Conditional Du-Escanciano (DE) expected shortfall (ES) backtest
TestResults = conditionalDE(ebtde)
[
specifies options using one or more name-value pair arguments in addition to the
input argument in the previous syntax.TestResults,SimTestStatistic] = conditionalDE(___,Name,Value)
esbacktestbyde Object and Run a ConditionalDE TestCreate an esbacktestbyde object for a t model with 10 degrees of freedom and 2 lags, and then run a conditionalDE test.
load ESBacktestDistributionData.mat rng('default'); % For reproducibility ebtde = esbacktestbyde(Returns,"t",... 'DegreesOfFreedom',T10DoF,... 'Location',T10Location,... 'Scale',T10Scale,... 'PortfolioID',"S&P",... 'VaRID',["t(10) 95%","t(10) 97.5%","t(10) 99%"],... 'VaRLevel',VaRLevel); conditionalDE(ebtde,'NumLags',2)
ans=3×13 table
PortfolioID VaRID VaRLevel ConditionalDE PValue TestStatistic CriticalValue AutoCorrelation Observations CriticalValueMethod NumLags Scenarios TestLevel
___________ _____________ ________ _____________ __________ _____________ _____________ _______________ ____________ ___________________ _______ _________ _________
"S&P" "t(10) 95%" 0.95 reject 3.2121e-09 39.113 5.9915 0.11009 1966 "large-sample" 2 NaN 0.95
"S&P" "t(10) 97.5%" 0.975 reject 1.6979e-07 31.177 5.9915 0.087348 1966 "large-sample" 2 NaN 0.95
"S&P" "t(10) 99%" 0.99 reject 9.1526e-05 18.598 5.9915 0.076814 1966 "large-sample" 2 NaN 0.95
ebtde — esbacktestbyde objectesbacktestbyde object, which contains a copy of the
data (the PortfolioData, VarData,
and ESData properties) and all combinations of
portfolio ID, VaR ID, and VaR levels to be tested. For more information
on creating an esbacktestbyde object, see esbacktestbyde.
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 =
conditionalDE(ebtde,'CriticalValueMethod','simulation','NumLags',10,'TestLevel',0.99)'CriticalValueMethod' — Method to compute critical values, confidence intervals, and p-values'large-sample'
(default) | character vector with values of 'large-sample'
or 'simulation' | string with values of "large-sample" or
"simulation"Method to compute critical values, confidence intervals, and
p-values, specified as the comma-separated
pair consisting of 'CriticalValueMethod' and a
character vector or string with a value of
'large-sample' or
'simulation'.
Data Types: char | string
'NumLags' — Number of lags in conditionalDE test1
(default) | positive integerNumber of lags in the conditionalDE test,
specified as the comma-separated pair consisting of
'NumLags' and a positive integer.
Data Types: double
'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
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:
'PortfolioID' — Portfolio ID for
the given data
'VaRID' — VaR ID for each of the
VaR levels
'VaRLevel' — VaR level
'ConditionalDE'— Categorical
array with the categories 'accept' and
'reject', which indicate the result
of the conditional DE test
'PValue'—
P-value of the conditional DE test
'TestStatistic'— Conditional DE
test statistic
'CriticalValue'— Critical value
for the conditional DE test
'AutoCorrelation'—
Autocorrelation for the reported number of lags
'Observations'— Number of
observations
'CriticalValueMethod'— Method
used to compute confidence intervals and
p-values
'NumLags'— Number of lags
'Scenarios'— Number of scenarios
simulated to get the p-values
'TestLevel'— Test confidence
level
Note
If you specify CriticalValueMethod as
'large-sample', the function reports the
number of 'Scenarios' as
NaN.
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 the test statisticsSimulated values of the test statistics, returned as an
NumVaRs-by-NumScenarios
numeric array.
The conditional DE test is a one-sided test to check if the test statistic is much larger than zero.
The test statistic for the conditional DE test is derived in several steps. First, define the autocovariance for lag j:
where
ɑ = 1- VaRLevel.
Ht is the cumulative failures or violations process: Ht = (α - Ut)I(Ut < α) / α, where I(x) is the indicator function.
Ut are the ranks or mapped returns Ut = Pt(Xt), where Pt(Xt) = P(Xt | θt) is the cumulative distribution of the portfolio outcomes or returns Xt over a given test window t = 1,...N and θt are the parameters of the distribution. For simplicity, the subindex t is both the return and the parameters, understanding that the parameters are those used on date t, even though those parameters are estimated on the previous date t-1, or even prior to that.
The exact theoretical mean α/2, as opposed to the sample mean, is used in the autocovariance formula, as suggested in the paper by Du and Escanciano [1].
The autocorrelation for lag j is then
The test statistic for m lags is
Significance of the Test
The test statistic CES is a random variable and a function of random return sequences or portfolio outcomes X1,…,XN:
For returns observed in the test window 1,…,N, the test statistic attains a fixed value:
In general, for unknown returns that follow a distribution of Pt, the value of CES is uncertain and it follows a cumulative distribution function:
This distribution function computes a confidence interval and a
p-value. To determine the distribution
PC, the esbacktestbyde class
supports the large-sample approximation and simulation methods. You can specify
one of these methods by using the optional name-value pair argument
CriticalValueMethod.
For the large sample approximation method, the distribution PC is derived from an asymptotic analysis. If the number of observations N is large, the test statistic is approximately distributed as a chi-square distribution with m degrees of freedom:
Note that the limiting distribution is independent of α.
If αtest = 1 - test confidence level, then the critical value CV is the value that satisfies the equation
The p-value is determined as
The test rejects if pvalue < αtest.
For the simulation method, the distribution PCis estimated as follows
Simulate M scenarios of returns as
Compute the corresponding test statistic as
Define PC as the empirical distribution of the simulated test statistic values as
where I(.) is the indicator function.
In practice, simulating ranks is more efficient than simulating returns and
then transforming the returns into ranks. simulate.
For the empirical distribution, the value of 1-PC(x) may be different than P[CES ≥ x] because the distribution may have nontrivial jumps (simulated tied values). Use the latter probability for the estimation of confidence levels and p-values.
If ɑtest = 1 - test confidence level, then the critical value of levels CV is the value that satisfies the equation
The reported critical value CV is one of the simulated test statistic values CsES that approximately solves the preceding equation.
The p-value is determined as
The test rejects if pvalue < αtest.
[1] Du, Z., and J. C. Escanciano. "Backtesting Expected Shortfall: Accounting for Tail Risk." Management Science. Vol. 63, Issue 4, April 2017.
[2] Basel Committee on Banking Supervision. "Minimum Capital Requirements for Market Risk". January 2016 (https://www.bis.org/bcbs/publ/d352.pdf).
esbacktestbyde | esbacktestbysim | runtests | simulate | summary | unconditionalDE
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