A Hypothesis Test for the Long-Term Calibration in Rating Systems with Overlapping Time Windows
Abstract
:1. Introduction
2. Setup and Preliminaries
2.1. Setup and Notation
2.2. Formal Description of the Test
2.3. Distribution of the Long-Run Default Rate
2.4. Covariance between Default States
2.5. Covariance between Default Rates
2.6. Variance of the Long-Run Default Rate
3. Hypothesis Test for Long-Term Calibration
3.1. Statistical Test per Rating Grade
3.2. Statistical Test on Portfolio Level
4. Discussion and Further Considerations
4.1. Effect of Persisting Customers on the Variance of Z
4.2. Effect of Persisting Customers on Acceptance Range
4.3. Some Thoughts on the Rate of Convergence
4.4. An Alternative Way to Bound the Variance
4.5. Additional Conditions on the Rating Distribution
4.6. Impact of Simplification on the Acceptance Range
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
List of Symbols and Abbreviations
probability of default | |
LGD | loss given default |
Bernoulli distribution with probability p | |
default probability of rating grade k in the underlying master scale | |
equal to , i.e., default probability of the best rating grade | |
equal to , i.e., default probability of the worst rating grade | |
reference date number t | |
N | number of reference dates |
set of natural numbers | |
number of obligors on reference date | |
set of natural numbers including zero | |
minimal number (larger than zero) of existing obligors at any reference date | |
maximal number of existing obligors at any reference date | |
minimum of a set A | |
maximum of a set A | |
q | number of reference dates within a one-year time horizon starting from an arbitrary reference date |
size of the overlap of observation periods with reference dates and | |
M | total number of all customers during the history |
set of customers at reference date | |
one-year default rate at reference date | |
one-year default state of an unspecified customer at reference date | |
one-year default state of customer j at reference date | |
probability of default over a one-year time horizon of customer j at reference date | |
realized default rate on reference date | |
set of indices for reference dates, where the portfolio contains at least one customer | |
cardinality of | |
Z | long-run default rate |
realized long-run default rate | |
estimated long-run default rate (long-run central tendency) | |
expected value of the long-run default rate | |
variance of the long-run default rate | |
null hypothesis | |
alternative hypothesis | |
lower and upper bound of the acceptance range of the hypothesis test | |
covariance of the random variables X and Y | |
Normal distribution with expected value and variance | |
expected value of a random variable X | |
probability measure | |
set of real numbers | |
indicator function of set A | |
cumulative distribution function for the standard normal distribution | |
length of an interval or cardinality of a finite set I | |
number of persisting customers with respect to reference dates and | |
∅ | empty set |
solution of the minimization problem | |
minimal value of the minimization problem | |
absolute value of a real number x |
Appendix A. Minimization Problem
Appendix B. Test on Portfolio Level without Solving the Minimization Problem
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Kurth, P.; Nendel, M.; Streicher, J. A Hypothesis Test for the Long-Term Calibration in Rating Systems with Overlapping Time Windows. Risks 2024, 12, 131. https://doi.org/10.3390/risks12080131
Kurth P, Nendel M, Streicher J. A Hypothesis Test for the Long-Term Calibration in Rating Systems with Overlapping Time Windows. Risks. 2024; 12(8):131. https://doi.org/10.3390/risks12080131
Chicago/Turabian StyleKurth, Patrick, Max Nendel, and Jan Streicher. 2024. "A Hypothesis Test for the Long-Term Calibration in Rating Systems with Overlapping Time Windows" Risks 12, no. 8: 131. https://doi.org/10.3390/risks12080131