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minBiasRelative

Minimally biased relative test for Expected Shortfall (ES) backtest by Acerbi-Szekely

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Syntax

TestResults = minBiasRelative(ebts)
[TestResults,SimTestStatistic] = minBiasRelative(ebts,Name,Value)

Description

example

TestResults= minBiasRelative(ebts)运行的相关版本的最低限度的偏见Expected Shortfall (ES) back test by Acerbi-Szekely (2017) using theesbacktestbysimobject.

example

[TestResults,SimTestStatistic] = minBiasRelative(ebts,Name,Value)specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

Examples

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Open Live Script

Create anesbacktestbysimobject.

loadESBacktestBySimDatarng('default');% for reproducibilityebts = esbacktestbysim(Returns,VaR,ES,"t",...'DegreesOfFreedom',10,...'Location',Mu,...'Scale',Sigma,...'PortfolioID',"S&P",...'VaRID',["t(10) 95%","t(10) 97.5%","t(10) 99%"],...'VaRLevel',VaRLevel);

Generate theTestResultsand theSimTestStatisticreports for theminBiasRelativeES backtest.

[TestResults,SimTestStatistic] = minBiasRelative(ebts)
TestResults=3×10 tablePortfolioID VaRID VaRLevel MinBiasRelative PValue TestStatistic CriticalValue Observations Scenarios TestLevel ___________ _____________ ________ _______________ ______ _____________ _____________ ____________ _________ _________ "S&P" "t(10) 95%" 0.95 reject 0.003 -0.10509 -0.056072 1966 1000 0.95 "S&P" "t(10) 97.5%" 0.975 reject 0 -0.15603 -0.073324 1966 1000 0.95 "S&P" "t(10) 99%" 0.99 reject 0 -0.26716 -0.104 1966 1000 0.95
SimTestStatistic =3×10000.0860 0.0284 -0.0480 0.0176 0.0262 0.0309 -0.0107 0.0361 -0.0171 -0.0154 -0.0247 0.0047 0.0055 0.0217 0.0073 0.0519 0.0388 0.1023 0.0516 -0.0326 -0.0203 0.0192 -0.0022 -0.0198 -0.0205 0.0036 0.0285 0.0462 -0.0134 -0.0335 -0.0301 0.0223 -0.0291 -0.0494 -0.0246 -0.0075 0.0060 0.0516 0.0498 -0.0020 -0.0008 -0.0060 -0.1238 -0.0222 0.0447 0.0352 -0.0422 -0.0667 0.0429 0.0079 0.1145 0.0177 -0.0741 0.0357 0.0505 0.0275 -0.0136 0.0421 -0.0190 -0.0230 -0.0074 0.0098 0.0209 0.0229 -0.0012 0.0561 0.0421 0.1078 0.0530 -0.0306 -0.0167 0.0193 0.0014 -0.0214 -0.0214 -0.0224 0.0185 0.0730 -0.0089 -0.0278 -0.0458 0.0348 -0.0066 -0.0522 -0.0304 -0.0095 -0.0073 0.0490 0.0575 -0.0118 -0.0051 0.0058 -0.1318 -0.0280 0.0349 0.0473 -0.0522 -0.0894 0.0420 0.0120 0.1435 -0.0195 -0.0915 0.0478 0.0796 0.0397 -0.0022 0.0282 -0.0055 -0.0587 0.0631 0.0314 0.0446 0.0340 0.0034 0.0706 0.0652 0.1414 0.0783 0.0148 -0.0196 0.0057 0.0395 -0.0479 -0.0352 -0.0644 0.0034 0.0960 -0.0064 -0.0081 -0.0651 0.0436 0.0241 -0.0357 -0.0170 0.0242 -0.0282 0.0730 0.0449 -0.0388 0.0169 0.0506 -0.1160 -0.0663 0.0338 0.0610 -0.0815 -0.1285 0.0363 0.0209

Input Arguments

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esbacktestbysim(ebts) object, which contains a copy of the given data (thePortfolioData,VarData,ESData, andDistributionproperties) and all combinations of portfolio IDs, VaR IDs, and VaR levels to be tested. For more information on creating anesbacktestbysimobject, seeesbacktestbysim.

Name-Value Arguments

Specify optional pairs of arguments asName1=Value1,...,NameN=ValueN, whereNameis the argument name andValueis the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and encloseNamein quotes.

Example:minBiasRelative(于本TestLevel, 0.99)

Test confidence level, specified as the comma-separated pair consisting of'TestLevel'and a numeric value between0and1.

Data Types:double

Output Arguments

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Results, returned as a table where the rows correspond to all combinations of portfolio IDs, VaR IDs, and VaR levels to be tested. The columns correspond to the following information:

  • 'PortfolioID'— Portfolio ID for the given data

  • 'VaRID'— VaR ID for each of the VaR data columns provided

  • 'VaRLevel'— VaR level for the corresponding VaR data column

  • 'MinBiasRelative'— Categorical array with categories'accept'and'reject'that indicate the result of theminBiasRelativetest

  • 'PValue'—p-value for theminBiasRelativetest

  • 'TestStatistic'—minBiasRelativetest statistic

  • 'CriticalValue'— Critical value forminBiasRelativetest

  • 'Observations'— Number of observations

  • 'Scenarios'— Number of scenarios simulated to obtainp-values

  • 'TestLevel'— Test confidence level

Note

For the test results, the terms'accept'and'reject'are used for convenience. Technically, a test does not accept a model; rather, a test fails to reject it.

Simulated values of the test statistic, returned as aNumVaRs-by-NumScenariosnumeric array.

More About

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Minimally Biased Test, Relative Version by Acerbi and Szekely

Therelative versionof the Acerbi-Szekely test ([1]) computes theTestStatisticin the units of data.

The absolute version of the minimally biased test statistic is given by

Z m i n b i a s r e l = 1 N ∑ t = 1 N 1 E S t ( E S t − V a R t − 1 p V a R ( X t + V a R t ) _ )

where

Xtis the portfolio outcome, that is, the portfolio return or portfolio profit and loss for periodt.

VaRtis the essential VaR for periodt.

EStis the expected shortfall for periodt.

pVaRis the probability of VaR failure, defined as 1 - VaR level.

Nis the number of periods in the test window (t= 1,...N).

(x)_ is the negative part function defined as (x)_ =max(0,-x).

Significance of the Test

Negative values of the test statistic indicate risk underestimation.

The minimally biased test is a one-sided test that rejects the model when there is evidence that the model underestimates risk (for technical details, see Acerbi-Szekely [1] and [2]). The test rejects the model when thep-value is less than1minus the test confidence level. For more information on the steps to simulate the test statistics and details on the computation of thep-values and critical values, seesimulate.

ES backtests are necessarily approximated in that they are sensitive to errors in the predicted VaR. However, the minimally biased test has only a small sensitivity to VaR errors and the sensitivity is prudential, in the sense that VaR errors lead to a more punitive ES test. For details, see Acerbi-Szekely ([1] and [2]). When distribution information is available using the minimally biased test is recommended.

References

[1] Acerbi, Carlo, and Balazs Szekely. "General Properties of Backtestable Statistics."SSRN Electronic Journal.(January, 2017).

[2] Acerbi, Carlo, and Balazs Szekely. "The Minimally Biased Backtest for ES."Risk.(September, 2019).

[3] Acerbi, C. and D. Tasche. "On the Coherence of Expected Shortfall."Journal of Banking and Finance.Vol. 26, 2002, pp. 1487-1503.

[4] Rockafellar, R. T. and S. Uryasev. "Conditional Value-at-Risk for General Loss Distributions."Journal of Banking and Finance.Vol. 26, 2002, pp. 1443-1471.

Version History

Introduced in R2020b

See Also

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