conditional

Conditional expected shortfall (ES) backtest by Acerbi and Szekely

Syntax

TestResults = conditional(ebts)

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

Description

TestResults = conditional(ebts) runs the conditional ES backtest of Acerbi-Szekely (2014). The conditional test has two underlying tests, a preliminary Value-at-Risk (VaR) backtest that is specified using the name-value pair argument VaRTest, and the standalone conditional ES backtest. A 'reject' result on either underlying test produces a 'reject' result on the conditional test.

[TestResults,SimTestStatistic] = conditional(ebts,Name,Value) adds optional name-value pair arguments for TestLevel and VaRTest.

Examples

Run an ES Conditional Test

Create an esbacktestbysim object.

load ESBacktestBySimData
rng('default'); % for reproducibility
ebts = 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 the ES conditional test report.

TestResults = conditional(ebts)

TestResults=3×14 table
    PortfolioID        VaRID        VaRLevel    Conditional    ConditionalOnly    PValue    TestStatistic    CriticalValue    VaRTest    VaRTestResult    VaRTestPValue    Observations    Scenarios    TestLevel
    ___________    _____________    ________    ___________    _______________    ______    _____________    _____________    _______    _____________    _____________    ____________    _________    _________

       "S&P"       "t(10) 95%"        0.95        reject           reject             0       -0.092302        -0.043941       "pof"        accept           0.70347           1966          1000         0.95   
       "S&P"       "t(10) 97.5%"     0.975        reject           reject         0.001        -0.11714        -0.052575       "pof"        accept           0.40682           1966          1000         0.95   
       "S&P"       "t(10) 99%"        0.99        reject           reject         0.003        -0.14608        -0.085433       "pof"        accept           0.11536           1966          1000         0.95   

Input Arguments

ebts — esbacktestbysim object

object

esbacktestbysim (ebts) object, which contains a copy of the given data (the PortfolioData, VarData, ESData, and Distribution properties) and all combinations of portfolio ID, VaR ID, and VaR levels to be tested. For more information on creating an esbacktestbysim object, see esbacktestbysim.

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [TestResults,SimTestStatistic] = conditional(ebts,'TestLevel',0.99)

TestLevel — Test confidence level

0.95 (default) | numeric value between 0 and 1

Test confidence level, specified as the comma-separated pair consisting of 'TestLevel' and a numeric value between 0 and 1.

Data Types: double

VaRTest — Indicator for VaR back test

'pof' (default) | character vector with a value of 'tl', 'bin', 'pof', 'tuff', 'cc', 'cci', 'tbf', or 'tbfi' | string array with a value of 'tl', 'bin', 'pof', 'tuff', 'cc', 'cci', 'tbf', or 'tbfi'

Indicator for VaR back test, specified as the comma-separated pair consisting of 'VaRTest' and a character vector or string array with a value of 'tl', 'bin', 'pof', 'tuff', 'cc', 'cci', 'tbf', or 'tbfi'. For more information on these VaR backtests, see varbacktest.

Note

The specified VaRTest is run using the same TestLevel value that is specified with the TestLevel name-value pair argument in the conditional function.

Data Types: char | string

Output Arguments

TestResults — Results

table

Results, returned as a table where the rows correspond to all combinations of portfolio ID, VaR ID, 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.

• 'Conditional'— Categorical array with categories 'accept' and 'reject' indicating the result of the conditional test. This result combines the outcome of the 'ConditionalOnly' column and the VaR test.

• 'ConditionalOnly'— Categorical array with categories 'accept' and 'reject' indicating the result of the standalone conditional test, independent of the VaR test outcome.

• 'PValue'— P-value of the standalone conditional test (for the'ConditionalOnly' column).

• 'TestStatistic'— Conditional test statistic (for the'ConditionalOnly' column).

• 'CriticalValue'— Critical value for the conditional test.

• 'VaRTest'— String array indicating the selected VaR test as specified by the VaRTest argument.

• 'VaRTestResult'— Categorical array with categories 'accept' and 'reject' indicating the result of the VaR test selected with the 'VaRTest' argument.

• 'VaRTestPValue'— P-value for the VaR backtest. If the traffic-light test (tl) is used, this is 1 minus the traffic-light test's 'Probability' column value.

• 'Observations'— Number of observations.

• 'Scenarios'— Number of scenarios simulated to get the p-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.

SimTestStatistic — Simulated values of test statistic

numeric array

Simulated values of the test statistic, returned as a NumVaRs-by-NumScenarios numeric array.

More About

Conditional Test by Acerbi and Szekely

The conditional test is also known as the first Acerbi-Szekely test.

The conditional test statistic is based on the conditional relationship

$$E{S}_t=-E_t\left[X_t|X_t<-Va{R}_t\right]$$

where

X_t is the portfolio outcome, that is the portfolio return or portfolio profit and loss for period t.

VaR_t is the estimated VaR for period t.

ES_t is the estimated expected shortfall for period t.

The number of failures is defined as

$$NumFailures={\displaystyle \sum _{t=1}^N{I}_t}$$

where

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

I_t is the VaR failure indicator on period t with a value of 1 if X_t < -VaR, and 0 otherwise.

The conditional test statistic is defined as:

$$Z_{cond}=\frac{1}{NumFailures}{\displaystyle \sum _{t=1}^N\frac{X_t{I}_t}{E{S}_t}+1}$$

The conditional test has two parts. A VaR backtest, specified by the VaRTest name-value pair argument, must be run for the number of failures (NumFailures), and a standalone conditional test is performed for the conditional test statistic Z_cond. The conditional test accepts the model only when both the VaR test and the standalone conditional test accept the model.

Significance of the Test

Under the assumption that the distributional assumptions are correct, the expected value of the test statistic Z_cond, assuming at least one VaR failure, is 0.

This is expressed as:

$$E[Z_{cond}|NumFailures>0]=0$$

Negative values of the test statistic indicate risk underestimation. The conditional test is a one-sided test that rejects when there is evidence that the model underestimates risk (for technical details on the null and alternative hypotheses, see Acerbi-Szekely, 2014). The conditional test rejects the model when the p-value is less than 1 minus the test confidence level.

For more information on the steps to simulate the test statistics and the details for the computation of the p-values and critical values, see simulate.

Edge Cases

The conditional test statistic is undefined (NaN) when there are no VaR failures in the data (NumFailures = 0).

The p-value is set to NaN in these cases, and test result is to 'accept', because there is no evidence of risk underestimation.

Likewise, the simulated conditional test statistic is undefined (NaN) for scenarios with no VaR failures. These scenarios are discarded for the estimation of the significance of the test. Under the assumption that the distributional assumptions are correct, $$E[Z_{cond}|NumFailures>0]=0$$, so the significance is computed over scenarios with at least one failure (NumFailures > 0). The number of scenarios reported by the conditional test function is the number of scenarios with at least one VaR failure. The number of scenarios reported can be smaller than the total number of scenarios simulated. The critical value is estimated over the scenarios with at least one VaR failure. If the simulated test statistic is NaN for all scenarios, the critical value is set to NaN. Scenarios with no failures are more likely as the expected number of failures Np_VaR gets smaller.

References

[1] Acerbi, C. and Szekely, B. Backtesting Expected Shortfall. MSCI Inc. December, 2014.