risk.validation.herfindahlIndexTest

Herfindahl index test

Since R2026a

Syntax

hHITest = risk.validation.herfindahlIndexTest(BaselineFrequency,TargetFrequency)

hHITest = risk.validation.herfindahlIndexTest(BaselineFrequency,TargetFrequency,ConfidenceLevel=confidenceLevel)

[hHITest,Output] = risk.validation.herfindahlIndexTest(___)

Description

hHITest = risk.validation.herfindahlIndexTest(BaselineFrequency,TargetFrequency) returns the result of a Herfindahl index (HI) test, which compares the dispersion of the frequencies in a baseline and target portfolio. The output is 1 if the test rejects the null hypothesis at the 95% confidence level, or 0 otherwise.

hHITest = risk.validation.herfindahlIndexTest(BaselineFrequency,TargetFrequency,ConfidenceLevel=confidenceLevel) specifies the confidence level for the HI test.

[hHITest,Output] = risk.validation.herfindahlIndexTest(___) also returns a structure Output that contains summary metrics.

Examples

Apply Herfindahl Index Test on Rating Grade Data

Perform a Herfindahl index (HI) test on rating grade data to determine whether the HI for a baseline portfolio is larger than the HI for a target portfolio. The PDRatingGradeData file contains a table with variables that include RatingGrade and ObservationYear. These variables, respectively, contain data for customer rating grades and the observation year.

Load the data.

data = readtable("PDRatingGradeData.csv")

pdData=500×4 table
    0.4074     9    2020    0
    0.4529    10    2020    0
    0.0635     2    2020    0
    0.4567    10    2020    1
    0.3162     7    2020    0
    0.0488     1    2020    0
    0.1392     3    2020    0
    0.2734     6    2020    0
    0.4788    10    2020    0
    0.4824    10    2020    0
    0.0788     2    2020    0
    0.4853    10    2020    0
    0.4786    10    2020    0
    0.2427     5    2020    0
      ⋮

Use the groupcounts function to get counts for each rating grade in years 2023 and 2024.

idx23 = data.ObservationYear==2023;
idx24 = data.ObservationYear==2024; 
Frequencies1 = groupcounts(data(idx23,:),"RatingGrade");
Frequencies2 = groupcounts(data(idx24,:),"RatingGrade");

Frequencies1 contains the baseline portfolio frequency data for 10 rating grades. Frequencies2 contains data for the target portfolio. Each column represents a rating grade.

Perform a Herfindahl index (HI) test to test the null hypothesis that the HI for the baseline portfolio is larger than the HI for the target portfolio.

risk.validation.herfindahlIndexTest(Frequencies1.GroupCount,Frequencies2.GroupCount)

ans = logical
   1

The output indicates that enough evidence exists to reject the null hypothesis.

Input Arguments

BaselineFrequency — Baseline portfolio frequencies

nonnegative numeric vector

Baseline portfolio frequencies, specified as a numeric vector. For probability of default models, BaselineFrequency often contains frequencies or dollar exposures for each rating grade.

The baseline frequencies are sometimes called the initial frequencies.

Data Types: single | double

TargetFrequency — Target portfolio frequencies

nonnegative numeric vector

Target portfolio frequencies, specified as a numeric vector. For probability of default models, TargetFrequency often contains frequencies or dollar exposures for each rating grade.

The target frequencies are sometimes called the current frequencies.

Data Types: single | double

confidenceLevel — Confidence level

0.95 (default) | numeric scalar in the range (0,1)

Confidence level of the hypothesis test, specified as a numeric scalar in the range (0,1).

Output Arguments

hHITest — Hypothesis test result

numeric scalar containing values of 1 or 0

Hypothesis test results, returned as a numeric scalar.

• A value of 1 rejects the null hypothesis at the specified confidence level.

• A value of 0 fails to reject the null hypothesis at the specified confidence level.

Output — Output metrics

structure

Output metrics, returned as a structure with the following fields:

• RejectTest — Numeric scalar indicating whether the null hypothesis was rejected. This field represents the same values as hHITest.

• PValue — p-value for the hypothesis test returned as a scalar in the range [0,1]. A small value indicates that the null hypothesis might not be valid.

• TestStatistic — Value of the test statistic for the hypothesis test, returned as a numeric scalar.

• CriticalValue — Minimum value of the test statistic at which the test rejects the null hypothesis for the given probability and confidence level, returned as a numeric scalar.

• BaselineCoefficientOfVariation — Coefficient of variation for the baseline portfolio, returned as a numeric scalar.

• BaselineHerfindahlIndex — Herfindahl index for the baseline portfolio, returned as a numeric scalar.

• TargetCoefficientOfVariation — Coefficient of variation for the target portfolio, returned as a numeric scalar.

• TargetHerfindahlIndex — Herfindahl index for the target portfolio, returned as a numeric scalar.

• ConfidenceLevel — Confidence level for the hypothesis test, returned as a numeric scalar.

For more information about the Herfindahl Index test and its corresponding statistics, see More About.

More About

Herfindahl Index Test

The Herfindahl index (HI) test [1] compares the concentration level of a baseline portfolio with the concentration level of a target portfolio. The null hypothesis for the HI test is that the HI for the baseline portfolio is larger than the HI for the target portfolio. The HI for a portfolio is given by

$$HI=\frac{1+\log \left(\frac{C{V}^2+1}{K}\right)}{\log (K)},$$

where K is the number of rating grades for nondefaulted exposures and CV represents the coefficient of variation (CV). The CV for a portfolio is given by

$$CV=\sqrt{K{\displaystyle \sum _{i=1}^K{\left(R_i-\frac{1}{K}\right)}^2}},$$

where R_i is the portfolio's relative frequency of rating grade i.

The test statistic for the HI test is

$$S=\frac{\sqrt{K-1}(C{V}_t-C{V}_b)}{C{V}_t{}^2\left(\frac{1}{2}+C{V}_t^2\right)},$$

where CV_b and CV_t are the CVs for the baseline and target portfolios, respectively. The software calculates the p-value as $$1-F(S)$$ using the normal distribution cdf F.

For more information about the HI test, see Section 2.5.5.3 in [1].

References

[1] European Central Bank, “Instructions for reporting the validation results of internal models.” February, 2019. https://www.bankingsupervision.europa.eu/activities/internal_models/shared/pdf/instructions_validation_reporting_credit_risk.en.pdf.

Version History

Introduced in R2026a