Conditional expected shortfall (ES) backtest by Acerbi and Szekely
TestResults = conditional(ebts)VaRTest, and
the standalone conditional ES backtest. A 'reject' result on
either underlying test produces a 'reject' result on the
conditional test.
[
adds optional name-value pair arguments for TestResults,SimTestStatistic] = conditional(ebts,Name,Value)TestLevel and
VaRTest.
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
ebts — esbacktestbysim objectesbacktestbysim (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.
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.
[TestResults,SimTestStatistic] =
conditional(ebts,'TestLevel',0.99)'TestLevel' — Test confidence level0.95
(default) | numeric value between 0 and
1Test 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
TestResults — ResultsResults, 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 statisticSimulated values of the test statistic, returned as a
NumVaRs-by-NumScenarios
numeric array.
The conditional test is also known as the first Acerbi-Szekely test.
The conditional test statistic is based on the conditional relationship
where
Xt is the portfolio outcome, that is
the portfolio return or portfolio profit and loss for period
t.
VaRt is the estimated VaR for period
t.
ESt is the estimated expected
shortfall for period t.
The number of failures is defined as
where
N is the number of periods in the test window
(t = 1,…,N).
It is the VaR failure indicator on
period t with a value of 1 if
Xt < -VaR, and
0 otherwise.
The conditional test statistic is defined as:
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
Zcond. The conditional test
accepts the model only when both the VaR test and the standalone conditional
test accept the model.
Under the assumption that the distributional assumptions are
correct, the expected value of the test statistic
Zcond, assuming at least one VaR
failure, is 0.
This is expressed as:
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.
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, , 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
NpVaR gets smaller.
[1] Acerbi, C. and Szekely, B. Backtesting Expected Shortfall. MSCI Inc. December, 2014.
bin | cc | cci | esbacktestbyde | esbacktestbysim | minBiasAbsolute | minBiasRelative | pof | quantile | runtests | simulate | summary | tbf | tbfi | tl | tuff | unconditional
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