FRM Part I · FRM Exam Part I

Calculating and Applying VaR for FRM Part I

Value at Risk (VaR) is the loss level that a portfolio is not expected to exceed over a set horizon at a chosen confidence level. You solve it by picking a method (parametric, historical simulation or Monte Carlo), setting horizon and confidence, and converting the loss distribution into a quantile.

What this chapter covers

This chapter teaches you to measure market risk with one number and to judge how far that number can be trusted. You start with the definition: VaR depends on a confidence level, a horizon and a loss distribution. You then learn three ways to build that distribution: parametric (delta-normal), historical simulation and Monte Carlo simulation.

The chapter then moves beyond VaR. Expected shortfall (ES) averages the losses beyond the VaR quantile, and coherent risk measures give a formal test for what a good measure looks like. You also handle nonlinear positions such as options, where delta alone is not enough and you need gamma or full revaluation. The chapter closes with limitations, backtesting and uses of VaR.

This chapter ties into much of the rest of Part I. It uses the normal distribution, quantiles, volatility and correlation from Quantitative Analysis. It uses option sensitivities from Financial Markets and Products. It also links to Foundations of Risk Management, where risk limits, capital and governance depend on these measures.

VaR is the most common risk number in banks, and the exam tests it with calculations and with judgement questions. You can score reliably here because the calculations follow fixed steps: a z-value, a volatility, a horizon scaling and a position size. The conceptual questions (which method suits which portfolio, why ES is coherent, what a backtest exception count means) reward clear understanding. Time matters in a 100-question, 4-hour exam, so fast and accurate VaR arithmetic frees time for harder questions elsewhere.

Calculating and Applying VaR: topics in the order to study them

1Value at Risk (VaR) Definition and ParametersEvery later method depends on confidence level, horizon and the quantile idea, so lock these in first.
2Parametric (Delta-Normal) VaRIt is the simplest method and the most calculation-heavy in the exam, and it builds your z-value and scaling habits.
3Historical Simulation VaRIt needs no distribution assumption, so it is easy to learn once you know what a quantile of losses means.
4Monte Carlo Simulation VaRIt builds on the historical approach but generates scenarios from a model, so compare it directly with the previous two.
5Expected Shortfall and Coherent Risk MeasuresES is defined relative to VaR, so you need the VaR quantile clear before you average the tail.
6Delta, Gamma and Full Revaluation for Nonlinear PositionsThis shows where the linear methods break down, and it uses the earlier methods as the base case.
7VaR Limitations, Backtesting and ApplicationsFinish with critique and testing, which make most sense once you know how each VaR number is produced.

How to prepare Calculating and Applying VaR

Treat this chapter as a method toolkit. Learn each method's steps, then practise choosing between them.

Write the core formula once: parametric VaR = z × σ × √t × position value, less any expected return term if the question includes it. Memorise common z-values: 1.645 at 95% and 2.326 at 99% (one-tailed).
Practise horizon scaling. Convert daily volatility to 10-day by multiplying by √10, and confirm that the question asks for a loss, not a return.
Work historical simulation by hand on a small data set. Sort losses, find the right rank for the confidence level and read off the VaR.
Compare the three methods in a table you build yourself: assumptions, speed, handling of options, handling of fat tails. Reuse it to answer concept questions.
Compute ES from a short list of tail losses, and learn the four coherence properties: monotonicity, translation invariance, positive homogeneity and subadditivity. Know that VaR can fail subadditivity.
For options, practise the delta approximation, then the delta-gamma adjustment, and note when full revaluation is needed.
Finish with backtesting: count exceptions against the expected number, and know the basic idea of the traffic-light zones. Then attempt mixed timed question sets.

Common mistakes in Calculating and Applying VaR

Using the wrong z-value, such as two-tailed 1.96 for a one-tailed 95% VaR.
Fix: For VaR, use the one-tailed value: 1.645 for 95% and 2.326 for 99%, unless the question gives a different figure.
Forgetting to scale volatility to the VaR horizon.
Fix: Underline the horizon first, and write σ × √t as a separate line before you multiply by the position.
Treating VaR as the maximum possible loss.
Fix: Remember it is a quantile: losses beyond it occur with probability 1 minus the confidence level, and VaR is silent on how large they are.
Picking the wrong rank in historical simulation.
Fix: Sort losses from worst to best, compute the number of tail observations as (1 − confidence) × N, and follow the question's stated convention.
Applying delta-normal VaR to option positions without adjustment.
Fix: Check for nonlinear payoffs. If present, use delta-gamma or full revaluation, and expect the linear result to be unreliable for large moves.
Confusing ES with VaR or saying ES is less than VaR.
Fix: ES averages losses at or beyond the VaR quantile, so for the same confidence level it is at least as large as VaR.

Last-day revision: Calculating and Applying VaR

VaR = loss threshold at a stated confidence level and horizon; it says nothing about the size of losses beyond it.
One-tailed z-values: 1.645 at 95%, 2.326 at 99%.
Square-root-of-time scaling: σ over t days = σ daily × √t, assuming independent returns and constant volatility.
Parametric VaR assumes normal returns and linear exposure to risk factors.
Historical simulation uses actual past changes; no distribution is assumed, but the past must represent the future.
Monte Carlo is the most flexible and the most computationally costly method; results depend on the model.
Expected shortfall is the average loss given that the loss exceeds VaR.
Coherent measures satisfy monotonicity, translation invariance, positive homogeneity and subadditivity.
VaR is not always subadditive; ES is subadditive.
Delta-only VaR can misstate risk for options; gamma adds curvature and full revaluation reprices the position.
Backtesting compares exceptions with the expected count: at 99% over 250 days, about 2.5 exceptions are expected.
Too many exceptions suggest the model understates risk.

Calculating and Applying VaR practice questions

Calculating and Applying VaR in other exams

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

Calculating and Applying VaR: frequently asked questions

Which VaR method is tested most in FRM Part I?

Parametric and historical simulation VaR often appear as calculations, while Monte Carlo is usually tested conceptually. Be ready to compute the first two and to compare all three. GARP publishes the learning objectives each year, so check the current ones.

Do I need a financial calculator for VaR questions?

Mostly you need basic arithmetic with a square root, so a standard calculator function is enough. Keep the z-values memorised so you do not waste time. A financial calculator helps more in other chapters, such as bond pricing.

Why is VaR not a coherent risk measure?

VaR can violate subadditivity, meaning the VaR of a combined portfolio can exceed the sum of the separate VaRs. This happens in some cases with skewed or discrete loss distributions. Expected shortfall does satisfy subadditivity.

How many backtesting exceptions should I expect?

Multiply the number of observations by one minus the confidence level. For 250 days at 99%, the expected count is 2.5. A count well above this suggests the model understates risk.

Should I learn the chapter before the Quantitative Analysis topics?

It helps to know the normal distribution, quantiles and volatility first. If those are weak, revise them briefly before you start. Then the VaR formulas will make sense and not be memorised blindly.