1-day 99% VaR = 2.326 × 600,000 = $1,395,600.
Scale with the square root of time: √10 ≈ 3.1623.
10-day VaR = 1,395,600 × 3.1623 ≈ $4,413,300.

Answer: About $4.41 million.

Example 2

A historical simulation uses 500 daily P&L observations for a portfolio. You sort the losses from largest to smallest. What is the 99% 1-day VaR, and how many observations exceed it?

Show the solution

Tail probability = 1 − 0.99 = 1%.
Tail observations = 1% × 500 = 5.
The 99% VaR is set at the boundary of the worst 5 outcomes. In a common convention it is the 5th largest loss (some conventions use the 6th, or interpolate, so follow the stated rule).
With the 5th-largest convention, 4 losses are larger than VaR and 1 equals it, so about 5 observations are at or beyond the cutoff.

Answer: VaR is the 5th-largest loss (99% confidence, 1% tail of 500 observations); about 5 observations lie in the tail.

Exam tips

Check whether the question gives annual or daily volatility before touching z. Unit mismatch is the most common trap.
Know the contrast table cold: delta-normal is fast but assumes normality and linearity; historical simulation is model-free but relies on the past window; Monte Carlo is flexible but model- and compute-heavy.
Questions often ask which change increases VaR. Higher confidence, longer horizon and higher volatility all do.
Watch for 'relative VaR' versus 'absolute VaR'. Relative VaR is measured from the mean, while absolute VaR subtracts the mean return from z × σ.
Use the calculator's square-root key for √t and keep at least four decimals until the final step.

Practice questions from Measures of Financial Risk

Value at Risk (VaR) Basics 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: frequently asked questions

What is the difference between parametric and historical simulation VaR?

Parametric VaR assumes a distribution, usually normal, and computes VaR from the mean, standard deviation and a z-value. Historical simulation applies actual past returns to today's portfolio and reads the loss percentile directly. It captures fat tails in the data but depends heavily on the chosen window.

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 independent with zero mean. If returns are autocorrelated or the mean is material, the rule is only approximate.

Does 99% VaR mean I lose that amount 1% of the time?

Not exactly. It means that with 99% confidence the loss will not exceed VaR over the horizon. A loss at least as large as VaR is expected about 1% of the time, but VaR gives no information on how large that loss could be.

When is Monte Carlo VaR better than delta-normal?

Monte Carlo is better for portfolios with non-linear payoffs such as options, or when returns are not normal. It revalues the portfolio in each simulated scenario. The cost is more computing time and reliance on the model you choose.