# Predicting Loss Distributions for Small-Size Defaulted-Debt Portfolios Using a Convolution Technique that Allows Probability Masses to Occur at Boundary Points

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## Abstract

To predict the loss distribution of a small-size defaulted-debt portfolio, this research applies the central limit theorem (CLT) to predicted loss given default (LGD) distributions and exposures of defaulted-debts in the portfolio. However, when the portfolio size is not large enough, the results from using the CLT can lead to the wrong inference. To overcome this problem, we propose a convolution procedure that iteratively combines predicted LGD distributions and exposures of defaulted-debts in the portfolio together. Our convolution procedure allows predicted LGD distributions to have probability masses at boundary points. To illustrate the proposed procedure, we collect 4962 defaulted-debts from Moody’s Default and Recovery Database and use the censored transformed beta model to predict their LGD distributions. Using an expanding rolling window approach, our empirical results confirm that the proposed convolution procedure has better and more robust out-of-sample performance than its alternative based on the CLT, in the sense of yielding more accurate predicted loss distributions of defaulted-debt portfolios. Thus, it is useful for pricing and managing defaulted-debt portfolios.

## Keywords

Central limit theorem Conditional independence Convolution Defaulted-debt portfolio Loss given default distribution Unconditional distribution## JEL classification

G21 G28## Notes

### Acknowledgements

The authors thank the reviewers and the editor for their valuable comments and suggestions that have greatly improved the presentation of this paper. The Ministry of Science and Technology of Taiwan provides support for this research.

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