Expected Shortfall

Expected shortfall measures the risk of a position, including the tail losses that VaR does not show. Under the internal models approach, in which a bank calculates its amount of risk with its own models, the bank calculates it daily for each trading desk as a value in the market risk capital rules.

What expected shortfall measures

Expected shortfall (ES) shows how large the loss is on average when it exceeds Value at Risk (VaR)¹. VaR only shows a loss amount that is exceeded with a set probability. It does not show how large the loss is when it is exceeded. Expected shortfall brings the losses in the tail of the distribution beyond VaR into the measure as an average. Some sources call it conditional VaR or other names, and Japan's capital adequacy notice for banks uses the katakana transliteration of expected shortfall².

Consider a hypothetical example in which two portfolios are compared on one-day profit and loss under 200 equally likely scenarios. The confidence level is 97.5%. Since 2.5% of 200 scenarios is 5, VaR is the fifth-largest loss, and expected shortfall is the average of the five largest losses, including VaR.

Both portfolios have a VaR of 16 million yen, so VaR alone shows no difference. Expected shortfall is 22 million yen and 26.2 million yen, and the higher figure for portfolio 2 reflects its scenario with a 60 million yen loss.

How to calculate expected shortfall

Let $L$ be the loss, $\alpha$ the confidence level (0.975 for 97.5%), and $\mathrm{VaR}_\alpha$ the VaR at that confidence level. If the loss distribution is continuous, expected shortfall is the expected loss given that the loss is at least VaR³.

$$\mathrm{ES}_\alpha = E[\,L \mid L \ge \mathrm{VaR}_\alpha\,]$$

If the distribution is not continuous, you cannot use this formula as it is, and the definition has to be made more rigorous⁴. When several losses equal VaR, the formula puts all of them into the average, so it averages more than the fraction $(1-\alpha)$ of the cases (see the example in the section on subadditivity).

With any of the three VaR methods (variance-covariance method, historical simulation, and the Monte Carlo method), you calculate expected shortfall from the same profit and loss distribution used for VaR. In historical simulation and the Monte Carlo method, you rank $n$ profit and loss figures (from past days or from generated scenarios) from the largest loss down, and let $k$ be $n(1-\alpha)$ rounded up to a whole number. The $k$th loss is VaR, and expected shortfall is the average of the $k$ largest losses. A sample of profit and loss figures is not a continuous distribution, so you use this average rather than the conditional expectation formula.

$$\mathrm{ES}_\alpha = \frac{L_{(1)} + L_{(2)} + \cdots + L_{(k)}}{k}$$

$L_{(i)}$ is the $i$th-largest loss. In the example in the previous section, $n = 200$ and $k = 5$. Sources sometimes differ by one in which rank they treat as VaR⁵.

The variance-covariance method assumes that profit and loss follow a normal distribution with a mean of 0, and calculates expected shortfall directly from the standard deviation of profit and loss, $\sigma$.

$$\mathrm{ES}_\alpha = \sigma \times \frac{\varphi(z_\alpha)}{1-\alpha}$$

$z_\alpha$ is the point on the standard normal distribution with $\alpha$ below it (the VaR factor, 1.9600 for 97.5%), and $\varphi$ is the probability density function of the standard normal distribution. The 97.5% factor is 2.3378, which is $\varphi(z_{0.975}) = 0.058445$ divided by 0.025 ($z_{0.975}$ is 1.95996 before rounding). At 99% the factor is 2.6652, which is larger than the 99% VaR factor of 2.3263⁶. Under a normal distribution, both VaR and expected shortfall are fixed multiples of the standard deviation.

Figure 1 shows, on a standard normal distribution, the VaR line and the position of the average of the tail beyond it (the 97.5% VaR factor of 1.960 and the expected shortfall factor of 2.338).

Figure description

The curve is the probability density of a normal distribution with mean 0 and standard deviation 1, drawn with profit or loss on the horizontal axis (the axis extends slightly to the right of 0 for readability). It is a bell shape that is symmetric around 0. A vertical dashed line labeled "VaR" at −1.960 on the loss side marks the position of 97.5% VaR. The shaded area to the left of that line is the 2.5% probability. Further left, a vertical dashed line labeled "ES" at −2.338 marks the position of expected shortfall. The "ES" line is the position of the average of the shaded tail losses, and it lies outside the VaR line. The vertical axis is probability density, and area corresponds to probability.

Deep diveDeriving expected shortfall for a normal distribution

For $Z$ following the standard normal distribution, the probability that $Z$ is $z_\alpha$ or more is $(1-\alpha)$. The conditional expectation is therefore the following integral divided by $(1-\alpha)$.

$$\int_{z_\alpha}^{\infty} x\,\varphi(x)\,dx = \varphi(z_\alpha)$$

The left-hand side equals $\varphi(z_\alpha)$ because the derivative of $\varphi(x) = e^{-x^2/2}/\sqrt{2\pi}$ is $-x\,\varphi(x)$. Since the loss is $L = \sigma Z$, expected shortfall is $\sigma\,\varphi(z_\alpha)/(1-\alpha)$.

Worked example with asset A and asset B

Take the position in the VaR simulator's initial values and set the confidence level to 97.5%. You hold 600 million yen of asset A (daily volatility of 1.2%) and 400 million yen of asset B (0.8%), the correlation between them is 0.3, and the holding period is 1 day. The standard deviation of one-day profit and loss is 8.7123 million yen ($\sqrt{7.2^2 + 3.2^2 + 2 \times 0.3 \times 7.2 \times 3.2}$), which combines 7.2 million yen for asset A (600 million yen × 1.2%) and 3.2 million yen for asset B (400 million yen × 0.8%) with a correlation of 0.3.

Each value is the standard deviation of 8.7123 million yen multiplied by the factor. The 97.5% expected shortfall of 20.37 million yen is almost the same as the 99% VaR of 20.27 million yen. If profit and loss are normally distributed, the factors are 2.3378 and 2.3263, so the two are almost the same size whatever the position or volatility.

Calculating the same position with historical simulation (250 days) and the Monte Carlo method (10,000 scenarios) gives the following. In the simulator, both the historical data and the Monte Carlo scenarios are generated with random numbers from a normal distribution, and neither is real market data.

In historical simulation, 2.5% of 250 days is 6.25, which rounds up to 7. The seventh-largest loss is VaR, and the average of the seven largest losses is expected shortfall. Values estimated from a sample vary around the variance-covariance values.

The gap with VaR widens under fat tails

A fat-tailed distribution is one in which values far from the mean, such as very large losses, are more likely than in a normal distribution⁷. If the profit and loss distribution has fatter tails than a normal distribution, VaR that assumes a normal distribution may understate losses⁸. Fat tails show up more strongly in expected shortfall than in VaR.

If you switch the returns for the same position to a t distribution with 5 degrees of freedom ("Student's t" on the simulator) (a fat-tailed distribution scaled to the same variance as the normal distribution), the 97.5% values become the following.

With the Monte Carlo method, switching to the t distribution raises VaR by 7%, from 16.72 million yen to 17.94 million yen, and raises expected shortfall by 17%, from 20.16 million yen to 23.60 million yen. The standard deviation is unchanged, but the losses beyond VaR get larger. The variance-covariance method assumes a normal distribution whatever the actual distribution, so its values do not change.

Diversification and subadditivity

A risk measure is subadditive if the risk of two positions combined does not exceed the sum of the risks of each position⁹. The property means that diversifying does not increase the amount of risk. VaR is subadditive when profit and loss are normally distributed, but it may not be under other distributions¹⁰. Expected shortfall is subadditive when the profit and loss distribution is continuous. When the distribution is not continuous, it has been shown to be subadditive if expected shortfall is defined more rigorously¹¹. After the example below, this section checks that the average of a fixed number of the largest losses, which is what you use with a sample, is subadditive.

Consider a hypothetical example in which two corporate bonds (bond 1 and bond 2) show VaR failing subadditivity. Each bond defaults in 4 (2%) of 200 equally likely scenarios, with no recovery. The two bonds never default in the same scenario, and interest is ignored. The confidence level is 97.5%, and as in the first section, VaR is the fifth-largest loss and expected shortfall is the average of the five largest losses, including VaR.

If you hold only one bond, there are 4 scenarios with a loss, and the fifth-largest loss is 0, so VaR is 0. If you split the money into 5 billion yen in each bond, there are 8 scenarios with a loss, and VaR rises to 5 billion yen. The combined VaR (5 billion yen) is larger than the sum of the individual VaRs (0 + 0), so under VaR, putting all 10 billion yen into one bond looks less risky. Under expected shortfall, the combined 5 billion yen does not exceed the sum of the individual values (4 + 4 = 8 billion yen), and it is smaller than the 8 billion yen for holding one bond. Large losses with a probability below 2.5% do not show up in VaR, but they enter expected shortfall through the average¹².

This example is a discrete distribution with many scenarios at a loss of 0, so the conditional expectation formula from the calculation section does not work. For 10 billion yen in bond 1, averaging every scenario with a loss at or above VaR (0), as the formula says, takes in all 200 scenarios and gives 40 billion yen ÷ 200 = 200 million yen. For 5 billion yen in each bond, each bond alone has an expected shortfall of 100 million yen, and the combined 5 billion yen exceeds their sum of 200 million yen. If you average the same number of the largest losses across equally likely scenarios, subadditivity always holds. Take the 5 scenarios with the largest losses for the combined holding: the sum of bond 1's losses in those scenarios cannot exceed the sum of bond 1's 5 largest losses, and the same is true for bond 2.

Weaknesses of expected shortfall

The more fat-tailed the distribution, the more estimates of expected shortfall vary compared with VaR. The average moves depending on whether a rare large loss is in the sample, so matching the precision of VaR takes more simulations¹³. With the t distribution setting in the fat-tail section, the 250-day historical simulation expected shortfall is 28.22 million yen, 4.62 million yen away from the 23.60 million yen from the Monte Carlo method with 10,000 scenarios. The historical simulation value is the average of only seven losses, so it changes a lot depending on which days are in the sample.

Checking expected shortfall is also a challenge. For VaR, you can count the days on which the loss exceeded VaR and compare them with actual profit and loss. For expected shortfall, establishing a backtesting method was listed as an open problem as of 2001¹⁴. Japan's capital adequacy notice for banks also bases backtesting for each trading desk on 97.5% and 99% VaR, not on expected shortfall¹⁵. How to check the measures is covered on the Backtesting page.

History in market risk regulation

Bank market risk rules include the internal models approach, under which a bank calculates its amount of risk with its own models. The internal models approach began using VaR with the 1996 amendment, and moved to expected shortfall with the Fundamental Review of the Trading Book (FRTB) after the global financial crisis. The trading book is the category made up of positions held for purposes such as short-term resale or profiting from short-term price moves, along with their hedges¹⁶.

In January 1996, the Basel Committee on Banking Supervision published an amendment that, from the end of 1997, required capital for market risk in addition to credit risk. Under the internal models approach, VaR was calculated daily at a one-tailed 99% confidence level, with a holding period of at least 10 business days and an observation period of at least one year. The capital requirement was VaR multiplied by a multiplier¹⁷. The multiplier under the current notice, and how backtesting results raise it, are covered on the Backtesting page. Japan applied this framework from 1998¹⁸.

In the global financial crisis, trading book losses at most banks far exceeded their market risk capital. In July 2009, the Basel Committee published Basel 2.5, which added stressed VaR on top of VaR¹⁹. Stressed VaR is VaR calculated with data from a continuous one-year stress period, at a one-tailed 99% confidence level with a 10-day holding period²⁰. Under the international agreement, Basel 2.5 took effect from the end of 2011²¹.

As the FRTB, the Basel Committee then published "Minimum capital requirements for market risk" in January 2016, and published the final revised rules in January 2019²². Because VaR does not capture tail losses, the internal models approach adopted expected shortfall, and the dual framework of VaR and stressed VaR was merged into a single expected shortfall that uses stress-period data²³.

In Japan, the capital adequacy notice sets the following requirements for banks that use the internal models approach. Expected shortfall is calculated daily at a one-tailed 97.5% confidence level. The stress period is the 12 months with the most severe market conditions in an observation period going back at least to 2007²⁴. The holding period is no longer a uniform 10 days. Instead, one of the liquidity horizons of 10, 20, 40, 60, or 120 days is used, depending on the market liquidity of each risk factor (a variable that moves the value of a position)²⁵. The formula that reflects liquidity horizons in expected shortfall is covered on the Fundamental Review of the Trading Book (FRTB) page.

The notice finalizing Basel III, including the FRTB, has applied to international-standard institutions and domestic-standard institutions that use internal models since March 31, 2024. It has applied since March 31, 2025 to domestic-standard institutions that do not use internal models. From March 31, 2023, institutions could also apply it earlier by filing a notification²⁶. Elsewhere, as of December 2025, the EU had postponed implementation of the market risk rules to 2027, and the UK planned to postpone applying internal models for market risk to 2028. The US had not set an implementation date²⁷.

Notes

1. Yasuhiro Yamai and Toshinao Yoshiba, "バリュー・アット・リスクのリスク指標としての妥当性について" (On the validity of value at risk as a risk measure), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2001), "ロ．期待ショートフォール" (b. Expected shortfall), page 38 (in Japanese). ↩

2. Yasuhiro Yamai and Toshinao Yoshiba, "期待ショートフォールによるポートフォリオのリスク計測" (Measuring portfolio risk with expected shortfall), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (December 2001), page 55, on names such as conditional VaR (in Japanese). For the notice's term, see Financial Services Agency, "銀行法第十四条の二の規定に基づき、銀行がその保有する資産等に照らし自己資本の充実の状況が適当であるかどうかを判断するための基準" (Standards for determining whether a bank's capital adequacy is appropriate in light of the assets it holds, under Article 14-2 of the Banking Act, our translation), Article 1, item 96 (page 25) (in Japanese). ↩

3. Yasuhiro Yamai and Toshinao Yoshiba, "期待ショートフォールによるポートフォリオのリスク計測" (December 2001), equation (1) on page 55 (in Japanese). The paper writes the loss as X and the confidence level as 100(1−α)%, using α for the probability of exceeding VaR. On this page, α is the confidence level, which corresponds to 1−α in the paper. ↩

4. Yasuhiro Yamai and Toshinao Yoshiba, "リスク指標の性質に関する理論的整理" (A theoretical review of the properties of risk measures), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (December 2001), note 32 on page 119 (in Japanese). The note says, "if the distribution function of the random variable is not continuous, expected shortfall needs to be redefined more rigorously" (our translation). The paper assumes a continuous distribution on page 96. ↩

5. Yasuhiro Yamai and Toshinao Yoshiba, "期待ショートフォールによるポートフォリオのリスク計測" (December 2001), note 9 on page 56 (in Japanese), takes the loss at rank n(1−confidence level) + 1 among n simulated losses as the VaR estimate, and the average of the losses up to that rank as the expected shortfall estimate. The examples and simulator on this page use the rank given by n(1−confidence level) rounded up. ↩

6. Yasuhiro Yamai and Toshinao Yoshiba, "バリュー・アット・リスクのリスク指標としての妥当性について" (April 2001), the main text and note 8 on page 38 (in Japanese), which give the formula for expected shortfall under a normal distribution and, at 99%, 2.33 and 2.67 times the standard deviation. The 97.5% factor was calculated from the same formula. ↩

7. Yasuhiro Yamai and Toshinao Yoshiba, "期待ショートフォールによるポートフォリオのリスク計測" (December 2001), page 57 (in Japanese). If losses follow a fat-tailed distribution, very large losses are more likely than under a normal distribution. ↩

8. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (II. Understanding and managing market risk) (June 2012), the weaknesses listed under "分散共分散法" (The variance-covariance method), page 23 (in Japanese). ↩

9. Yasuhiro Yamai and Toshinao Yoshiba, "バリュー・アット・リスクのリスク指標としての妥当性について" (April 2001), note 1 on page 34 (in Japanese). ↩

10. Yasuhiro Yamai and Toshinao Yoshiba, "バリュー・アット・リスクのリスク指標としての妥当性について" (April 2001), page 39 (in Japanese), on the subadditivity of VaR under a normal distribution and the problem when the distribution is not normal. ↩

11. Yasuhiro Yamai and Toshinao Yoshiba, "リスク指標の性質に関する理論的整理" (December 2001), theorem 17 on page 119 (subadditivity of expected shortfall when the distribution function is continuous) and notes 32 and 34 (in Japanese). The notes say that the non-continuous case needs a redefinition, and that the treatment of the discrete case and the proof of subadditivity are due to Acerbi and Tasche [2001]. See also example 5 on page 102. ↩

12. The example adapts, to a size you can calculate by hand, the "大口与信がある場合の信用リスク計測" (Measuring credit risk with large exposures) example in Yasuhiro Yamai and Toshinao Yoshiba, "バリュー・アット・リスクのリスク指標としての妥当性について" (April 2001), page 41 (100 corporate bonds, a 1% default rate, and a 95% confidence level) (in Japanese). ↩

13. Yasuhiro Yamai and Toshinao Yoshiba, "期待ショートフォールによるポートフォリオのリスク計測" (December 2001), the abstract on page 53 and page 57 (in Japanese). ↩

14. Yasuhiro Yamai and Toshinao Yoshiba, "バリュー・アット・リスクのリスク指標としての妥当性について" (April 2001), the abstract on page 33 (in Japanese), which lists the stability of estimates and establishing a backtesting method as open problems. ↩

15. Financial Services Agency, "銀行法第十四条の二の規定に基づき、銀行がその保有する資産等に照らし自己資本の充実の状況が適当であるかどうかを判断するための基準", Article 275-3, paragraph 2 (page 461) (in Japanese). ↩

16. 磯部昌吾, "自己資本規制におけるマーケット・リスク計測" (Market risk measurement in capital adequacy regulation), FSA Institute Discussion Paper DP2017-5, Financial Research Center, Financial Services Agency (October 2017), the summary on page 1 (the standardized approach and the internal models approach) and page 4 (the trading book) (in Japanese). ↩

17. Basel Committee on Banking Supervision, "マーケット・リスクを自己資本合意の対象に含めるための改定" (January 1996, Japanese translation by the Bank of Japan), item 1 of the preface on page 1, and items (a) to (d), (i), and (j) of "Ｂ．４ 定量的基準" (B.4 Quantitative standards) on pages 56–58 (in Japanese). The English original is Basel Committee on Banking Supervision, "Amendment to the Capital Accord to Incorporate Market Risks" (January 1996), http://www.bis.org/publ/bcbs24.pdf; the page numbers and item labels above are those of the translation. ↩

18. FSA Institute Discussion Paper DP2017-5 (October 2017), page 5 (in Japanese). ↩

19. FSA Institute Discussion Paper DP2017-5 (October 2017), pages 2 and 9 (in Japanese). ↩

20. FSA Institute Discussion Paper DP2017-5 (October 2017), page 9 and the "ストレス VaR の算出" (Calculating stressed VaR) row of the table on page 11 (in Japanese). ↩

21. Financial Services Agency and Bank of Japan, "「マーケット・リスクの最低所要自己資本」の概要" (Overview of "Minimum capital requirements for market risk") (April 2016), "最終文書公表の経緯" (Background to the publication of the final document), page 2 (in Japanese). ↩

22. Financial Services Agency and Bank of Japan, "「マーケット・リスクの最低所要自己資本」の概要" (January 2019), "経緯" (Background), page 2 (in Japanese). ↩

23. Financial Services Agency and Bank of Japan, "「マーケット・リスクの最低所要自己資本」の概要" (April 2016), items ① and ② on page 9 (in Japanese). ↩

24. Financial Services Agency, "銀行法第十四条の二の規定に基づき、銀行がその保有する資産等に照らし自己資本の充実の状況が適当であるかどうかを判断するための基準", Article 276-2, paragraph 1 (one-tailed 97.5%, daily; page 467), and Article 1, item 97 (for market stress, the 12 months with the most severe market conditions in an observation period going back at least to 2007 (Heisei 19); page 25) (in Japanese). ↩

25. The same notice, Article 276-2, paragraphs 2 and 3 (pages 468–469) (in Japanese). The liquidity horizons of 10 to 120 days are also on page 14 of Financial Services Agency and Bank of Japan, "「マーケット・リスクの最低所要自己資本」の概要" (April 2016) (in Japanese). ↩

26. Financial Services Agency, "「自己資本比率規制（第１の柱・第３の柱）に関する告示の一部改正（案）」等に対するパブリックコメントの結果等について" (Results of the public comments on the draft partial amendments to the notices on capital adequacy regulation (Pillar 1 and Pillar 3) and related matters) (April 28, 2022), "５．公布・適用日等について" (5. Promulgation and application dates) (in Japanese). ↩

27. Financial Services Agency, "バーゼル規制の概要" (Overview of the Basel framework) (August 2026), "バーゼルⅢ最終化の実施時期" (Implementation dates of the Basel III finalization), page 10 (in Japanese). ↩

28. Financial Services Agency, "銀行法第十四条の二の規定に基づき、銀行がその保有する資産等に照らし自己資本の充実の状況が適当であるかどうかを判断するための基準", Article 276-2, paragraphs 1 to 3 (pages 467–469), and Article 276-3, paragraph 1 (expected shortfall assuming market stress; page 470) (in Japanese). ↩