# Default probabilities

1. Jun 25, 2004

### JosephFrank

Hello ,

I have a transition matrix
Panel A:Moody's
Aaa Aa A Baa Ba B Caa default
(%) (%) (%) (%) (%) (%) (%) (%)
Aaa 91.897 7.385 0.718 0 0 0 0 0
Aa 1.131 91.264 7.091 0.308 0.206 0 0 0
A 0.102 2.561 91.189 5.328 0.615 0.205 0 0
Baa 0 0.206 5.361 87.938 5.464 0.825 0.103 0.103
Ba 0 0.106 0.425 4.995 85.122 7.333 0.425 1.594
B 0 0.109 0.109 0.543 5.972 82.193 2.172 8.903
Caa 0 0.437 0.437 0.873 2.511 5.895 67.795 22.052
Default 0 0 0 0 0 0 0 100

It indicates one-year ratings migration probabilities. For example, based upon the matrix, a Baa-rated bond has a 5.464% probability of being downgraded to a Ba-rating by the end of one year.To use a ratings transition matrix to calculate the one year default probabilities, we simply take the default probabilities indicated in the last column and ascribe them to bonds of the corresponding credit ratings. For example, with this approach, we would ascribe an Baa-rated bond a 0.103% probability of default within one year.

My problem is with the computation of the two year default probabilities. Normally, If we want two-year default probabilities, we simply multiply the matrix by itself once (i.e. employ matrix multiplication as defined in linear algebra) to obtain a two-year ratings transition matrix.

Panel A:Moody's
Aaa Aa A Baa Ba B Caa Default
(%) (%) (%) ( %) (%) (%) (%) (%)
Aaa 84.535 13.545 1.838 0.061 0.020 0.001 0.000 0.000
Aa 2.079 83.557 12.963 0.940 0.424 0.032 0.001 0.004
A 0.216 4.692 83.625 9.584 1.393 0.444 0.013 0.034
Baa 0.008 0.514 9.642 77.895 9.541 1.821 0.202 0.377
Ba 0.002 0.218 1.035 8.711 73.182 12.336 0.814 3.703
B 0.001 0.209 0.261 1.247 10.077 68.128 3.284 16.795
Caa 0.005 0.717 0.790 1.542 4.243 9.034 46.101 37.57
Default 0.000 0.000 0.000 0.000 0.000 0.000 0.000 100.00

The last column of that matrix will provide the desired two year default probabilities. For example, based upon the matrix, a Aa-rated bond has a 0.004% probability of default within two years period.
I came across a paper by Elton and Gruber where the default probabilities were reported to be the following:

Aaa Aa A Baa Ba B Caa
0.000 0.004 0.034 0.274 2.143 8.664 19.906

the first three are exactly those predicted by squarring the matrix but after that the results don't coincide. They say that the defaults probabilities are the last column of the squared transition matrix divided by one minus the
probability of default in period 1.

Does anyone know how these default probabilities were calculated and why it is not just squarring the matrix? I spent too much time trying to figure out the results but without success. I really appreciate any help (attached is the excel file).
thx

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