simulate

Simulate Du-Escanciano (DE) expected shortfall (ES) test statistics

Syntax

ebtde = simulate(ebtde)

ebtde = simulate(___,Name,Value)

Description

ebtde = simulate(ebtde) performs a simulation of the Du-Escanciano (DE) [1] expected shortfall (ES) test statistics. simulate simulates scenarios and calculates the supported test statistics for each scenario. The function uses the simulated test statistics to estimate the significance of the ES backtests when the CriticalValueMethod name-value pair argument for unconditionalDE or conditionalDE is set to 'simulation'.

ebtde = simulate(___,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 a Simulation

Create an esbacktestbyde object for a t model with 10 degrees of freedom. First, run a conditionalDE test based on 1000 scenarios and then use the simulate function to run a second simulation with 5000 scenarios.

load ESBacktestDistributionData.mat
  rng('default'); % For reproducibility
    % Constructor runs simulation with 1000 scenarios
  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);
% Run conditionalDE tests
conditionalDE(ebtde,'CriticalValueMethod','simulation')

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        0.003        15.285           3.2822           0.088175            1966           "simulation"           1         1000         0.95   
       "S&P"       "t(10) 97.5%"     0.975         reject        0.006        16.177           3.9304           0.090711            1966           "simulation"           1         1000         0.95   
       "S&P"       "t(10) 99%"        0.99         reject        0.037        6.9975           4.1995            0.05966            1966           "simulation"           1         1000         0.95   

The tests report 1000 scenarios, see the Scenarios column.

Run a second simulation with 5000 scenarios

ebtde = simulate(ebtde,'NumScenarios',5000);
conditionalDE(ebtde,'CriticalValueMethod','simulation')

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        0.0016       15.285           3.2535           0.088175            1966           "simulation"           1         5000         0.95   
       "S&P"       "t(10) 97.5%"     0.975         reject        0.0046       16.177           3.7668           0.090711            1966           "simulation"           1         5000         0.95   
       "S&P"       "t(10) 99%"        0.99         reject        0.0362       6.9975           3.8144            0.05966            1966           "simulation"           1         5000         0.95   

The tests show 5000 scenarios and updated p-values and critical values.

Input Arguments

ebtde — esbacktestbyde object

object

esbacktestbyde object, which contains a copy of the 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 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: ebtde = simulate(ebtde,'NumLags',10,'NumScenarios',1000000,'BlockSize',10000,'TestList','conditionalDE')

'NumLags' — Number of lags in the conditionalDE test statistic

5 (default) | positive integer

Number of lags in the conditionalDE test statistic, specified as the comma-separated pair consisting of 'NumLags' and a positive integer. The simulated test statistics are stored for all lags from 1 to NumLags, so that the conditionalDE test results are available for any number of lags between 1 and NumLags after running the simulate function.

Data Types: double

'NumScenarios' — Number of scenarios to simulate

1000 (default) | scalar positive integer

Number of scenarios to simulate, specified using the comma-separated pair consisting of 'NumScenarios' and a scalar positive integer.

Data Types: double

'BlockSize' — Number of scenarios to simulate in single simulation block

1000 (default) | scalar positive integer

Number of scenarios to simulate in a single simulation block, specified using the comma-separated pair consisting of 'BlockSize' and a scalar positive integer.

Data Types: double

'TestList' — Indicator for which test statistics to simulate

["conditionalDE","unconditionalDE"] (default) | character vector with a value of 'conditionalDE' or 'unconditionalDE' | string with a value of "conditionalDE" or "unconditionalDE"

Indicator for which test statistics to simulate, specified as the comma-separated pair consisting of 'TestList' and a cell array of character vectors or a string array with the value 'conditionalDE', 'unconditionalDE'.

Data Types: cell | string

Output Arguments

ebtde — Updated esbacktestbyde object

object

ebtde is returned as an updated esbacktestbyde object. After you run simulate, the updated esbacktestbyde object stores the simulated test statistics, which unconditionalDE uses to calculate p-values and generate test results.

For more information on the esbacktestbyde object, see esbacktestbyde.

More About

Simulation of Test Statistics

The simulation of test statistics requires simulating scenarios of returns, assuming the distribution of returns X_t ~ P_t is correct (null hypothesis), and computing the corresponding tests statistics for each scenario.

More specifically, the following steps describe the simulation process. The description uses the conditional test statistic C_ES for concreteness, but the same steps apply to the unconditional test statistic U_ES.

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

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

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

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

   where I(.) is the indicator function.

To compute the test statistic in step 2, the ranks or mapped returns U_t = P_t(X_t) need to be computed (see the definition of the test statistics for unconditionalDE and conditionalDE). Assuming that the model distribution is correct, the ranks U_t are always uniformly distributed in the unit interval. Therefore, in practice, directly simulating ranks is more efficient than simulating returns and then transforming the returns into ranks.

The simulate function implements the simulation process more efficiently as follows:

1. Simulated M scenarios of returns as

   $$\begin{array}{l}{U}^s=(U_1^s,\mathrm{...},U_N^s),s=1,\mathrm{...},M,\\ \text{with }{U}_t^s~Uniform(0,1).\end{array}$$

2. Compute the corresponding test statistic C_ES using the simulated ranks U^s as

   $$C_{ES}^s=C_{ES}(U_1^s,\mathrm{...},U_N^s),s=1,\mathrm{...},M.$$

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

   $$P_C=P\left[C_{ES}\le x\right]=\frac{1}{M}I(C_{ES}^s\le x).$$

After you determine the empirical distribution of the test statistic P_C in step 3, the significance of the test follows the descriptions provided for unconditionalDE and conditionalDE. The same steps apply to the unconditional test statistic U_ES and its distribution function P_U.

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).