Value at risk for the FRM® Exam Part I: methods and worked examples

Value at risk (VaR) is a loss threshold at a stated confidence level over a stated holding period. Historical simulation, parametric models and Monte Carlo simulation estimate that threshold differently. VaR does not tell you how severe losses beyond it could be. (RBA, VaR methods, §§2.1–2.7 (secondary source))

This guide supports study for the FRM® Exam Part I, run by the Global Association of Risk Professionals (GARP). (GARP, FRM Exam information) Our questions complement GARP’s own study materials; they do not replace them. See the course page for course information. All example amounts, returns and scenarios below are illustrative assumptions, not market observations.

What does a VaR number mean?

Let $L$ denote loss, with gains negative, and $c$ the confidence level. We use the smallest loss threshold whose cumulative probability reaches that level: (Acerbi and Tasche, expected shortfall)

An exact one-day 99% VaR threshold has at most 1% model probability of being exceeded. With a continuous distribution, that probability is 1%; rounding the threshold can change it. (Acerbi and Tasche, expected shortfall) VaR is neither the maximum possible loss nor a prediction of exactly one breach in every hundred days. Always state currency, horizon and confidence together. The estimate depends on the portfolio and the model’s assumptions. (RBA, VaR methods, §§2.1–2.7 (secondary source))

How does historical simulation work?

Apply past joint risk-factor changes to today’s holdings, revalue the portfolio under each scenario, then rank the resulting losses. Keep each day’s shocks together to retain their observed dependence. Full revaluation can capture option curvature without assuming normal returns. The selected history still limits the scenarios represented. (RBA, VaR methods, §§2.1–2.7 (secondary source))

Suppose 40 equally weighted daily scenarios produce losses whose four largest values, in ascending order, are £25,000, £36,000, £48,000 and £72,000. The other 36 losses are below £25,000. Using our empirical quantile convention, choose rank $\lceil cN\rceil$ from smallest to largest, where the brackets mean round upwards:

Two scenarios exceed the threshold: 2 ÷ 40 = 5%. This small sample illustrates the calculation. Other quantile conventions interpolate between observations, so state the convention before comparing answers. (RBA, VaR methods, §§2.1–2.7 (secondary source))

How do you calculate parametric VaR with a delta-normal model?

Delta-normal VaR combines a linear approximation to portfolio value with jointly normal risk-factor changes. Delta captures first-order sensitivity; it omits curvature. Volatilities and correlations determine portfolio variance, so simply adding individual VaRs generally misses diversification. (IMF, risk measurement, Appendix II (secondary source))

For portfolio value $V$ and a normally distributed portfolio return with mean $\mu$ and standard deviation $\sigma$, loss is minus value times return. Transforming the normal return quantile gives:

Here $z_c$ is the standard normal quantile: approximately 2.33 for a one-sided 99% confidence level. (RBA, VaR methods, §§2.1–2.7 (secondary source)) Assume a £1,800,000 linear position, daily volatility of 1.25% and zero mean daily return:

If mean daily return is instead 0.02%, expected daily profit is £360. Subtract that amount to obtain £52,065. The sign matters: positive expected profit reduces the loss quantile. Both results use the rounded multiplier 2.33. Heavy tails and nonlinear positions can make the normal, linear approximation unreliable. (IMF, VaR approaches, pp. 19–20 (secondary source))

What changes with Monte Carlo VaR?

Monte Carlo generates random risk-factor scenarios from a specified joint distribution, revalues the portfolio and takes a loss quantile. Unlike historical simulation, it can generate scenarios absent from the observation window. It can accommodate nonlinear valuation and non-normal distributions, but only if these are included in the model. More simulations do not repair inappropriate assumptions. (IMF, VaR approaches, pp. 19–20 (secondary source))

For a worked simulation, retain the £1,800,000 position, zero mean and 1.25% daily volatility. Draw a standard normal value $Z$ for each scenario and calculate:

In our illustrative run of 100,000 draws, rank 99,000 gives a one-day 99% VaR of approximately £52,867. The difference from £52,425 reflects simulation variation and rounding of the analytic normal multiplier. Using fresh random draws can change the estimate. With dependent risk factors, simulate their dependence too; replacing it with independent draws would change the model. (IMF, risk measurement, Appendix II (secondary source))

When can you use the square-root-of-time rule?

For independent, identically distributed normal daily profit-and-loss increments, variances add and standard deviation scales with the square root of time. With zero mean, unchanged exposures and the same confidence level, this gives the familiar VaR rule. (IMF, risk measurement, Appendix II (secondary source))

Our zero-mean example gives a ten-day VaR of £52,425 × √10 ≈ £165,782. With non-zero mean, however, the expected profit scales linearly. Under the same additive model, ignoring return compounding:

Using the earlier 0.02% daily mean gives approximately £162,182 after subtracting ten days’ expected profit, £3,600. Multiplying the mean-adjusted one-day VaR by √10 would be incorrect.

Serial dependence, changing volatility, non-normal tails and nonlinear exposures undermine these assumptions. Even when variance scales with time, a non-normal loss quantile need not scale with its square root. Longer horizons therefore require scrutiny of the distribution and exposure dynamics, not just a larger multiplier. (IMF, risk measurement, Appendix II (secondary source))

How does expected shortfall differ from VaR?

Expected shortfall (ES) averages losses in the worst fraction of the distribution. At 95% confidence it averages the worst 5%; VaR identifies the boundary. For integrable losses, the quantile representation is: (Acerbi and Tasche, expected shortfall)

In our historical example, the worst 5% comprises the two largest losses:

Increase only the worst loss to £144,000: VaR stays £36,000, but ES rises to £96,000. This shows why knowing the threshold alone cannot describe tail severity.

Including the £36,000 boundary observation would average three losses instead of the required two, producing £52,000 rather than £60,000. The tail’s probability, not just the comparison with the threshold, determines which observations belong in ES.

With a continuous loss distribution, ES equals expected loss conditional on exceeding VaR. For discrete distributions, include only the probability mass needed at the boundary to fill the specified tail. Simply averaging every loss at or above VaR can give the wrong answer. (Acerbi and Tasche, expected shortfall)

Why did the Basel market-risk framework move to ES?

ES is a risk measure, not a fourth simulation method. Historical simulation, Monte Carlo and suitable analytical models can all estimate ES. Changing from VaR to ES changes the statistic extracted from the loss distribution; it does not dictate a single modelling technique. (BCBS, MAR33)

The Basel Committee identified better capture of tail risk as a reason for moving from VaR to ES in its fundamental review of the trading book. (BCBS, tail-risk rationale) The revised market-risk standard uses ES in its internal models approach, alongside separate treatment of non-modellable risk factors. (BCBS, revised market-risk standard)

Basel’s MAR33 specifies one-tailed 97.5% ES, calibration to stress and liquidity horizons from 10 to 120 days. Its base ten-day ES must use ten-day risk-factor changes, without scaling from a shorter horizon. This is more specific than multiplying one-day VaR by √10. VaR also remains relevant to backtesting; the change did not eliminate every regulatory use of VaR. (BCBS, MAR33)

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