Expected shortfall in practice · RiskAtoms Academy

By the end you will be able to

1Estimate expected shortfall from historical simulation and Monte Carlo, and state it with its confidence level, horizon and currency.
2Read the shape of the tail from the gap between ES and VaR.
3Compare how stable ES and VaR estimates are across windows, seeds and resamples.
4Apply liquidity horizons to ES with the cascade used in the FRTB.

Why it matters

Under the Fundamental Review of the Trading Book, internal-model capital for market risk rests on 97.5% expected shortfall, not 99% VaR. On RiskAtoms Bank's desk the two are close: £2.27m and £2.39m. But 11 of the 19 days behind that ES come from one 60-day spell, and a single day contributes £262k. Is ES a better number, or just a jumpier one?

Core explanation

Averaging the tail

Illustrative dataRiskAtoms Bank plc's Markets desk

One made-up trading book on one made-up valuation day, in pounds: eight positions in four books, eight risk factors and 750 days of simulated factor moves. The levels (UK index 8,000, GBP/USD 1.30, Brent $80 and so on) are round illustrative numbers, not market data. The same book runs through every lesson in Module 3.2, so the figures agree from lesson to lesson.

ES and VaR here are positive one-day losses in pounds at the stated confidence level, from today's eight positions revalued in full over the 750 days of history (L3.2.10: Historical simulation in practice) or the module's Monte Carlo run of 10,000 paths (L3.2.11: Monte Carlo VaR).

ES from historical simulation

L2.5.08: Expected shortfall defined expected shortfall as the average loss beyond the VaR. With 750 days at 97.5%, the tail holds 750 × 2.5% = 18.75 days, rounded up to 19. The 97.5% ES is the average of the 19 worst:

day 488−£4.98m

day 494−£3.44m

day 515−£2.93m

day 383−£2.92m

day 520−£2.90m

day 524−£2.86m

day 486−£2.34m

day 513−£2.27m

day 505−£2.26m

day 60−£2.16m

day 498−£2.08m

day 374−£1.95m

day 134−£1.86m

day 679−£1.82m

day 510−£1.79m

day 483−£1.75m

day 691−£1.74m

day 453−£1.70m

day 207−£1.68m

In the stressed spell (days 480–539)Other days

Their average is £2,391,489, a little above the 99% VaR of £2,274,006 (the 8th worst loss). 11 of the 19 days come from the stressed spell. The worst, day 488 at £4.98m, adds £262k to ES on its own; take it out of the history and ES falls to £2.217m. VaR only knows that the day was worse than the 8th; ES knows how much worse.

ES from Monte Carlo, and the shape of the tail

From the module's simulation the 97.5% ES is the average of the 250 worst of 10,000 paths: £1.852m. Line the methods up:

One day, £	VaR 99%	ES 97.5%	ES 97.5% vs VaR 99%	ES 99% ÷ VaR 99%
Delta-normal	£1.944m	£1.954m	+0.5%	1.146
Monte Carlo, 10,000 paths	£1.848m	£1.852m	+0.2%	1.135
Historical simulation, 750 days	£2.274m	£2.391m	+5.2%	1.354
Monte Carlo, stressed covariance	£4.160m	£4.165m	+0.1%	1.134

For a normal distribution, 97.5% ES and 99% VaR almost coincide (£1.954m against £1.944m), which is why the Basel Committee chose 97.5%. Monte Carlo, built on normal moves, keeps that relationship. The history does not: its ES is 5.2% above its VaR, and its 99% ES is 1.35 times its 99% VaR, against 1.15 for a normal tail. That tail ratio is a quick test for fat tails (M-C08: Skewness and fat tails). The stressed calibration multiplies the Monte Carlo figures by more than two (£4.16m) but leaves the ratio at 1.13: scaling volatilities makes a tail wider, not fatter.

Is ES less stable than VaR?

ES is often said to be noisier, because it averages the most extreme scenarios. Three tests of stability on the desk say it depends on what you compare.

Rolling windows. Over the 500 days on which a 250-day window can be formed, the 99% VaR changes on 13 days with jumps of up to £582k. The 97.5% ES changes more often (30 days, since any of its 7 tail days can enter or leave) but by less: at most £479k. The 99% ES, an average of only 3 days, jumps by up to £1.01m.

250-day VaR 99%250-day ES 97.5%

Seeds. In the seed study of L3.2.11: Monte Carlo VaR, the same runs give these spreads:

Paths per run	Runs	sd of VaR 99%	sd of ES 97.5%	sd of ES 99%	ES 97.5% formula
1,000	160	£98.8k	£82.9k	£110.6k	£85.2k
4,000	40	£50.9k	£39.4k	£49.7k	£42.6k
16,000	10	£25.3k	£19.4k	£24.5k	£21.3k

Resampling history. A bootstrap draws 750 days with replacement from the 750, 2,000 times. The standard deviations: VaR 99% £343k, ES 97.5% £243k, ES 99% £383k.

What the three tests agree on

At the regulatory pairing, 97.5% ES is steadier than 99% VaR: it averages 19 days where VaR rests on one. At the same confidence level ES is the noisier of the two, and the fatter the tail, the bigger the gap: in the bootstrap of the fat-tailed history, ES 99% is 12% more variable than VaR 99%.

Liquidity horizons

A ten-day loss assumes every position can be exited or hedged in ten days. A corporate bond in a stressed market cannot. The FRTB gives each risk factor a liquidity horizon of 10, 20, 40, 60 or 120 days by category, and adds the extra risk of the slower factors in a cascade. An illustrative mapping for the desk:

Factor	Category (illustrative)	Horizon, days
UK index	Equity price, large cap	10
US index	Equity price, large cap	10
2y rate	Interest rate, specified currency	10
10y rate	Interest rate, specified currency	10
GBP/USD	FX rate, specified currency pair	10
Brent	Energy price	20
UK vol	Equity volatility, large cap	20
IG spread	Credit spread, corporate investment grade	40

The book's 10-day ES, scaled here from one day by √10 for illustration (£7.56m; the limits of that scaling are in L3.2.13: Scaling and aggregating risk, and the Basel rules require 10-day moves measured directly, without scaling from a shorter horizon), becomes £8.13m once Brent, UK vol and the credit spread carry their longer horizons: an uplift of 7.5%. If the two equity indices were small-cap stocks (20 days, their volatility 60), it would be £11.03m, 46% more. The worked example below builds the cascade step by step; market liquidity itself is in L3.2.17: Market liquidity risk.

The formal version

Estimators. With losses sorted worst first, $L_{(1)} \ge L_{(2)} \ge \dots$ , and $m = n(1-\alpha)$ the exact tail size:

\begin{aligned} \widehat{\text{ES}}_{\text{simple}} &= \frac{1}{k}\sum_{i=1}^{k} L_{(i)}, \quad k = \lceil m \rceil = 19 \\ &= \text{£}2{,}391{,}489 \\[4pt] \widehat{\text{ES}}_{\text{exact}} &= \frac{1}{m}\Big(\sum_{i=1}^{\lfloor m \rfloor} L_{(i)} \\ &\qquad + (m - \lfloor m \rfloor)\,L_{(\lfloor m \rfloor + 1)}\Big) \\ &= \text{£}2{,}401{,}017 \quad (m = 18.75) \end{aligned}

The second gives the 19th loss a weight of 0.75 so that the tail holds exactly 2.5% of the days. The difference here is £9.5k; the module uses the simple version throughout.

Standard error. Writing ES as $q + \mathbb{E}[(L-q)^+]/(1-\alpha)$ makes the estimate a sample mean, and the variance of the excess $(L-q)^+$ splits into the spread inside the tail and the distance of the tail beyond $q$ :

\operatorname{se}^2\big(\widehat{\text{ES}}\big) \approx \frac{\operatorname{var}(L \mid L \ge q) + \alpha\,(\text{ES} - q)^2}{N(1-\alpha)}

For a normal P&L with standard deviation $\sigma$ , $q = z\sigma$ , $\text{ES} = \sigma\varphi(z)/(1-\alpha)$ and $\mathbb{E}[L^2 \mid L \ge q] = \sigma^2\big(1 + z\varphi(z)/(1-\alpha)\big)$ . With the module's run at 97.5%:

\begin{aligned} \sigma &= 842{,}436 \\ q &= 1{,}651{,}144 \\ \text{ES} &= 1{,}969{,}449 \\ \operatorname{sd}(L \mid L \ge q) &= 287{,}772 \\ \text{ES} - q &= 318{,}305 \\ N(1-\alpha) &= 10{,}000 \times 0.025 = 250 \\ V &= 287{,}772^2 \\ &\quad + 0.975 \times 318{,}305^2 \\ \operatorname{se} &= \sqrt{V / 250} = \text{£}26{,}952 \end{aligned}

against £31,450 for the 99% VaR (L3.2.11: Monte Carlo VaR) and £38,654 for the 99% ES. The formula assumes a normal tail; the seed study's spreads come in a little below it because the puts thin the desk's simulated tail.

Liquidity horizons. With base horizon $T = 10$ days, horizons $LH_1 = 10 < LH_2 = 20 < \dots < LH_5 = 120$ , and $\text{ES}_T(P, j)$ the ES when only the factors with horizon at least $LH_j$ move:

\begin{aligned} \text{ES} &= \sqrt{\text{ES}_T(P)^2 + \textstyle\sum_{j \ge 2} X_j^2} \\ X_j &= \text{ES}_T(P, j)\,\sqrt{(LH_j - LH_{j-1})/T} \end{aligned}

Each slower group is charged for the extra days it needs, and the terms add as squares, as if the extra periods were independent.

Try it

Pick a set of scenarios and slide the confidence level. Watch ES ÷ VaR: for a normal distribution it is about 1.25 at 95% and 1.15 at 99%, and the Monte Carlo sets stay close to that. The history with the stressed spell in it runs well above.

Try itRead VaR and ES off five sets of scenarios

Scenarios

Confidence α97.50%

97.50% one-day ES£2.391m

97.50% one-day VaR£1.677m

ES ÷ VaR1.426

Scenarios in the tail (k)19 of 750

Tail days from the stress11

Normal ES with the same σ£2.261m

σ of the scenarios£967k

VaR is the k-th worst loss and ES the average of the k worst, with k = n(1 − α) rounded up. The Monte Carlo sets are the module's run (seed 32110), simulated in your browser the first time you pick them.

Now the cascade. Move the credit spread to 120 days, or the equity indices to 20, and see which changes the liquidity-adjusted ES most.

Try itGive each factor a liquidity horizon

UK equity index (equity price, large cap)

US equity index (equity price, large cap)

GBP 2-year swap rate (interest rate, specified currency)

GBP 10-year swap rate (interest rate, specified currency)

GBP/USD (dollars per pound) (fx rate, specified currency pair)

Brent crude (energy price)

UK index implied volatility (equity volatility, large cap)

Investment-grade credit spread (credit spread, corporate investment grade)

LH ≥	Factors shocked	10-day ES	Weight	Term
10	UKX, SPX, R2Y, R10Y, GBPUSD, BRENT, UKXVOL, IG	£7.56m	1.000	£7.56m
20	BRENT, UKXVOL, IG	£2.66m	1.000	£2.66m
40	IG	£0.96m	1.414	£1.35m
60	none	£0.00m	1.414	£0.00m
120	none	£0.00m	2.449	£0.00m

Liquidity-adjusted ES£8.13m

10-day ES, all factors£7.56m

Uplift7.5%

With the lesson's horizons£8.13m

Each 10-day ES is the one-day 97.5% historical ES over 750 days, shocking only the factors listed, scaled by √10. Illustrative only: the capital rules (M4.3) also calibrate to a stressed period and use a reduced set of factors.

Worked example

The liquidity-adjusted ES of the desk

The cascade, step by step

Step 1 of 5

1
Assign the horizons
Ten days for both equity indices, both swap rates and GBP/USD; 20 for Brent and the UK index volatility; 40 for the investment-grade spread (the table above). No factor has 60 or 120.

Regulatory lens · looking ahead

Under the FRTB internal models approach, capital rests on 97.5% ES with liquidity horizons, calibrated to a 12-month period of significant stress, replacing the Basel 2.5 pairing of 99% VaR and stressed VaR. Backtesting still counts VaR exceptions (at 97.5% and 99% for each trading desk), and desks must also pass the P&L attribution test of L3.2.08: P&L attribution and explain. How the stressed calibration, the reduced set of risk factors and the capital multiplier work is in M4.3: Market risk capital (FRTB) and CVA.

Spark Atoms

Spark AtomOne measure, four names

Expected shortfall also goes by conditional VaR, tail VaR and average VaR. In 2000 R. Tyrrell Rockafellar and Stanislav Uryasev showed that a portfolio's conditional VaR can be minimised by linear programming, using the same rewriting of ES as

q + \mathbb{E}[(L-q)^+]/(1-\alpha)

as the formal version above. VaR offers no such shortcut.

Spark AtomThe road to 97.5%

The Basel Committee's first consultation on its fundamental review of the trading book, in May 2012, proposed moving from VaR to expected shortfall. The second, in October 2013, proposed 97.5%, and the final standard followed in January 2016, revised in January 2019.

Key takeaways

Historical ES is the average of the k worst days: £2.39m at 97.5% on the desk, from 19 days, 11 of them in the stressed spell.
For a normal distribution 97.5% ES matches 99% VaR. A gap between them, and an ES 99% ÷ VaR 99% ratio above 1.15, signals a fat tail.
A stressed calibration widens a normal tail without fattening it.
97.5% ES is steadier than 99% VaR here; ES at 99% is the least stable of the three, most of all on fat-tailed history.
The standard error of ES grows with the spread of the tail and its distance beyond the VaR.
Liquidity horizons add the extra risk of slow factors in a cascade: £7.56m becomes £8.13m over ten days.

Reaction

Check what stuck

Question 1 · objective 1

A desk has 400 days of history. Its ten worst losses, in £k, are 950, 820, 760, 700, 640, 610, 580, 560, 540, 520. What is its one-day 97.5% historical ES, in £k?

£k

Question 2 · objective 2

On the desk, ES 99% ÷ VaR 99% is 1.35 from history but 1.13 from Monte Carlo. What does that say?

AThe history has a fatter tail than the normal model behind the simulationBMonte Carlo has fewer scenarios, so its ES is less accurateCHistory always gives a higher VaRDNothing: the ratio depends only on the confidence level

Question 3 · objective 3

Which estimate moved least across the bootstrap resamples of the desk's 750 days?

AES at 97.5%BVaR at 99%CES at 99%DThey moved the same

Question 4 · objective 4

A book's 10-day ES is £6m. With only the factors of horizon 20 days or more shocked, the 10-day ES is £2m; with only those of 40 days or more, £1m. No factor has a longer horizon. What is the liquidity-adjusted ES, in £m?

£m

Question 5 · objective 4

In the cascade with a 10-day base, what weight multiplies the ES of the factors whose horizon is 120 days?

Glossary

Expected shortfall (historical): The average loss on the worst (1 − α) share of scenarios: here the average of the k worst, with k = n(1 − α) rounded up.
Tail ratio: ES divided by VaR at the same confidence level. For a normal distribution at 99% it is 1.146; a larger ratio means a fatter tail.
Stability: How much an estimate moves when the data change a little: a new day in the window, a new seed, a resample.
Bootstrap: Resampling the observed days with replacement many times and recomputing the estimate each time, to see how much it depends on which days happened.
Liquidity horizon: The time assumed to exit or hedge a risk factor's exposure in stressed markets without moving prices much: 10, 20, 40, 60 or 120 days in the FRTB.
Cascade: Adding the extra risk of slower factors in steps: the 10-day ES of the whole book, then the ES of only the factors with longer horizons, each scaled for the extra time.

Deep Dives & Origins

Deep Dives

Y. Yamai and T. Yoshiba, “Value-at-risk versus expected shortfall: a practical perspective”, Journal of Banking & Finance 29(4), 2005. Intermediate.
C. Acerbi and D. Tasche, “On the coherence of expected shortfall”, Journal of Banking & Finance 26(7), 2002. Advanced.

Origins

Basel Committee on Banking Supervision, Minimum capital requirements for market risk, BIS, January 2019 (liquidity horizons and the ES formula; now chapter MAR33 of the Basel Framework).
R. T. Rockafellar and S. Uryasev, “Optimization of conditional value-at-risk”, Journal of Risk 2(3), 2000.
Basel Committee on Banking Supervision, Fundamental review of the trading book, consultative document, BIS, May 2012; and Fundamental review of the trading book: a revised market risk framework, October 2013.

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