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FRM Part I · FRM Exam Part I

Measures of Financial Risk for FRM Part I

Measures of financial risk turn a loss distribution into numbers you can compare and limit. VaR gives a loss quantile, expected shortfall averages losses beyond it, and coherence tests judge whether a measure behaves sensibly. To solve questions, identify the confidence level, the distribution and the horizon, then apply the formula.

What this chapter covers

This chapter asks one question: how do you summarise the possible losses of a portfolio in a few numbers? You start with Value at Risk (VaR), the loss that is not expected to be exceeded at a given confidence level over a given horizon. You then move to Expected Shortfall (ES), which looks at the average loss in the tail beyond VaR. After that you study the properties a good risk measure should have, and how VaR and ES fare against them.

The middle of the chapter is more theoretical. Coherent risk measures must satisfy monotonicity, translation invariance, positive homogeneity and subadditivity. Spectral and distortion measures generalise ES by weighting different parts of the tail. Standard deviation and other measures, such as semi-deviation and downside measures, are compared with VaR and ES. The chapter closes with how quantiles are estimated from data and how large the estimation errors are.

This chapter connects to the whole paper. Quantitative Analysis supplies the normal distribution, quantiles and sampling error. Valuation and Risk Models builds on it with historical simulation, parametric VaR, and backtesting. Foundations of Risk Management uses these measures when discussing capital and limits. If you understand this chapter well, those later topics become much easier.

VaR and ES are the core language of market risk, so the exam returns to them in several forms: direct calculations, conceptual comparisons and links to other chapters. Calculation questions are usually quick marks if you know the formulas, such as VaR = μ + z × σ for losses under a normal assumption, or VaR = z × σ when the mean is zero. Conceptual questions, like whether VaR is subadditive, are easy to lose if you only memorise definitions. Time spent here pays off across a 100-question, 4-hour paper, and the ideas help you in later topics too. GARP publishes no weightage in marks, so use the annual Study Guide and Learning Objectives to confirm what is in scope.

Measures of Financial Risk: topics in the order to study them

  1. 1Value at Risk (VaR) BasicsEverything else in the chapter is defined relative to VaR, so learn the definition, confidence level, horizon and the normal-distribution formula first.
  2. 2Expected Shortfall (ES)ES is the natural next step: it fixes VaR's blind spot by averaging tail losses, and you can compare the two side by side.
  3. 3Standard Deviation and Other Risk MeasuresOnce you know VaR and ES, comparing them with standard deviation and downside measures is easy and shows why tail measures were developed.
  4. 4Coherent Risk Measures and SubadditivityYou need VaR, ES and standard deviation in mind to test each against the four coherence axioms and see why VaR can fail subadditivity.
  5. 5Spectral and Distortion Risk MeasuresThese generalise ES using weights on quantiles, so they make sense only after you know ES and coherence.
  6. 6Estimating VaR and ES Quantiles and ErrorsFinish with estimation, since it applies the earlier measures to real data and needs sampling-error ideas from Quantitative Analysis.

How to prepare Measures of Financial Risk

Treat this chapter as a short calculation toolkit plus a set of properties to compare. Practise both, and do not skip either.

  1. Learn the VaR definition in words first: a loss quantile at a stated confidence level and horizon. Then drill the normal formula, VaR = μ + z × σ for losses, with z = 1.645 at 95% and 2.326 at 99% for one-tailed normal quantiles.
  2. Practise horizon scaling. Under the usual assumptions of independent returns with zero mean, multiply by √T, for example 10-day VaR = 1-day VaR × √10. Note the conditions each time you apply it.
  3. Compute ES for simple discrete loss distributions by hand: find the tail outcomes beyond VaR and take their probability-weighted average. Check that ES is never below VaR at the same confidence level.
  4. Build a table of measures against the four coherence properties, and know which ones VaR, ES and standard deviation satisfy. Create one counterexample for VaR subadditivity that you can recall quickly.
  5. Read spectral and distortion measures as weighted averages of quantiles. Remember that a spectral measure is coherent when its weights are non-negative and increase with the loss severity.
  6. For estimation, practise standard error ideas: more observations and less extreme quantiles give more precise estimates, and extreme tail quantiles have larger errors. Finish with mixed timed question sets.

Common mistakes in Measures of Financial Risk

  • Using the wrong z-value or the wrong tail, for example 1.96 for a one-tailed 95% VaR.

    Fix: Decide first whether the question is one-tailed. VaR is a one-tailed loss quantile, so 95% uses 1.645 and 99% uses 2.326.

  • Scaling VaR by √T without checking the conditions.

    Fix: Use it only when returns are independent and identically distributed with zero mean, as the question implies. Otherwise read the stated assumptions.

  • Saying VaR is coherent or that it is never subadditive.

    Fix: Remember: VaR can violate subadditivity in general, though it is subadditive for normal (elliptical) distributions. ES is coherent.

  • Computing ES by averaging the wrong outcomes or mixing up the tail probability.

    Fix: List the tail outcomes, weight each by its probability, and divide by the total tail probability (for example 5%). Check that ES ≥ VaR.

  • Treating standard deviation as a complete risk measure for non-normal losses.

    Fix: Remember that it ignores tail shape and skew. Use the comparison with VaR and ES when questions mention fat tails or asymmetric payoffs.

  • Assuming estimated tail quantiles are as reliable as central ones.

    Fix: Link precision to sample size and quantile position. Extreme confidence levels have few observations beyond them, so standard errors are larger.

Last-day revision: Measures of Financial Risk

  • VaR is a loss quantile at a given confidence level and horizon; it says nothing about losses beyond that point.
  • Normal VaR (loss) = μ + z × σ; with zero mean, VaR = z × σ.
  • One-tailed normal z values: 1.645 at 95% and 2.326 at 99%.
  • Scaling to T days with iid returns and zero mean: VaR(T) = VaR(1) × √T.
  • ES is the expected loss given that the loss exceeds VaR; at the same confidence level ES ≥ VaR.
  • Coherent means monotonicity, translation invariance, positive homogeneity and subadditivity.
  • Subadditivity: risk of a combined portfolio ≤ sum of the separate risks; it reflects diversification.
  • VaR is not always subadditive; ES is coherent.
  • Spectral measures are weighted averages of quantiles; they are coherent if the weights are non-negative and non-decreasing in the tail.
  • Standard deviation treats gains and losses alike and is not a tail measure; it is not coherent in general.
  • Quantile estimates are less precise further into the tail and with fewer observations.
  • Check whether a question gives losses or returns before choosing the sign and tail.

Measures of Financial Risk practice questions

Measures of Financial Risk in other exams

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

Measures of Financial Risk: frequently asked questions

What is the difference between VaR and expected shortfall?

VaR is the loss threshold at a chosen confidence level. Expected shortfall is the average loss given that the loss is beyond that threshold. At the same confidence level, ES is at least as large as VaR and captures tail severity.

Why is subadditivity important for a risk measure?

Subadditivity means combining portfolios cannot increase measured risk beyond the sum of the parts, which matches the idea of diversification. A measure that breaks it can make a merged portfolio look riskier than its parts, which distorts limits and capital allocation.

Do I need to memorise normal z-values for the exam?

Yes, the common one-tailed values are worth knowing: 1.645 at 95% and 2.326 at 99%. Check each question's wording, since the exam may give a value or ask you to use another confidence level.

How should I study this chapter in limited time?

Start with VaR and ES calculations, then learn the coherence properties and which measures satisfy them. Spectral measures and estimation errors then become short additions. Finish with timed practice questions and review the current GARP Study Guide and Learning Objectives.