unconditionalDE

Unconditional Du-Escanciano (DE) expected shortfall (ES) backtest

Syntax

TestResults = unconditionalDE(ebtde)

[TestResults,SimTestStatistic] = unconditionalDE(___,Name,Value)

Description

TestResults = unconditionalDE(ebtde) runs the unconditional Du-Escanciano (DE) expected shortfall (ES) backtest [1] . The unconditional test supports critical values by large-scale approximation and by finite-sample simulation.

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

Examples

Create an esbacktestbyde Object and Run an UnconditionalDE Test

Create an esbacktestbyde object for a t model with 10 degrees of freedom, and then run an unconditionalDE 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);
    unconditionalDE(ebtde)

ans=3×14 table
    PortfolioID        VaRID        VaRLevel    UnconditionalDE     PValue     TestStatistic     LowerCI      UpperCI     Observations    CriticalValueMethod    MeanLS      StdLS      Scenarios    TestLevel
    ___________    _____________    ________    _______________    ________    _____________    _________    _________    ____________    ___________________    ______    _________    _________    _________

       "S&P"       "t(10) 95%"        0.95          accept            0.181       0.028821       0.019401     0.030599        1966          "large-sample"        0.025    0.0028565       NaN         0.95   
       "S&P"       "t(10) 97.5%"     0.975          accept         0.086278       0.015998      0.0085028     0.016497        1966          "large-sample"       0.0125    0.0020394       NaN         0.95   
       "S&P"       "t(10) 99%"        0.99          reject         0.016871      0.0080997      0.0024575    0.0075425        1966          "large-sample"        0.005    0.0012972       NaN         0.95   

Input Arguments

ebtde — esbacktestbyde object

object

esbacktestbyde (ebtde) 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.

Name-Value Pair 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 = unconditionalDE(ebtde,'CriticalValueMethod','large-sample','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

'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

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:

• 'PortfolioID' — Portfolio ID for the given data

• 'VaRID' — VaR ID for each of the VaR levels

• 'VaRLevel' — VaR level

• 'UnconditionalDE'— Categorical array with the categories 'accept' and 'reject', which indicate the result of the unconditional DE test

• 'PValue'— P-value of the unconditional DE test

• 'TestStatistic'— Unconditional DE test statistic

• 'LowerCI'— Confidence-interval lower limit for the unconditional DE test statistic

• 'UpperCI'— Confidence-interval upper limit for the unconditional DE test statistic

• 'Observations'— Number of observations

• 'CriticalValueMethod'— Method for computing confidence intervals and p-values

• 'MeanLS'— Mean of the large-sample normal distribution; if CriticalValueMethod is 'simulation', 'MeanLS' is reported as NaN

• 'StdLS'— Standard deviation of the large-sample normal distribution; if CriticalValueMethod is 'simulation', 'StdLS' is reported as NaN

• 'Scenarios'— Number of scenarios simulated to get the p-values; if CriticalValueMethod is 'large-sample', the number of scenarios is reported as NaN

• '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 the test statistics

numeric array

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

More About

Unconditional DE Test

The unconditional DE test is a two-sided test to check if the test statistic is close to an expected value of ɑ/2, where ɑ = 1- VaRLevel.

The test statistic for the unconditional DE test is

$$U_{ES}=\frac{1}{N}{\displaystyle {\sum }_{t=1}^N{H}_t}$$

where

• H_t is the cumulative failures or violations process; H_t = (α - U_t)I(U_t < α) / α, where I(x) is the indicator function.

• U_t are the ranks or mapped returns U_t = P_t(X_t), where P_t(X_t) = P(X_t | θ_t) is the cumulative distribution of the portfolio outcomes or returns X_t over a given test window t = 1,...N and θ_t 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.

Significance of the Test

The test statistic U_ES is a random variable and a function of random return sequences:

$$U_{ES}=U_{ES}(X_1,\mathrm{...},X_N).$$

For returns observed in the test window 1,…,N, the test statistic attains a fixed value:

$$U_{ES}^{obs}=U_{ES}(X_1^{obs},\mathrm{...},X_N^{obs}).$$

In general, for unknown returns that follow a distribution of P_t, the value of U_ES is uncertain and follows a cumulative distribution function:

$$P_U(x)=P\left[U_{ES}\le x\right].$$

This distribution function computes a confidence interval and a p-value. To determine the distribution P_U, 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 P_U is derived from an asymptotic analysis. If the number of observations N is large, the test statistic U_ES is distributed as

$$U_{ES}\underset{dist}{\to }N\left(\frac{\alpha }{2},\frac{\alpha (1/3-\alpha /4)}{N}\right)=P_U$$

where N(μ,σ²) is the normal distribution with mean μ and variance σ².

Because the test statistic cannot be smaller than 0 or greater than 1, the analytical confidence interval limits are clipped to the interval [0,1]. Therefore, if the analytical value is negative, the test statistic is reset to 0, and if the analytical value is greater than 1, it is reset to 1.

The p-value is

$$p_{value}=2\ast \min \left\{P_U(U_{ES}^{obs}),1-P_U(U_{ES}^{obs})\right\}.$$

The test rejects if p_value < α_test.

For the simulation method, the distribution P_Uis estimated as follows

1. Simulate M scenarios of returns as

   $$X^s=(X_1^s,\mathrm{...},X_N^s),s=1,\mathrm{...},M.$$

2. Compute the corresponding test statistic as

   $$U_{ES}^s=U_{ES}^s(X_1^s,\mathrm{...},X_N^s),s=1,\mathrm{...},M.$$

3. Define P_U as the empirical distribution of the simulated test statistic values as

   $$P_U=P\left[U_{ES}\le x\right]=\frac{1}{M}I(U_{ES}^s\le x),$$

   where I(.) is the indicator function.

In practice, simulating ranks is more efficient than simulating returns and then transforming the returns into ranks. For more information, see simulate.

For the empirical distribution, the value of 1-P_U(x) can differ from the value of P[U_ES ≥ 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 ɑ_test = 1 - test confidence level, then the confidence intervals levels CI_lower and CI_upper are the values that satisfy equations:

$$\begin{array}{l}{P}_U(C{I}_{lower})=P\left[C{I}_{lower}\le U_{ES}\right]=\frac{{\alpha }_{test}}{2},\\ P\left[U_{ES}\ge C{I}_{upper}\right]=\frac{{\alpha }_{test}}{2}.\end{array}$$

The reported confidence interval limits CI_lower and CI_upper are simulated test statistic values U^s_ES that approximately solve the preceding equations.

The p-value is determined as

$$p_{value}=2\ast \min \left\{P\left[U_{ES}\le U_{ES}^{obs}\right],P\left[U_{ES}\ge U_{ES}^{obs}\right]\right\}.$$

The test rejects if p_value < α_test.

References

[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).