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FRM Exam Part II · Fundamental Review of the Trading Book

FRTB Internal Models Approach and 97.5% Expected Shortfall

Updated 11 October 2026 · Fact-checked

Under the FRTB internal models approach, banks replace 99% VaR with 97.5% expected shortfall, the average loss beyond the 97.5th percentile. ES is scaled for liquidity horizons of 10 to 120 days and calibrated to a stress period. Capital is computed from desk-level ES, with non-modellable risk factors added separately.

Understand Internal Models Approach and Expected Shortfall

Before FRTB, Basel 2.5 used 10-day 99% VaR plus stressed VaR. VaR says nothing about losses beyond the cutoff. A bank could have a tame VaR and a huge tail. The internal models approach (IMA) in FRTB fixes this by using expected shortfall (ES).

ES at 97.5% is the average loss in the worst 2.5% of outcomes. For a normal distribution, 97.5% ES is close to 99% VaR (about 2.34 standard deviations versus 2.33). So switching the confidence level alone is roughly neutral for normal data. Capital still changes through liquidity horizons and stress calibration. For fat-tailed data, ES rises relative to VaR, because ES responds to tail size and VaR does not. ES is also a coherent risk measure; VaR is not always subadditive.

Basel also replaced the flat 10-day horizon with liquidity horizons. Each risk factor is assigned to a liquidity horizon: 10, 20, 40, 60 or 120 days. This is the time needed to exit or hedge the position in stress without moving the market. Large-cap equity price and major interest rates sit at 10 days. Credit spreads of lower quality names, or some exotic or complex volatilities, sit at much longer horizons such as 60 or 120 days.

The ES is built in steps. You compute a base 10-day ES using all risk factors, then ES with only factors whose horizon is at least 20 days, at least 40 days, and so on. Each is scaled by a square root of time factor and combined. Then the ES is calibrated to stress: the reduced-set stressed ES is multiplied by the ratio of full-set current ES to reduced-set current ES, with the ratio floored at 1. Each ES term in this calibration is itself the liquidity-adjusted ES, computed on the stated factor set (full or reduced) and period (current or stress). The stress period is the most stressful 12-month period for the portfolio since 2007. The reduced factor set used for the stressed ES must explain at least 75% of the full-set ES.

Only desks approved by backtesting and P&L attribution tests may use IMA. Failing desks fall back to the standardised approach. For the approved desks, the aggregate ES capital is a combination of two measures. One is an unconstrained ES across all IMA-approved desks together. The other is the sum of constrained ES measures, one for each broad risk class. The aggregate, called the internally modelled capital charge (IMCC), gives a 0.5 weighting to the unconstrained ES and the other 0.5 to the constrained sum.

The bank's capital requirement is then the higher of two figures. One is the previous day's IMCC plus SES. The other is mc × the 60-day average IMCC plus the 60-day average SES. mc is a multiplier of at least 1.5 that depends on backtesting results. The stress scenario capital requirement (SES) covers non-modellable risk factors (NMRFs). On top of this comes a separate, model-based default risk charge (DRC). Total IMA capital is also subject to the output floor against the standardised approach.

Key formulas to remember

Expected shortfall at 97.5%
ES(97.5%) = average of losses beyond the 97.5th percentile loss
Worst 2.5% tail. Basel uses 97.5% ES instead of 99% VaR.
Liquidity-horizon scaled ES
ES = √[ ES(P, T)² + Σ over j≥2 ( ES(P, Qj) × √((LHj − LHj−1) ÷ T) )² ]
T = 10 days. ES(P, Qj) uses only factors with horizon at least LHj. LH values: 10, 20, 40, 60, 120 days.
Stressed calibration
ES = ES(R,S) × max[ ES(F,C) ÷ ES(R,C), 1 ]
R = reduced factor set, F = full set, S = stress period, C = current period. Each ES term is the liquidity-adjusted ES computed on that factor set and period. The ratio is floored at 1. The reduced set must explain at least 75% of the full-set ES.
Internally modelled capital charge (IMCC)
IMCC = 0.5 × unconstrained ES + 0.5 × Σ risk-class constrained ES
ES terms are over IMA-approved desks only (desks that pass backtesting and P&L attribution).
IMA capital
Capital = max[ IMCC(t−1) + SES(t−1), mc × IMCC(avg) + SES(avg) ] + DRC
(t−1) = previous day's figure. avg = 60-day average. mc is the backtesting-based multiplier, at least 1.5. SES is the stress scenario capital requirement for NMRFs. DRC is the separate default risk charge. Total IMA capital is also subject to the output floor.

How to solve Internal Models Approach and Expected Shortfall questions

Use this order for any IMA or ES question. It keeps the Basel terms straight and the arithmetic short.

  1. 1Identify what is asked: a risk measure comparison, a liquidity horizon assignment, a scaling calculation, or a capital or eligibility question.
  2. 2State the measure: 97.5% ES, 10-day base horizon, not 99% VaR.
  3. 3For tail-average questions, count the tail observations (2.5% of scenarios) and average the losses in the tail.
  4. 4For liquidity horizons, assign each risk factor to its horizon bucket (10, 20, 40, 60 or 120 days) and note which factors remain in each nested ES.
  5. 5For scaling, apply √(ΔLH ÷ 10) to each incremental ES, square each term, sum, and take the square root.
  6. 6For stressed calibration, compute the ratio of full-set current ES to reduced-set current ES, floor at 1, and multiply by reduced-set stressed ES.
  7. 7Check desk eligibility and NMRFs: failing backtesting or P&L attribution sends the desk to the standardised approach, and NMRFs get a separate stress capital charge.
  8. 8Sanity-check the arithmetic and the interpretation. The liquidity-adjusted ES will typically be larger than the base 10-day ES, but check your working rather than assume it.

Quickest way: Shortcut for tail average and horizon scaling

When to use it: When the question gives you a list of scenario losses or a few ES numbers and asks for a single figure.

  1. For tail average: take 2.5% of the scenario count, pick the worst losses of that number, and average them. Do not use the cutoff value alone.
  2. For horizon scaling: if only one extra horizon exists, ES = √(A² + (B × √(ΔLH ÷ 10))²).
  3. If a factor has a 10-day horizon only, no scaling term is added.
  4. For calibration, look at the ratio first. If it is below 1, treat it as 1.
  5. Eliminate options that mention 99% VaR as the IMA measure or a horizon outside 10, 20, 40, 60, 120.

Common mistakes in Internal Models Approach and Expected Shortfall

  • Saying FRTB uses 99% VaR with a longer horizon.

    Basel 2.5 and earlier rules used 99% VaR, so old habits stick.

    Fix: Remember that IMA uses 97.5% ES. Under normality it is roughly equal to 99% VaR, but it captures tail size.

  • Using the 97.5th percentile loss (the VaR) as the ES, or averaging the wrong observations.

    Students confuse ES with VaR.

    Fix: ES is the mean of losses beyond the cutoff. It is always at least as large as VaR at the same confidence.

  • Multiplying the whole ES by √(LH ÷ 10) for a single horizon.

    Mimics the old square-root-of-time rule for VaR.

    Fix: Use the cascading formula: each incremental horizon step ΔLH scales its own ES, then combine in root sum of squares.

  • Letting the stress ratio fall below 1.

    If the current ratio is below 1 students use it as calculated.

    Fix: The multiplier is max(ratio, 1). Stress calibration can only raise capital.

  • Applying one ES to the whole bank and ignoring desk approval.

    Confusing the approach with a firm-wide VaR model.

    Fix: IMA approval is desk by desk. A desk failing backtesting or P&L attribution reverts to the standardised approach.

  • Treating non-modellable risk factors as part of ES.

    All risk factors look alike in a model.

    Fix: NMRFs fail the modellability test on observability. They carry a separate capital charge, the stress scenario capital requirement (SES).

Worked examples

Example 1

A desk has 400 simulated 10-day P&L scenarios. The worst 10 losses (₹ crore) are 14, 12, 11, 10, 9, 9, 8, 8, 7, 7. Estimate the 97.5% expected shortfall from these data.

Show the solution
  1. The tail is 2.5% of 400 scenarios = 10 scenarios.
  2. The worst 10 losses are those listed.
  3. Sum: 14 + 12 + 11 + 10 + 9 + 9 + 8 + 8 + 7 + 7 = 95.
  4. Average: 95 ÷ 10 = 9.5.

Answer: 97.5% ES = ₹9.5 crore.

Example 2

A desk has 10-day ES with all factors of $30 million. ES using only factors with liquidity horizon of at least 20 days is $20 million. No factor has a horizon above 20 days. Compute the liquidity-adjusted ES.

Show the solution
  1. Base term: ES(P,T)² = 30² = 900.
  2. Incremental horizon: LH of 20 vs previous 10 gives ΔLH = 10.
  3. Scaling factor: √(10 ÷ 10) = 1, so term = 20 × 1 = 20 and squared = 400.
  4. Terms for horizons of 40, 60 and 120 days are zero, because no factors have those horizons.
  5. Sum = 900 + 400 + 0 + 0 + 0 = 1,300.
  6. ES = √1,300 ≈ 36.06.

Answer: About $36.06 million.

Exam tips

  • Know the numbers cold: 97.5%, 10 days base, horizons 10, 20, 40, 60, 120, stress period since 2007.
  • Expect conceptual questions on why ES is preferred: it captures tail severity and is coherent. Do not say it is always higher than VaR at a different confidence level; say it is at least VaR at the same level.
  • Watch the stress-ratio floor of 1 and which factor set (reduced vs full) goes in each part of the ratio.
  • Link IMA to desk eligibility: backtesting and P&L attribution decide who may use the model.
  • In calculations, square, add, then root. Do not add the ES figures directly.

Practice questions from Fundamental Review of the Trading Book

Internal Models Approach and Expected Shortfall in other exams

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

Internal Models Approach and Expected Shortfall: frequently asked questions

Why did FRTB move from 99% VaR to 97.5% expected shortfall?

VaR ignores losses beyond its cutoff and is not always subadditive. ES averages the tail and is coherent. At 97.5% it gives a similar tail measure to 99% VaR for normal data, but a larger one for fat tails.

What are FRTB liquidity horizons?

They are the time, in days, assumed necessary to exit or hedge a risk factor in stress. The set is 10, 20, 40, 60 and 120 days. Less liquid factors get longer horizons and so higher capital.

How is IMA capital calculated?

A bank computes liquidity-adjusted, stress-calibrated ES for approved desks. It combines them into the IMCC: 0.5 × unconstrained ES + 0.5 × the sum of risk-class constrained ES. Capital is the higher of the previous day's IMCC plus SES, and mc × the 60-day average IMCC plus the 60-day average SES. Here mc is a backtesting-based multiplier of at least 1.5 and SES is the stress scenario capital requirement for non-modellable risk factors. A separate default risk charge is then added. Desks that fail backtesting or P&L attribution use the standardised approach.

What is stressed calibration in FRTB?

The ES is calibrated to the most stressful 12-month period for the portfolio since 2007. Banks measure ES on a reduced factor set with a long data history in the stress period, and the reduced set must explain at least 75% of the full-set ES. They then multiply it by max(full-set current ES ÷ reduced-set current ES, 1).