comments on “the empirical relationship between average asset correlation, firm probability of...

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: [email protected])

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]

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 Yosh

iba [2000]):

iAA

i vX 1 , ...)1,0(~, diivXi,

where A is the (average) asset correlation.

)( ),( jiCor Aji


• The simultaneous default rate for company i and company j which has the same default rate as company i is as follows.

jijiA

jiA

p p

Aij dxdxxxxxp

2)1(2

1exp

12

1 22

2

)( )(

2

1 1

(A)


• The correlation coefficient between the default events of company i and company j (the <average> default correlation) is calculated using the following relationship:

.)1()1(

2

pppp

ppijD

(B)


• 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).

A

D p


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

DC=0.01 DC=0.03 DC=0.05

Asset correlation

Default rate


• Default correlation can be estimated using ratings-based bond default data published by rating agencies.

.)1(

~2

ppD

where p is the default rate, the standard deviation of p, and default correlation.

D~

Relationships between Asset Correlation, Default Correlation and Default Rate

D

Source: Keenan, Shtogrin and Sobehart [1999]

Aaa Aa A Baa Ba B1970 0.00% 0.00% 0.00% 0.27% 4.12% 23.38%1971 0.00% 0.00% 0.00% 0.00% 0.42% 4.00%1972 0.00% 0.00% 0.00% 0.00% 0.00% 7.41%1973 0.00% 0.00% 0.00% 0.45% 0.00% 3.92%1974 0.00% 0.00% 0.00% 0.00% 0.00% 10.34%1975 0.00% 0.00% 0.00% 0.00% 1.02% 6.15%1976 0.00% 0.00% 0.00% 0.00% 1.01% 0.00%1977 0.00% 0.00% 0.00% 0.27% 0.52% 3.39%1978 0.00% 0.00% 0.00% 0.00% 1.08% 5.56%1979 0.00% 0.00% 0.00% 0.00% 0.49% 0.00%1980 0.00% 0.00% 0.00% 0.00% 0.00% 5.06%1981 0.00% 0.00% 0.00% 0.00% 0.00% 4.60%1982 0.00% 0.00% 0.26% 0.30% 2.73% 2.41%1983 0.00% 0.00% 0.00% 0.00% 0.91% 6.36%1984 0.00% 0.00% 0.00% 0.36% 0.83% 6.78%1985 0.00% 0.00% 0.00% 0.00% 1.75% 8.28%1986 0.00% 0.00% 0.00% 1.33% 2.05% 11.80%1987 0.00% 0.00% 0.00% 0.00% 2.72% 5.86%1988 0.00% 0.00% 0.00% 0.00% 1.24% 6.02%1989 0.00% 0.61% 0.00% 0.60% 2.98% 9.17%1990 0.00% 0.00% 0.00% 0.00% 3.32% 16.11%1991 0.00% 0.00% 0.00% 0.28% 5.25% 14.66%1992 0.00% 0.00% 0.00% 0.00% 0.30% 9.00%1993 0.00% 0.00% 0.00% 0.00% 0.55% 5.76%1994 0.00% 0.00% 0.00% 0.00% 0.23% 3.81%1995 0.00% 0.00% 0.00% 0.00% 0.67% 4.84%1996 0.00% 0.00% 0.00% 0.00% 0.00% 1.45%1997 0.00% 0.00% 0.00% 0.00% 0.19% 2.10%1998 0.00% 0.00% 0.00% 0.12% 0.61% 4.08%

0.00% 0.02% 0.01% 0.14% 1.21% 6.63%

0 1.28E-06 2.33E-07 8.01E-06 1.88E-04 2.49E-03

— 0.0061 0.0026 0.0058 0.0158 0.0402

2~p


Aaa Aa A Baa Ba B

p ———0.14%1.21%6.63%D ———0.00580.01580.0402A ———0.17340.13620.1302


• 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.

Discussion on the Intuitive Explanation of Relationships between Parameters by Lopez [2002]

“In the general finance theory of portfolio diversification, as the number of

different stocks within a portfolio increases, the portfolio becomes more

diversified, and the idiosyncratic element of the portfolio’s return becomes

less important. An analogous view could be taken with respect to a firm’s

asset size; that is a firm becomes larger and comes to contain more assets, its

risk and return characteristics should more closely resemble the overall

asset market and be less dependent on the idiosyncratic elements of the

individual business lines.” (Lopez[2002] p.12)

(Asset Correlation versus Asset Size)


• The explanation is very clear.

• Thus, it is expected that asset correlation is an increasing

function of default rate.


(Asset Correlation versus Default Rate)

“Within the ASRF framework, we expect to see a negative relationship

between firm asset correlation and their probabilities of default (PD). That is,

as a firm’s PD increases due to worsening conditions and its approaching

possible default, it is reasonable to think that idiosyncratic factors begin to

take on a more important role relative to the common systematic risk factor.”

(Lopez [2002] p.10)


• 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.

HiiHiHi vRXR ,2

, 1

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.

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.

References

Ieda, A., K. Marumo and T. Yoshiba; “A Simplified Method for Calculating the

Credit Risk of Lending Portfolios,” Monetary and Economic Studies, Vol.18,

No.2, Bank of Japan, pp.49-82, 2000.

Keenan, S. C., I . Shtogrin and J . Sobehart: “Historical Default Rates of

Corporate Bond Issues, 1920 – 98”, Moody’s Special Comment,

March 1999.

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