tuff

Time until first failure test for value-at-risk (VaR) backtesting

Syntax

TestResults = tuff(vbt)

TestResults = tuff(vbt,Name,Value)

Description

TestResults = tuff(vbt) generates the time until first failure (TUFF) test for value-at-risk (VaR) backtesting.

TestResults = tuff(vbt,Name,Value) adds an optional name-value pair argument for TestLevel.

Examples

Generate TUFF Test Results

Create a varbacktest object.

load VaRBacktestData
vbt = varbacktest(EquityIndex,Normal95)

vbt = 
  varbacktest with properties:

    PortfolioData: [1043x1 double]
          VaRData: [1043x1 double]
             Time: [1043x1 double]
      PortfolioID: "Portfolio"
            VaRID: "VaR"
         VaRLevel: 0.9500

Generate the tuff test results.

TestResults = tuff(vbt)

TestResults=1×9 table
    PortfolioID    VaRID    VaRLevel     TUFF     LRatioTUFF    PValueTUFF    FirstFailure    Observations    TestLevel
    ___________    _____    ________    ______    __________    __________    ____________    ____________    _________

    "Portfolio"    "VaR"      0.95      accept      1.7354       0.18773           58             1043          0.95   

Run the TUFF Test for VaR Backtests for Multiple VaRs at Different Confidence Levels

Use the varbacktest constructor with name-value pair arguments to create a varbacktest object.

load VaRBacktestData
    vbt = varbacktest(EquityIndex,...
       [Normal95 Normal99 Historical95 Historical99 EWMA95 EWMA99],...
       'PortfolioID','Equity',...
       'VaRID',{'Normal95' 'Normal99' 'Historical95' 'Historical99' 'EWMA95' 'EWMA99'},...
       'VaRLevel',[0.95 0.99 0.95 0.99 0.95 0.99])

vbt = 
  varbacktest with properties:

    PortfolioData: [1043x1 double]
          VaRData: [1043x6 double]
             Time: [1043x1 double]
      PortfolioID: "Equity"
            VaRID: ["Normal95"    "Normal99"    "Historical95"    "Historical99"    "EWMA95"    "EWMA99"]
         VaRLevel: [0.9500 0.9900 0.9500 0.9900 0.9500 0.9900]

Generate the tuff test results using the TestLevel optional input.

TestResults = tuff(vbt,'TestLevel',0.90)

TestResults=6×9 table
    PortfolioID        VaRID         VaRLevel     TUFF     LRatioTUFF    PValueTUFF    FirstFailure    Observations    TestLevel
    ___________    ______________    ________    ______    __________    __________    ____________    ____________    _________

     "Equity"      "Normal95"          0.95      accept      1.7354       0.18773           58             1043           0.9   
     "Equity"      "Normal99"          0.99      accept     0.36686       0.54472          173             1043           0.9   
     "Equity"      "Historical95"      0.95      accept      1.5348        0.2154           55             1043           0.9   
     "Equity"      "Historical99"      0.99      accept     0.36686       0.54472          173             1043           0.9   
     "Equity"      "EWMA95"            0.95      accept     0.13304        0.7153           28             1043           0.9   
     "Equity"      "EWMA99"            0.99      accept     0.14596       0.70243          143             1043           0.9   

Input Arguments

vbt — varbacktest object

object

varbacktest (vbt) object, contains a copy of the given data (the PortfolioData and VarData properties) and all combinations of portfolio ID, VaR ID, and VaR levels to be tested. For more information on creating a varbacktest object, see varbacktest.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: TestResults = tuff(vbt,'TestLevel',0.99)

TestLevel — Test confidence level

0.95 (default) | numeric between 0 and 1

Test confidence level, specified as the comma-separated pair consisting of 'TestLevel' and a numeric between 0 and 1.

Data Types: double

Output Arguments

TestResults — tuff test results

table

tuff test 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 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

• 'TUFF' — Categorical array with the categories accept and reject that indicate the result of the tuff test

• 'LRatioTUFF' — Likelihood ratio of the tuff test

• 'PValueTUFF' — P-value of the tuff test

• 'FirstFailure' — Number of periods until the first failure

• 'Observations' — Number of observations

• 'TestLevel' — Test confidence level

Note

For tuff test results, the terms accept and reject are used for convenience, technically a tuff test does not accept a model. Rather, the test fails to reject it.

More About

Time Until First Failure (TUFF) Test

The tuff function performs Kupiec's time until first failure test.

The TUFF test is a likelihood ratio test proposed by Kupiec (1995) to assess if the number of periods until the first failure is consistent with the VaR confidence level.

Algorithms

The likelihood ratio (test statistic) of the tuff test is given by

$$LRatioTUFF=-2\log \left(\frac{pVaR{\left(1-pVaR\right)}^{n-1}}{\left(\frac{1}{n}\right){\left(1-\frac{1}{n}\right)}^{n-1}}\right)=-2(\log (pVaR)+(n-1)\log (1-pVaR)+n\log (n)-(n-1)\log (n-1))$$

where n is the number of periods until the first failure and pVaR = 1 - VaRLevel. By the properties of the logarithm (if n = 1),

$$LRatioTUFF=-2\log (pVaR)$$

This is asymptotically distributed as a chi-square distribution with 1 degree of freedom.

The p-value of the tuff test is the probability that a chi-square distribution with 1 degree of freedom exceeds the likelihood ratio LRatioTUFF

$$PValueTUFF=1-F(LRatioTUFF)$$

where F is the cumulative distribution of a chi-square variable with 1 degree of freedom.

The result of the test is to accept if

$$F(LRatioTUFF)<F(TestLevel)$$

and reject otherwise, where F is the cumulative distribution of a chi-square variable with 1 degree of freedom.

If the sample has no failures, the test statistic is not defined. However, there are two cases distinguished here:

• If the number of observations is large enough that no matter when the first failure occurred it would be too late to pass the test, then the model is rejected. Technically, this happens if the number of observations N is larger than 1/pVaR (large enough relative to the VaR confidence level) and if the test fails when n = N + 1 (the earliest observation for the first VaR failure). In this case, the likelihood ratio is reported for n = N + 1, and the corresponding p-value.

• In all other cases, it is not possible to tell with certainty whether the result of the test would eventually be to accept or reject the model. There are ranges of possible first failure values that would result in accepting or rejecting the model. In these cases, the tuff function accepts the model and reports undefined (NaN) values for the likelihood ratio and p-value.

References

[1] Kupiec, P. "Techniques for Verifying the Accuracy of Risk Management Models." Journal of Derivatives. Vol. 3, 1995, pp. 73 – 84.

Version History

Introduced in R2016b