Expected shortfall · RiskAtoms Academy

By the end you will be able to

1Define expected shortfall as the average loss beyond VaR, and explain why it is never smaller than VaR.
2Compute ES from historical data and under the normal assumption.
3Show with an example that VaR can penalise diversification, and that ES does not.
4Explain why Basel moved from 99% VaR to 97.5% ES.

Why it matters

Two traders each report a 99% VaR of $1 million. One has sold far out-of-the-money options: on the worst day in a hundred she loses a little over $1 million; on the worst day in a thousand she loses $40 million. The other cannot lose more than $1.5 million. VaR cannot tell them apart.

Core explanation

What lies beyond the VaR

VaR (L2.5.06: Value at risk: the idea) marks where the bad tail starts and stops looking. It does not care whether the losses past the line are a little bigger or catastrophically bigger. Expected shortfall (ES) asks the next question: when we have a bad day, how bad is it on average?

What an ES figure says

“The fund's one-day 97.5% ES is $181,875” means: on the worst 2.5% of days, the fund loses $181,875 on average. Like VaR, it needs a confidence level, a horizon and a currency, and it is quoted as a positive loss.

ES is an expectation (M-C02: Expectation) of the tail, so it is never smaller than the VaR at the same confidence. How much bigger depends on the shape of the tail. For a normal distribution, 99% ES is only 15% above 99% VaR. On the fund's own 500 days it is 51% above, because the worst days are very much worse than the 5th worst:

One day, $10m fund	VaR 97.5%	ES 97.5%	VaR 99%	ES 99%
Normal (mean zero)	$124,728	$148,773	$148,044	$169,609
Historical, 500 days	$109,114	$181,875	$184,550	$279,443

Illustrative dataThe Module 2.5 data set

500 trading days of daily returns for an equity index fund, a bond fund and a bank share, generated for this module by a seeded model with a calm stretch, a turbulent spell and a few extreme days. It is not real market history. The same series runs through every lesson in the module, so the numbers line up. The fund we follow is worth $10 million: $6 million in the equity index fund and $4 million in the bond fund.

Diversification. A sensible risk measure should never say that combining two portfolios is riskier than holding them apart: the combined risk should be no more than the sum. That property is called subadditivity. Volatility has it. VaR, surprisingly, does not:

Two loans, 95% confidence

VaR fails here because the default risk hides just below its 5% cut-off for each loan alone. Combining them pushes it over. ES looks at the whole tail, so it cannot be fooled this way. Measures with subadditivity and three other common-sense properties are called coherent. ES is coherent; VaR is not. VaR is subadditive for normal distributions, which is why the problem shows up mainly with credit, options and other lopsided payoffs.

The formal version

With losses $L$ (minus P&L) and confidence $\alpha$ :

\text{ES}_{\alpha} = \mathbb{E}\big[\,L \mid L \ge \text{VaR}_{\alpha}\,\big] = \frac{1}{1-\alpha}\int_{\alpha}^{1} \text{VaR}_{u}\,du

The second form, the average of all VaRs beyond $\alpha$ , is the general definition and also works when the distribution has jumps (like the loans). For data and for the normal case it becomes:

\widehat{\text{ES}}_{\alpha} = \frac{1}{k}\sum_{i=1}^{k} L_{(i)}, \quad k = \lceil n(1-\alpha) \rceil \qquad\qquad \text{ES}_{\alpha} = \sigma V\,\frac{\varphi(z_{\alpha})}{1-\alpha}

Conventions on this page

Historical ES averages the same

k

worst losses whose last one is the historical VaR (L2.5.07: Historical VaR), so the two are always consistent. The RiskAtoms calculator's histes rounds

n(1-\alpha)

to the nearest whole number instead of up; for 500 days at 97.5% and 99% both give 13 and 5. Normal ES uses mean zero, like normal VaR (L2.5.06: Value at risk: the idea).

Try it

The explorer opens at 97.5% on the fund's historical losses. The red line is the VaR, the bold line the ES: the balance point of everything to its right. Slide towards 99.8% and watch ES sit far beyond VaR, because the tail is lumpy. Switch to Normal and see how close they stay.

Try itVaR and ES explorer

Confidence level97.5%

Loss distribution

Historical ES here is the average of the 13 worst of the 500 losses, the same 13 that set the VaR.

97.5% one-day ES

$181,875

VaR (the quantile)$109,114

ES ÷ VaR1.67

Days averaged, k13

Normal ES$148,773

Worked example

The fund's 97.5% ES

Try each step before revealing it.

Historical and normal ES

Step 1 of 5

1
How many days?
$k = \lceil 500 \times 0.025 \rceil = 13$ . The 97.5% VaR is the 13th worst loss: $109,114 on day 275.

Regulatory lens

In the Fundamental Review of the Trading Book (BCBS, January 2016, revised January 2019), banks using internal models compute a 97.5% expected shortfall instead of 99% VaR. The 97.5% level was picked because, for a normal distribution, it gives almost the same number as 99% VaR ($148,773 against $148,044 for the fund), so the switch rewards thin tails and penalises fat ones. ES starts from a 10-day horizon and is scaled up for less liquid risk factors using liquidity horizons of 10 to 120 days, and it is calibrated to a period of stress. Insurance regulators differ: Solvency II sets capital at a one-year 99.5% VaR, while the Swiss Solvency Test uses a one-year 99% ES (M4.6: Solvency II and insurance regulation). The detail is in M4.3: Market risk capital (FRTB) and CVA, part of Level 4: Regulation, Capital & Accounting.

Sparks

SparkFour axioms, 1999

In 1999 Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath published “Coherent measures of risk”. Instead of starting from a formula, they asked what any sensible risk measure should satisfy, wrote down four rules, and showed that VaR breaks one of them. The paper turned a practitioners' argument into mathematics.

SparkThe catch: backtesting

VaR is easy to check: count the days the loss exceeded it. ES is harder, because it is an average of rare events. In 2011 Tilmann Gneiting showed that ES lacks a property called elicitability, which some read as meaning it cannot be backtested at all. Later work found practical tests, but the FRTB still backtests with VaR at 97.5% and 99%.

Key takeaways

Expected shortfall is the average loss beyond VaR: E[L | L ≥ VaR]. It is always at least the VaR.
Historical ES: average the k worst losses, the same k as historical VaR. Normal ES: σV × φ(z) ÷ (1 − α).
ES is coherent and subadditive; VaR can say that diversifying increases risk.
With fat tails, ES is far above VaR; for a normal distribution, 97.5% ES ≈ 99% VaR.
Basel's FRTB uses 97.5% ES with liquidity horizons from 10 to 120 days.

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Glossary

Expected shortfall (ES): The average loss on the worst (1 − α) share of outcomes: the expected loss given that the loss is at least the VaR. Also called conditional VaR (CVaR) or expected tail loss.
Tail: The part of the loss distribution beyond the VaR.
Subadditivity: A risk measure is subadditive if the risk of a combined portfolio is never more than the sum of the risks of its parts. It rewards diversification.
Coherent risk measure: A risk measure with four properties set out by Artzner, Delbaen, Eber and Heath (1999): monotonic, subadditive, positively homogeneous and translation invariant.
Liquidity horizon: In FRTB, the time assumed to exit or hedge a position in stressed markets: 10, 20, 40, 60 or 120 days depending on the risk factor.

Deep Dives & Origins

Deep Dives

J. C. Hull, Risk Management and Financial Institutions, Wiley, the chapter on VaR and ES. Intermediate.
A. J. McNeil, R. Frey and P. Embrechts, Quantitative Risk Management, Princeton University Press, revised edition 2015, the chapter on risk measures. Advanced.

Origins

P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, “Coherent measures of risk”, Mathematical Finance 9(3), 1999, 203–228.
T. Gneiting, “Making and evaluating point forecasts”, Journal of the American Statistical Association 106(494), 2011, 746–762.
Basel Committee on Banking Supervision, Minimum capital requirements for market risk, January 2016, revised January 2019.

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