Internal models: expected shortfall · RiskAtoms Academy

By the end you will be able to

1Explain why the FRTB replaced 99% VaR with 97.5% expected shortfall, and compute ES from historical scenarios.
2Apply liquidity horizons to ES with the cascade of MAR33.4.
3Calibrate ES to stress with a reduced set of risk factors, the ratio and its floor of 1, and build IMCC with ρ = 0.5.
4Work the whole calculation on RiskAtoms Bank plc's desk and say which floor or maximum binds.

Why it matters

On 31 December 2025 RiskAtoms Bank plc's trading desk has a ten-day 99% VaR of £5.04m. Under the old internal models rules that number, scaled, was the capital. Under the FRTB it is only the first rung. The ten-day ES at 97.5% is £5.41m; stretched for slow-moving risk factors, £6.57m; calibrated to stress, £7.29m; with diversification across risk classes only half believed, £12.58m; and times the multiplier of 1.5, £18.87m. Each step is a rule with a reason, and this lesson takes them one at a time.

Core explanation

From one quantile to a stressed, horizon-weighted average

L4.3.02: The sensitivities-based method: delta and vega to L4.3.04: Default risk charge and residual risk add-on built the standardised approach, which every bank must be able to run. A bank may instead ask its supervisor to let some desks use the internal models approach. Approval is desk by desk and depends on two tests that L4.3.07: Backtesting and the P&L attribution test covers. This lesson is the model itself: the expected shortfall part, for risk factors with enough data to be modelled.

Illustrative dataRiskAtoms Bank plc's trading desk under the internal models approach

The same made-up Markets desk as Module 3.2: eight positions on four desks (Equities, Rates, FX and commodities, Credit), eight risk factors and 750 days of seeded history with a stressed spell, every position revalued in full. Figures are in £ millions at 31 December 2025.

Every internal-model figure is illustrative. The UK internal model approach for market risk starts on 1 January 2028 (PRA PS1/26, para 1.24), and the bank has no model permission: its capital is the standardised approach of the earlier lessons. The pages show what the bank's models would give if it had one.

This page uses the whole desk, all four of its desks together. L4.3.08: Total market risk capital keeps only the desks that pass the tests.

Why expected shortfall at 97.5%

A VaR is one quantile (M-C09: Quantiles): it says how often a loss is exceeded, not how large the losses beyond it are. The Basel Committee's 2016 market risk standard moved to an expected shortfall, which “will help to ensure prudent capture of ‘tail risk’” and keep capital adequate in stress (press release, 14 January 2016). The confidence level is 97.5% (MAR33.3), and at that level, for a normal distribution, the number barely changes: ES at 97.5% is 1.005 times VaR at 99% (L2.5.08: Expected shortfall). For a fat-tailed book it is more (M-C08: Skewness and fat tails). On the desk's last 250 ten-day P&Ls, VaR at 99% is the 3rd worst loss, £5.04m, and ES at 97.5% is the average of the 7 worst, £5.41m: 1.072 times the VaR. The rules do not prescribe the method: historical simulation, Monte Carlo or a parametric model may be used if it meets the standards (MAR33.9), the freedom L3.2.11: Monte Carlo VaR described. This desk uses historical simulation.

The empirical distribution (M-D02: Histograms and empirical distributions) of the 250 overlapping ten-day P&Ls ending on days 500 to 749, today's positions revalued in full. Red: the 7 losses the ES averages, from £7.44m down to £4.38m.

Liquidity horizons and the cascade

A ten-day loss assumes every position can be exited in ten days. L3.2.17: Market liquidity risk showed why that fails for less liquid markets, and L3.2.12: Expected shortfall in practice built the liquidity horizon cascade on this desk. The rules fix the horizon by risk factor category (MAR33.12, Table 2). The desk's eight factors map like this:

Risk factor	Category (MAR33.12)	Horizon	Reduced set
UK equity index	Equity price, large cap	10 days	yes
US equity index	Equity price, large cap	10 days	yes
GBP 2-year swap rate	Interest rate, specified currency (GBP)	10 days	yes
GBP 10-year swap rate	Interest rate, specified currency (GBP)	10 days	yes
GBP/USD (dollars per pound)	FX rate, specified currency pair	10 days	yes
Brent crude	Energy price	20 days	yes
UK index implied volatility	Equity volatility, large cap	20 days	no
Investment-grade credit spread	Credit spread, corporate IG	40 days	yes

Sort itWhich liquidity horizon?

Place each risk factor category in its MAR33.12 horizon.

GBP interest rate
Large-cap equity price
Brent crude price
Large-cap equity volatility
Investment-grade corporate spread
High-yield corporate spread
Sovereign IG spread

Today, the ten-day ES of the whole desk is £5.406m. Shocking only Brent, the UK volatility and the credit spread (the factors of 20 days or more) gives £3.615m, and the credit spread alone £0.657m. The cascade adds them in quadrature with weights √((LH_j − LH_j−1)/10): £6.569m, 21.5% above the plain ten-day ES. The 60 and 120-day steps are empty on this desk.

Calibrating to stress

Basel 2.5 added a stressed VaR next to the VaR, so that a calm data window could not shrink capital (L2.5.01: Measuring volatility, L2.5.07: Historical VaR). The FRTB keeps one measure but calibrates it to a stress period: the worst 12 months for the bank's current portfolio, found in an observation horizon that goes back at least to 2007 (MAR33.7). Many risk factors have no reliable history that far back, so the bank names a reduced set that does, which must explain at least 75% of the full model's ES (MAR33.5). Then:

\text{ES} = \text{ES}_{R,S} \times \max\!\left(\frac{\text{ES}_{F,C}}{\text{ES}_{R,C}},\ 1\right)

The reduced set's stressed ES, scaled by how much more the full set sees than the reduced set today, with that ratio floored at 1 (MAR33.6). The desk's history is 750 days (three years), not back to 2007, so the search runs over every 250-day window in it. The chart shows the reduced-set ES of each window.

Reduced set (no UK vol)Full set

Windows that include the desk's stressed spell (days 480 to 539) jump. The worst is days 317 to 566, at £7.289m; the calmest, ending on day 464, gives £4.09m. Today's window (days 500 to 749) still holds most of the spell, so today and the stress period are close.

Today the reduced set gives £7.225m and the full set £6.569m. The reduced set explains 110.0%, comfortably over 75%, but the ratio is 0.909, below 1. Leaving out the volatility factor removes a hedge: the desk's long puts gain when volatility jumps in a sell-off. The floor of 1 binds, so the stressed ES is simply £7.289m. The rule never lets a missing factor reduce capital.

IMCC: only half the diversification

This is the limit on diversification across risk classes that L3.2.13: Scaling and aggregating risk promised. A bank-wide ES lets the model use its own correlations across risk classes (MAR33.13), and those are the correlations most likely to break in a crisis. So the rules also run the stressed ES for each of the five broad risk classes alone, the partial ES, and add them up with no diversification at all. IMCC is the average of the two, with weight ρ = 0.5 (MAR33.15). On the desk the partials sum to £17.870m against £7.289m for the whole, and IMCC is £12.579m.

£m. Each stressed ES is the stressed reduced-set ES times max(full ÷ reduced today, 1); only equity has a factor outside the reduced set
Broad risk class	ES today, full	ES today, reduced	Stressed, reduced	Stressed ES
Interest rate	3.191	3.191	5.668	5.668
Credit spread	1.315	1.315	1.582	1.582
Equity	4.750	5.279	5.330	5.330
FX	0.568	0.568	0.568	0.568
Commodity	4.722	4.722	4.722	4.722
Sum of partials				17.870

Three questions about the stressed calibration

The 60-day average and the multiplier

Capital is the larger of yesterday's IMCC and the average of the last 60 business days times the multiplier m_c (MAR33.41), plus the stress scenario capital for factors that cannot be modelled (L4.3.06: Non-modellable risk factors and the internal DRC). The multiplier starts at 1.5 and rises with backtesting exceptions (L4.3.07: Backtesting and the P&L attribution test). On the desk, the 60-day average of IMCC equals today's £12.579m, because the positions are held constant, the floor of 1 binds and the stress window is the same on each of the 60 days (a feature of the illustration, not of the rule), so the averaged limb wins: 1.5 × 12.579 = £18.869m. The old rules multiplied a 99% VaR by at least 3 (L2.5.06: Value at risk: the idea); the new floor of 1.5 sits on a measure that is already stressed.

The formal version

With T = 10 days, LH_j the liquidity horizons 10, 20, 40, 60 and 120 days, P the portfolio, L_(i) the i-th worst ten-day loss, F and R the full and reduced sets of risk factors, C the current 12 months and S the stress period, C_i the broad risk classes, and SES the stress scenario capital of L4.3.06: Non-modellable risk factors and the internal DRC (MAR33.3–33.6, 33.15, 33.41; PRA IMA Part Art 325ba–325bc):

\begin{aligned} \text{ES}_T(P) &= \frac{1}{k}\sum_{i=1}^{k} L_{(i)}, \qquad k = \lceil n \times 2.5\% \rceil \\ \text{ES} &= \sqrt{\text{ES}_T(P)^2 + \sum_{j=2}^{5}\left(\text{ES}_T(P,j)\sqrt{\frac{LH_j - LH_{j-1}}{T}}\right)^2} \\ \text{IMCC}(C) &= \text{ES}_{R,S} \times \max\!\left(\frac{\text{ES}_{F,C}}{\text{ES}_{R,C}},\ 1\right), \qquad \frac{\text{ES}_{R,C}}{\text{ES}_{F,C}} \ge 75\% \\ \text{IMCC} &= \rho\,\text{IMCC}(C) + (1-\rho)\sum_{i=1}^{5}\text{IMCC}(C_i), \qquad \rho = 0.5 \\ C_A &= \max\!\big(\text{IMCC}_{t-1} + \text{SES}_{t-1},\ m_c\cdot\text{IMCC}_{\text{avg}} + \text{SES}_{\text{avg}}\big), \qquad m_c \ge 1.5 \end{aligned}

ES_T(P, j) is the ten-day ES with only the factors whose horizon is at least LH_j shocked, the others held at today's values. Averages are over the previous 60 business days. The 75% test is an average over the previous 12 weeks in Basel (MAR33.5).

Try it

Give the corporate spread a 60-day horizon, as if the bonds were high yield, and see the cascade grow. Put the UK volatility into the reduced set and watch the ratio go to exactly 1. The stress window moves too, to days 392–641, and IMCC falls to £11.952m. Then move ρ to 0, the rule with no credit for diversification across classes at all.

Try itBuild the IMCC

Corporate spread liquidity horizon

Equity index liquidity horizon

Reduced set of risk factors

Weight ρ on the unconstrained ES0.50

Backtesting exceptions (bank-wide, 250 days)0 → m_c 1.50

Stress window (days)317–566

Reduced set ÷ full set, today110.0% (≥ 75%: allowed)

ES full set, today (ES_F,C)£6.569m

ES reduced set, today (ES_R,C)£7.225m

ES reduced set, stress (ES_R,S)£7.289m

Ratio ES_F,C ÷ ES_R,C0.909 → floor of 1 binds

Unconstrained stressed ES£7.289m

Sum of the five partial ES£17.870m

IMCC£12.579m

m_c × IMCC (averaged limb)£18.869m

Stressed ES of each broad risk class on its own

Interest rate

£5.668m

Credit spread

£1.582m

Equity

£5.330m

FX

£0.568m

Commodity

£4.722m

Ten-day overlapping P&L, full revaluation, 97.5% ES as the average of the 7 worst of 250 days. Today is the window ending on day 749; the stress window is the 250 days, anywhere in the 750-day history, that give the reduced set its largest ES. The averaged limb leaves out SES (L4.3.06), and IMCC is taken as constant over the 60 days, as it is on the book. Illustrative, £ millions.

Worked example

RiskAtoms Bank plc's IMCC, step by step

The whole desk on 31 December 2025, £ millions, illustrative. Today's window is days 500 to 749.

From ten-day losses to the averaged limb

Step 1 of 7

1
ES at 97.5%
$\text{ES}_T(P) = \tfrac{1}{7}\big(7.44 + 6.42 + 5.04 + 4.97 + 4.85 + 4.74 + 4.38\big) = \text{£}5.406\text{m}$

Regulatory lens

Basel: MAR33.1–33.15 and 33.41–33.42, in force in the Basel Framework since 1 January 2023. Status as at 3 November 2026 as at 3 Nov 2026.

In the UK

The internal model approach of the Basel 3.1 rules starts on 1 January 2028, a year after the rest (PRA PS1/26, para 1.24); until then firms keep their existing VaR-based model permissions (paras 3.4 and 3.10). The ES, the cascade and IMCC with ρ = 50% are in the PRA Rulebook's Market Risk: Internal Model Approach (CRR) Part, Articles 325bb to 325bd. The PRA applies the 75% reduced-set test at the level of the whole portfolio only (PS17/23, para 3.37). Its consultation CP9/26 (19 June 2026, closed 18 September 2026) proposes an alternative coverage test to the 75% ratio, subject to notification (para 2.34); no final policy had been published as at 3 November 2026.

In the EU

The same measure is in CRR Articles 325ba to 325bd. The market risk rules apply from 1 January 2027, after two postponements by delegated acts (Delegated Regulation (EU) 2026/1221, recital 2). Until 31 December 2029, CRR Article 495m, inserted by that regulation, lets banks take the averages in Article 325ba(1)(b) over 12 weeks instead of 60 business days.

In the US

The agencies' proposals of 19 March 2026 would replace the VaR-based measure in the market risk rule with an expected shortfall-based measure that reflects different liquidity horizons (Federal Reserve fact sheet, 19 March 2026). Comments closed on 18 June 2026; no final rule had been issued as at 3 November 2026.

Spark Atoms

Spark AtomAbout 40% more capital, on average

When the Basel Committee issued the FRTB in January 2016 it estimated that the new framework would raise total market risk capital by a median of about 22%, and a weighted average of about 40%, against the rules then in force (press release, 14 January 2016). The 2019 revision was informed by a new impact study, on data as of end-December 2017.

Spark Atom97.5% sits close to 99%

For a normal distribution ES at 97.5% is 1.0049 times VaR at 99%, so the switch leaves a thin-tailed book's risk measure about where it was and charges a fat-tailed one more. On this desk the ratio is 1.072: the seven worst ten-day losses include two of over £6m.

Key takeaways

The IMA measures risk with ES at 97.5% on a ten-day base: the average of the 7 worst of 250 scenarios. On the desk, £5.41m against a 99% VaR of £5.04m.
Liquidity horizons of 10 to 120 days enter through the cascade; the desk's 20 and 40-day factors add 21.5%.
Capital is calibrated to the worst 12 months since 2007 with a reduced set of factors, scaled by today's full-to-reduced ratio, floored at 1.
IMCC averages the unconstrained ES and the sum of the five partial ES with ρ = 0.5: £12.58m on the desk, against £7.29m unconstrained.
C_A is the larger of yesterday's IMCC + SES and m_c × the 60-day average + SES, with m_c from 1.5 to 2.0: £18.87m before SES.
Every IMA figure here is illustrative: the UK IMA starts on 1 January 2028 and the bank has no permission.

Reaction

Check what stuck

Question 1 · objective 1

A desk's ES at 97.5% is estimated by historical simulation on 250 ten-day scenarios. How many worst losses are averaged?

Question 2 · objective 2

A portfolio's ten-day ES is £4m. The factors with a liquidity horizon of 20 days or more give a ten-day ES of £2m, and those of 40 days or more £1m. Nothing has a longer horizon. What is the cascaded ES, in £m (two decimals)?

£m

Question 3 · objective 3

The reduced set gives ES of £9m in the stress period and £5m today. The full set gives £6m today. What is the stressed ES used for capital, in £m?

£m

Question 4 · objective 3

A bank's unconstrained stressed ES is £10m. The five partial stressed ES are £6m, £4m, £3m, £1m and £0m. What is its IMCC, in £m?

£m

Question 5 · objective 4

On RiskAtoms Bank plc's desk the reduced set (all factors but the UK index volatility) gives a larger ES today than the full set. Why?

AThe volatility factor hedges the desk: the long puts gain when volatility rises in a sell-off, so leaving it out removes a hedgeBThe reduced set always gives the larger ES, by constructionCThe rules add a 25% buffer to the reduced setDVolatility has a 20-day liquidity horizon, which lowers the full-set ES

Glossary

Internal models approach (IMA): Internal models approach (IMA): the FRTB route that lets a bank use its own model for market risk capital, desk by desk, once the supervisor approves each desk (MAR30–MAR33).
Expected shortfall (97.5%): Expected shortfall at 97.5%: the average loss in the worst 2.5% of outcomes. On 250 historical scenarios, the average of the 7 worst.
Liquidity horizon: Liquidity horizon: the time assumed to exit or hedge a risk factor in stressed markets, 10, 20, 40, 60 or 120 days, set by category in MAR33.12 Table 2.
Reduced set: Reduced set of risk factors: the factors that have a long enough history to be run through a stressed period back to 2007. It must explain at least 75% of the full model's ES.
Stress period: Stress period: the 12-month window, from an observation horizon going back at least to 2007, in which the reduced set gives the portfolio its largest ES (MAR33.7).
Full-to-reduced ratio: The ratio ES_F,C ÷ ES_R,C: today's full-set ES over today's reduced-set ES. It scales the stressed reduced-set ES up to the full set, and is floored at 1.
Partial ES: Partial ES: the ES of one broad risk class (interest rate, credit spread, equity, FX, commodity) with every other factor held still (MAR33.14).
IMCC: IMCC: the capital for modellable risk factors, ρ times the unconstrained stressed ES plus (1 − ρ) times the sum of the five partial stressed ES, ρ = 0.5 (MAR33.15).
Multiplier m_c: Multiplier m_c: 1.5, plus 0 to 0.5 set by the bank-wide backtesting exceptions (MAR32.9, MAR33.42).

Deep Dives & Origins

Deep Dives

Basel Committee on Banking Supervision, Minimum capital requirements for market risk, January 2019 (d457), and its explanatory note: the standard behind MAR30–MAR33. Advanced.
Prudential Regulation Authority, CP9/26: Basel 3.1: Adjustments to the internal model approach (IMA) for market risk, June 2026: what the UK may still change before 2028. Intermediate.

Origins

Basel Committee on Banking Supervision, MAR33, paras 33.1–33.15 and 33.41–33.42, bis.org (read in the Swiss SIF copy for the formulas).
Basel Committee on Banking Supervision, press release, “Revised framework for market risk capital requirements issued by the Basel Committee”, 14 January 2016, and Minimum capital requirements for market risk (d352), January 2016.
Prudential Regulation Authority, PRA Rulebook: Market Risk: Internal Model Approach (CRR) Part, Articles 325ba–325bd, version in force from 1 January 2028.
Prudential Regulation Authority, PS1/26: Implementation of Basel 3.1: Final rules, January 2026, paras 1.24 and 3.4; PS17/23, December 2023, para 3.37; CP9/26, June 2026.
Commission Delegated Regulation (EU) 2026/1221 of 4 June 2026, recital 2 and Article 495m.
Board of Governors of the Federal Reserve System, press release and fact sheet, 19 March 2026.

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