Example 7.2 Portfolio Credit Risk

[enjofaes]

Don't get example 7.2 in the books:

Consider a portfolio containing five positions:

1. A five-year senior secured bond issued by Ford Motor Company
2. A five-year subordinatte unsecured bond issued by Ford Motor Company
3. Long protection in a five-year CDS on Ford Motor Credit Company
4. A five-year senior bond issued by General Motors company
5. A 10-year syndicated term loan to Starwood Resorts

If we set horizon measuring credit risk of t = 1 year, we need to have 4 default probabilities and 12 pairwise correlations.
I figured 4 default probabilities as 1 & 2 relate to Ford Motor Company. Logic..

However, I always thought you had to divide the N(N-1) by 2? so 4*3/2 i.e. 6 pairwise correlations...
E.g. for 3-credit portfolio: you have 3 pairwise correlations (i.e. 3*2/2) like just described a bit earlier in the textbook.

1 of the 2 seems thus incorrect no?

 

[David Harper CFA FRM]

Hi @enjofaes I agree, it's a mistake in the text and it should read ....

If we set a horizon for measuring credit risk of t = 1 year, we need to have four default probabilities and six pairwise default correlations, since there are only four distinct corporate entities represented in the portfolio. However, since the two Ford Motor Company bonds are at two different places in the capital structure, they will have two different recovery rates.

I'm fond of triangle numbers (see https://forum.bionicturtle.com/thre...d-concordant-discordant-pairs.8209/post-33057). If you have N credits, the diagonal separates an upper versus lower triangle. In a covariance/correlation matrix, of course, the upper and lower mirror each other. As you say, the number of cells in the upper/lower diagonal is N*(N-1)/2.

If we include the diagonal, then N*(N-1)/2 + N = N(N+1)/2. The correlation matrix diagonal is 1.0s, so it always uses N*(N-1)/2. But the covariance diagonal is variances, so we could say either: N*(N-1)/2 covariances plus N variances; or N(N+1)/2 covariances including the variances. Thanks!