Conditional Du-Escanciano (DE) expected shortfall (ES) backtest
esbacktestbydeObject and 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
esbacktestbyde object, which contains a copy of the
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
comma-separated pairs of
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
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
'simulation'| string with values of
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
'NumLags'— Number of lags in
1(default) | positive integer
Number of lags in the
specified as the comma-separated pair consisting of
'NumLags' and a positive integer.
'TestLevel'— Test confidence level
0.95(default) | numeric value between
Test confidence level, specified as the comma-separated pair
'TestLevel' and a numeric value
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:
'PortfolioID' — Portfolio ID for
the given data
'VaRID' — VaR ID for each of the
'VaRLevel' — VaR level
array with the categories
'reject', which indicate the result
of the conditional DE test
P-value of the conditional DE test
'TestStatistic'— Conditional DE
'CriticalValue'— Critical value
for the conditional DE test
Autocorrelation for the reported number of lags
'Observations'— Number of
used to compute confidence intervals and
'NumLags'— Number of lags
'Scenarios'— Number of scenarios
simulated to get the p-values
'TestLevel'— Test confidence
If you specify
'large-sample', the function reports the
For the test results, the terms
'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 statistics
Simulated values of the test statistics, returned as an
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:
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 .
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
supports the large-sample approximation and simulation methods. You can specify
one of these methods by using the optional name-value pair argument
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.
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.
 Du, Z., and J. C. Escanciano. "Backtesting Expected Shortfall: Accounting for Tail Risk." Management Science. Vol. 63, Issue 4, April 2017.
 Basel Committee on Banking Supervision. "Minimum Capital Requirements for Market Risk". January 2016 (https://www.bis.org/bcbs/publ/d352.pdf).