Akira Ieda Institute for Monetary and Economic Studies Bank of Japan

1. Comments on “The Empirical Relationship between Average Asset Correlation,Firm Probability of Default and Asset Size” Akira Ieda Institute for Monetary and Economic Studies Bank of Japan (e-mail: akira.ieda@boj.or.jp)

2. Structure of My Comments • Relationships between asset correlation, default correlation, and default rate. • Discussion on the intuitive explanation of relationships between parameters by Lopez [2002]. • Discussion on the empirical results by Lopez [2002]

3. Relationships between Asset Correlation, Default Correlation, and Default Rate • In the single factor approach, one can create a model that assumes that the rate of return on assets will have the standard normal distribution (see, for example, Ieda, Marumo, and Yoshiba [2000]):

4. Relationships between Asset Correlation, Default Correlation, and Default Rate • The simultaneous default rate for company i and company j which has the same default rate as company i is as follows. (A)

5. Relationships between Asset Correlation, Default Correlation, and Default Rate • The correlation coefficient between the default events of company i and company j (the <average> default correlation) is calculated using the following relationship: (B)

6. Relationships between Asset Correlation, Default Correlation, and Default Rate • Hence, if you already know figures for two of the parameters among asset correlation, default correlation, and default rate, you can automatically obtain that for the remaining one by solving equations (A) and (B).

7. Relationships between Asset Correlation, Default Correlation, and Default Rate Asset correlation Default rate

8. Relationships between Asset Correlation, Default Correlation, and Default Rate • Default correlation can be estimated using ratings-based bond default data published by rating agencies. where p is the default rate, the standard deviation of p, and default correlation.

9. Relationships between Asset Correlation, Default Correlation and Default Rate Source: Keenan, Shtogrin and Sobehart [1999]

10. Relationships between Asset Correlation, Default Correlation, and Default Rate

11. Relationships between Asset Correlation, Default Correlation, and Default Rate • This empirical result says that asset correlation seems to be a decreasing function of the default rate. • Thus, from the perspective of measurement of credit risk in a portfolio, it is at least not accurate to assume asset correlation is a constant number.

12. Discussion on the Intuitive Explanation of Relationships between Parameters by Lopez [2002] (Asset Correlation versus Asset Size)

13. Discussion on the Intuitive Explanation of Relationships between Parameters by Lopez [2002] • The explanation is very clear. • Thus, it is expected that asset correlation is an increasing function of default rate.

14. Discussion on the Intuitive Explanation of Relationships between Parameters by Lopez [2002] (Asset Correlation versus Default Rate)

15. Discussion on the Intuitive Explanation of Relationships between Parameters by Lopez [2002] • To the contrary, this explanation of asset correlation versus default rate is not very clear. • Thinking of a single factor model approach employed here (see the following equation), it does not necessarily seem reasonable to expect that the idiosyncratic factor begins to bear a more important role relative to the systematic factor as a firm’s PD increases.

16. Discussion on the Empirical Results by Lopez [2002] • “The empirical results indicate that average asset correlation is increasing in asset size.” This is very intuitive as previously mentioned.

17. Discussion on the Empirical Results by Lopez [2002] • “……calibration results indicate that average asset correlation is a decreasing function of the probability default.” These results do not seem to strongly support that asset correlation declines in PD. In bivariate analysis with respect to small and medium-sized firms, the fall in asset correlation with PD is very small. For large firms, though asset correlation seems to decline in PD in some portfolios, this might stem from differences in distributions of asset size.