tl

Traffic light test for value-at-risk (VaR) backtesting

Syntax

TestResults = tl(vbt)

Description

TestResults = tl(vbt) generates the traffic light (TL) test for value-at-risk (VaR) backtesting.

Examples

Generate Traffic Light Test Results

Create a varbacktest object.

load VaRBacktestData
vbt = varbacktest(EquityIndex,Normal95)

vbt = 
  varbacktest with properties:

    PortfolioData: [1043×1 double]
          VaRData: [1043×1 double]
             Time: [1043×1 double]
      PortfolioID: "Portfolio"
            VaRID: "VaR"
         VaRLevel: 0.9500

Generate the tl test results.

TestResults = tl(vbt)

TestResults=1×9 table
    PortfolioID    VaRID    VaRLevel     TL      Probability     TypeI     Increase    Observations    Failures
    ___________    _____    ________    _____    ___________    _______    ________    ____________    ________

    "Portfolio"    "VaR"      0.95      green      0.77913      0.26396       0            1043           57   

Run the TL 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: [1043×1 double]
          VaRData: [1043×6 double]
             Time: [1043×1 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 tl test results.

TestResults = tl(vbt)

TestResults=6×9 table
    PortfolioID        VaRID         VaRLevel      TL      Probability      TypeI      Increase    Observations    Failures
    ___________    ______________    ________    ______    ___________    _________    ________    ____________    ________

     "Equity"      "Normal95"          0.95      green       0.77913        0.26396          0         1043           57   
     "Equity"      "Normal99"          0.99      yellow      0.97991        0.03686    0.26582         1043           17   
     "Equity"      "Historical95"      0.95      green       0.85155        0.18232          0         1043           59   
     "Equity"      "Historical99"      0.99      green       0.74996        0.35269          0         1043           12   
     "Equity"      "EWMA95"            0.95      green       0.85155        0.18232          0         1043           59   
     "Equity"      "EWMA99"            0.99      yellow      0.99952      0.0011122    0.43511         1043           22   

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.

Output Arguments

TestResults — tl test results

table

tl 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

• 'TL' — Categorical (ordinal) array with the categories green, yellow, and red that indicate the result of the traffic light tl test

• 'Probability' — Cumulative probability of observing up to the corresponding number of failures

• 'TypeI' — Probability of observing the corresponding number of failures or more if the model is correct

• 'Increase' — Increase in the scaling factor

• 'Observations' — Number of observations

• 'Failures' — Number of failures

More About

Traffic Light Test

The tl function performs Basel's traffic light test, also known as three-zone test. Basel's methodology can be applied to any number of time periods and VaR confidence levels, as explained in Algorithms.

The Basel Committee reports, as an example, a table of the three zones for 250 time periods and a VaR confidence level of 0.99. The increase in scaling factor in the table reported by Basel has some ad-hoc adjustments (rounding, and so on) not explicitly described in the Basel document. The following table compares the increase in scaling factor reported in the Basel document for the case of 250 periods and 0.99% VaR confidence level, and the increase in the factors reported by the TL test.

Failures	Zone	Increase Basel	Increase TL
0	Green	0	0
1	Green	0	0
2	Green	0	0
3	Green	0	0
4	Green	0	0
5	Yellow	0.40	0.3982
6	Yellow	0.50	0.5295
7	Yellow	0.65	0.6520
8	Yellow	0.75	0.7680
9	Yellow	0.85	0.8791
10	Red	1	1

The tl function computes the scaling factor following the methodology described in the Basel document (see References) and is explained in the Algorithms section. The tl function does not apply any ad-hoc adjustments.

Algorithms

The traffic light test is based on a binomial distribution. Suppose N is the number of observations, p = 1 - VaRLevel is the probability of observing a failure if the model is correct, and x is the number of failures.

The test computes the cumulative probability of observing up to x failures, reported in the 'Probability' column,

$$Probability=Probability(X\le x|N,p)=F(x|N,p)$$

where $$F(x|N,p)$$ is the cumulative distribution of a binomial variable with parameters N and p, with p = 1 - VaRLevel. The three zones are defined based on this cumulative probability:

• Green: $$F(x|N,p)$$ ≤ 0.95

• Yellow: 0.95 < $$F(x|N,p)$$ ≤ 0.9999

• Red: 0.9999 < $$F(x|N,p)$$

The probability of a Type-I error, reported in the 'TypeI' column, is $$TypeI=TypeI(x|N,p)=1-F(X\ge x|N,p)$$.

This probability corresponds to the probability of mistakenly rejecting the model if the model were correct. Probability and TypeI do not sum up to 1, they exceed 1 by exactly the probability of having x failures.

The increase in scaling factor, reported in the 'Increase' column, is always 0 for the green zone and always 1 for the red zone. For the yellow zone, it is an adjustment based on the relative difference between the assumed VaR confidence level (VaRLevel) and the observed confidence level (x / N), where N is the number of observations andx is the number of failures. To find the increase under the assumption of a normal distribution, compute the critical values zAssumed and zObserved.

The increase to the baseline scaling factor is given by

$$Increase=Baseline\times \left(\frac{zAssumed}{zObserved}-1\right)$$

with the restriction that the increase cannot be negative or greater than 1. The baseline scaling factor in the Basel rules is 3.

The tl function computes the scaling factor following this methodology, which is also described in the Basel document (see References). The tl function does not apply any ad-hoc adjustments.

References

[1] Basel Committee on Banking Supervision, Supervisory Framework for the Use of 'Backtesting' in Conjunction with the Internal Models Approach to Market Risk Capital Requirements. January, 1996, https://www.bis.org/publ/bcbs22.htm.

Version History

Introduced in R2016b