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Backtesting Expected Shortfall Carlo Acerbi Balazs Szekely January 21, 2015

©2015 MSCI Inc. All rights reserved.

Outline       

The VaR vs ES Dilemma Elicitability Three Tests for ES Numerical Results Testing ES in Practice Back to Elicitability Conclusions

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The VaR or ES Dilemma The Importance of Backtest

VaR and ES  VaR: the best of worst x% losses; losses  ES: the average of worst x% losses

→ threshold of x%

→ expected x% loss

Profit and Loss Distribution

ES VaR

 ES multiple advantages: tail sensitivity, subadditivity (→ coherent), mathematical tractability, uniqueness, uses same risk models …  Last roadblock for ES toward Basel: backtesting  As of Oct13, no general consensus ES backtest

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Basel: VaR or ES?  1994: RiskMetrics Technical Document popularizes “Value at Risk” (VaR)  1996: Basel Committee internal-based approach to capital adequacy, based on VaR  1997: Artzner et al. “Coherent Measures of Risk”: axioms for sensible risk measures. VaR criticized for not complying  2001: Rockafellar and Uryasev, Acerbi and Tasche, define “Expected Shortfall” (ES, aka CVaR), a coherent measure of risk  2000s VaR and ES are widely adopted by financial institutions as complementary tools  2013: Basel Committee replaces VaR1% with ES2.5%  VaR is maintained for model backtesting

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Backtesting in a Nutshell predicted

predicted

predicted

predicted

realized realized

realized

t

t+1 realized

 Backtesting means checking whether realizations were in line with model forecasts  However, distributions (and statistics) do not materialize  Only one scenario at a time does

 Not all risk measures can be backtested  And it is not easy to say which ones can

 Opposite examples  VaR: just test whether losses exceed VaR x% of the times (model independent!)  Mode: probably impossible to backtest 6

Basel VaR Backtest: Null and Alternative Hypotheses 𝐻0

𝐻1

 “Fundamental review of the trading book, a revised market risk framework”, Basel Committee 2013 7

Basel VaR Backtest: Traffic Light System

 Same mechanism since 1996

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Time Independence

 Conditional coverage tests (Christoffersen ‘98)  However, in practice, independence of arrival of VaR exceptions is normally tested through visual inspection 9

Why so Difficult with ES?  Backtesting ES is less immediate  “How can I backtest ES, if it’s the average of the x% worst cases, but life is a single scenario only?”  It sounds like we need multiple scenarios to compute a realized ES

 As a matter of fact, literature on backtesting ES is scarce and fragmentary  Only one proposed model-independent ES backtest (Kerkhof and Melenberg 2004)

 Legitimate doubt: is ES backtestable at all?  A recent result (Gneiting 2011) seemed to have proven that it is not

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Elicitability

Eliciwhat?  A statistic 𝑌 is said to be elicitable if it solves 𝑌 = arg min 𝔼 𝑆(𝑦, 𝑋) 𝑦

for some scoring function 𝑆 𝑦, 𝑥  Popular examples:  Mean:

𝑦 = arg min 𝔼 (𝑦 − 𝑋)2

 Median:

𝑦 = arg min 𝔼 𝑦 − 𝑋

 𝛼-quantile:

𝑞𝛼 = arg min 𝔼 (𝑋 − 𝑞)(𝛼 − (𝑋 − 𝑞 < 0))

𝑦 𝑦

𝑞

→ VaR !

 If a statistic 𝑌 is elicitable, given forecasts 𝑦𝑡 and realizations 𝑥𝑡 the mean score 1 𝑆= 𝑇

𝑇

𝑆 𝑦𝑡 , 𝑥𝑡 𝑡=1

ranks predictive models: the lower the better  Natural test: think of mean squared errors, for instance 12

ES is not Elicitable  Gneiting 2011 proves that there is no 𝑆 𝑦, 𝑥 that elicits ES  Many risk experts concluded that ES cannot be backtested  ES cannot be backtested because it fails to satisfy elicitability ... If you held a gun to my head and said: ‘We have to decide by the end of the day if Basel 3.5 should move to ES, with everything we know now, or do we stick with VaR’, I would say: ‘Stick with VaR’ (Paul Embrechts, March 2013, RISK.net)

 Criticism followed the announced choice of the Basel Committee to adopt ES  Are there risk measures which are both coherent and elicitable?  Yes, expectiles (Ziegel, 2013, Bellini et al. 2014)  Explosion of interest for these measures

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Something is not Quite Right  If non-elicitable means non-backtestable  Why has VaR never been backtested by exploiting its elicitability?  The exceptions test has nothing to do with scoring functions

 How about variance? It’s not elicitable either! Should we drop it as well?  And what about Kerkhof and Melenberg 2004? “…contrary to common belief, ES is not harder to backtest than VaR... Furthermore, the power of the test for ES is considerably higher”

 We will answer all of these questions

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Three Tests for Expected Shortfall

Model Independent Setting  Basel choices:  1 year lookback window: 𝑡 = 1, … 𝑇 = 250  𝑉𝑎𝑅1% and 𝐸𝑆2.5%

 Independent (not iid) Profit-loss 𝑋𝑡  𝐹𝑡 real (unknown) distribution  𝑃𝑡 forecast (model) distribution

 Only assumption: continuous distributions  → we can write:

𝐸𝑆𝛼,𝑡 = −𝔼 𝑋𝑡 𝑋𝑡 + 𝑉𝑎𝑅𝛼,𝑡 < 0]

 Standard hypothesis testing framework (the below is generic)  𝐻0 :

𝐹𝑡 = 𝑃𝑡

null: model is perfect

 𝐻1 :

𝐹 𝑃 𝐸𝑆𝛼,𝑡 > 𝐸𝑆𝛼,𝑡

alternative: ES underestimated (one sided test)

 Proposed tests must be model-independent  No distribution families, no location-scale, no asymptotic convergence, …

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Test 1: Testing ES After VaR 𝔼

 From

𝑋𝑡 𝐸𝑆𝛼,𝑡

𝑋𝑡 + 𝑉𝑎𝑅𝛼,𝑡 < 0 + 1 = 0

𝑍1 𝑋 =

 Define

𝑇 𝑋𝑡 𝐼𝑡 𝑡=1𝐸𝑆 𝛼,𝑡 𝑇 𝐼 𝑡=1 𝑡

+1

where 𝐼𝑡 = 𝟏{𝑋𝑡+𝑉𝑎𝑅𝛼,𝑡 1

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𝐻0 : Student-t, 𝜈 = 100; 𝐻1: Scaled Distributions

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26


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𝐻0 : Student-t; 𝐻1: ES Coverage 95%, 90%  𝐻0 : 𝐹𝑡 = 𝑃𝑡 , Student-t distributions  𝐻1 : 𝐹𝑡 (𝑥) = 𝑃𝑡 (𝑥/ 𝛾), scaled distribution, 𝛾 > 1, but labeled in terms of ES coverage  𝐸𝑆𝛼𝑃 = 𝐸𝑆𝛼𝐹′ with 𝛼 = 2,5% and 𝛼 ′ = 5%, 10%  Analogous to the Basel VaR coverage tables

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𝐻0 : Student-t , 𝜈 = 100; 𝐻1: ES Coverage 95%, 90%

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𝐻0 : Student-t , 𝜈 = 5; 𝐻1: ES Coverage 95%, 90%

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𝐻0 : Student-t , 𝜈 = 100; 𝐻1: 𝜈 = 10, 5, 3  𝐻0 : 𝐹𝑡 = 𝑃𝑡 , Student-t distributions  𝐻1 : Student-t distributions with lower 𝜈  Notice that standard deviation is also larger 𝜎 =

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𝜈/(𝜈 − 2)

𝐻0 : Student-t , 𝜈 = 100; 𝐻1: 𝜈 = 10, 5, 3

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𝐻0 : Student-t , 𝜈 = 100; 𝐻1: 𝜈 = 10, 5, 3

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𝐻0 : Student-t , 𝜈 = 10; 𝐻1: 𝜈 = 5, 3

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𝐻0 : Normalized Student-t , 𝜈 = 100; 𝐻1: 𝜈 = 10, 5, 3  𝐻0 : 𝐹𝑡 = 𝑃𝑡 , Normalized Student-t distributions with 𝜎 = 1  𝐻1 : Normalized Student-t distributions with lower 𝜈 and 𝜎 = 1

35

𝐻0 : Normalized Student-t , 𝜈 = 100; 𝐻1: 𝜈 = 10, 5, 3

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𝐻0 : Normalized Student-t , 𝜈 = 100; 𝐻1: 𝜈 = 10, 5, 3

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𝐻0 : Normalized Student-t , 𝜈 = 10; 𝐻1: 𝜈 = 5, 3

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𝐻0 : Normalized Student-t ; 𝐻1: fixed VaR2.5%    

𝐻0 : 𝐹𝑡 = 𝑃𝑡 , Normalized Student-t distributions with 𝜎 = 1 𝐻1 : Normalized Student-t distributions with lower 𝜈 and 𝜎 = 1 Distributions offset so that they have equal VaR2.5% Alternative hypotheses built to analyze Test 1

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𝐻0 : Norm. Student-t 𝜈 = 100; 𝐻1: fixed VaR2.5%

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41


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Summary of Results  All tests for ES 2.5% generally display more power than the Basel test for VaR 1% in identical conditions  Test 1 is subordinated to testing VaR first, but has strong power for model misspecifications in the tail index  Test 2 and test 3 excel in different cases: test 2 is more powerful on scaled distributions; test 3 is more powerful on distributions with different tail index

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Testing ES in Practice Implementing Test 2

Implementing Test 2  Test 2 can be adopted without storing forecast distributions  95% and 99.99% significance level thresholds are fixed values 𝑍2 = −0.70 and 𝑍2 = −1.8

 Every day, it is sufficient to record the quantities  𝑋𝑡 𝐼𝑡 :

magnitude of exceptions, or zero

 𝐸𝑆𝑡 :

predicted ES

 The graph

𝑠↦

𝑋𝑡 𝐼𝑡 𝑠 𝑡=1 𝑇 𝛼 𝐸𝑆

𝛼,𝑡

+

𝑠 𝑇

for 𝑠 = 1, … , 𝑇 allows us to

visualize the time evolution of the contributions to the final 𝑍2 and check time independence

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𝑍2 Past History (of a Poor Risk Model) Z stat (ES) Z stat (VaR) Critical values (ES) Critical values (VaR) jpmembig/hist1y - confidence level: 97.5% 1

Backtest stats (rolling window)

0.5

0

-0.5

-1

-1.5

-2

-2.5

Oct13

Jan14

Apr14

46

Jul14

𝑍2 Is Not Rejected (But Some Exceptions Cluster) P&L - jpmembig/hist1y - confidence level: 97.5%

Return [%]

2 VaR bands ES bands

0 -2 -4

Jul13

Oct13

Jan14

Apr14

Contribution to ZES statistic 0.2 0 -0.2 -0.4 -0.6

Jul13

Oct13

Jan14

Apr14

Cumulative contribution to ZES statistic 1 0 -1 -2

Jul13

Oct13

Jan14

47

Apr14

𝑍2 Too High – Risk Overestimation P&L - jpmembig/hist1y - confidence level: 97.5%

Return [%]


1 0 -1 -2

Oct13

Jan14

Apr14

Jul14


Oct13

Jan14

Apr14

Jul14

Cumulative contribution to ZES statistic 1 0 -1 -2

Oct13

Jan14

Apr14

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Jul14

𝑍2 Far Too Low – Model failure P&L - jpmembig/hist1y - confidence level: 97.5%

Return [%]


0 -2 -4

Feb13

May13

Aug13

Nov13


Feb13

May13

Aug13

Nov13

Cumulative contribution to ZES statistic 1 0 -1 -2 -3

Feb13

May13

Aug13

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Nov13

Back to Elicitability Where Is The Catch?

Elicitable ≠ Backtestable  We have shown that ES can be backtested without being elicitable  Therefore backtestable ⇒ elicitable  We are tempted to conclude that elicitability is not the only way to backtest

backtestable elicitable

VaR

 Actually, there is even more…

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ES

Elicitability: Model Selection, Not Model Testing  If a measure is elicitable, we can rank models by their mean score  However, this is a relative, not an absolute scale  A mean score alone doesn’t tell us anything about the validity of a single model

 A mean score allows to choose the best model among several ones which forecast the same random process  Ex: Bank A has three VaR forecast models and runs a contest to select the best one  → Model selection

 Statistical test instead provides a validation with absolute significance  Ex: Bank A wants to validate the model  Ex: Regulators want to compare models of Banks A, B, C, …Z against the same scale  → Model testing (a.k.a. validation)

 This key observation has been completely overlooked so far in the public debate

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Elicitability Has Nothing To Do with Backtestability  So, in reality the situation is as follows:  Elicitability is irrelevant for model testing!

Mode

backtestable elicitable

ES

VaR

 As a matter of fact, the Mode is elicitable  It is not by chance that VaR is not backtested via elicitability 53

By the Way, ES is Elicitable (at Least in Practice)  A way to see why ES is not elicitable is to realize that there is no expression like 𝔼 𝐿 𝑋, 𝐸𝑆 = 0  If that were the case, we could interpret 𝜕𝑆 𝑋, 𝑒 𝐿 𝑋, 𝐸𝑆 = 𝜕𝑒 𝑒=𝐸𝑆

and build a scoring function 𝑆  However, there do exist null expectations involving both ES and VaR 𝔼 (𝑋 + 𝐸𝑆)(𝑋 + 𝑉𝑎𝑅 < 0) = 0 𝔼 𝑋 𝑋 + 𝑉𝑎𝑅 < 0 + 𝛼𝐸𝑆 = 0

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By the Way, ES is Elicitable (at Least in Practice)  It is (tedious but) straightforward to build a scoring function 𝑆𝑊

𝛼𝑒 2 𝑊𝛼𝑣 2 𝑊 𝑥2 − 𝑣2 𝑣, 𝑒, 𝑥 = + − 𝛼𝑒𝑣 + 𝑒 𝑣 + 𝑥 + 2 2 2

(𝑥 + 𝑣 < 0)

which jointly elicits ES and VaR if 𝐸𝑆 < 𝑊 𝑉𝑎𝑅, for a chosen 𝑊 ∈ ℝ 𝑉𝑎𝑅, 𝐸𝑆 = arg min 𝔼 𝑆 𝑊 (𝑣, 𝑒, 𝑋) 𝑣,𝑒

 Some 𝑊 ∼ 2 will do for any concrete application  Perfectly analogous to the variance, which is elicitable only jointly with the mean (Lambert et al. 2008)  … we do not know any 𝔼 𝐿 𝑋, 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒

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=0

Model Selection via ES-VaR Elicitability  Example of elicitability contest for different Riskmetrics models on different test portfolios  Data: RiskMetrics Year in Review 2014

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On Davis’ Consistency  Davis (2014): a risk measure 𝑋 is consistent if we can write 1 𝑛→∞ 𝑏𝑛

lim

𝑛 𝑖=1

𝑙(𝑋𝑖 , 𝑌𝑖 ) = 0 a.s.

 where 𝑋𝑖 and 𝑌𝑖 are model predictions for the price process and for the risk measure respectively and 𝑏𝑛 is a suitable divergent sequence  𝑙 is called a calibration function  Also in this case, if we extend the notion of consistency to vector statistics , we can prove it for 𝑌 = (𝐸𝑆, 𝑉𝑎𝑅) choosing  𝑏𝑛 = 𝑛

 𝑙 𝑋, 𝐸𝑆, 𝑉𝑎𝑅 =

𝑋(𝑋+𝑉𝑎𝑅