Default Probabilities: Calculating with Moody's Transition Matrix

[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

 

[haruspex]

The transition matrices shown appear to be transposed from what I would expect. Normally, it acts on a column vector of prior states, so the ith column of the matrix lists the probabilities of transitions from the ith state. Hence each column of the matrix should add to 1, but in the matrices shown each row adds to 1.

Likewise, the default probabilities in any given year should also add to 1, but the Elton and Gruber numbers add to about 30%. You say they divide by (1- some probability), but that would increase the value, not decrease it.

Setting that aside, I suspect that what they are calculating is a different quantity. I would need to see the paper to comment further.

 

[pbuk]

Hello ,

I have a transition matrix...

Note you can copy and paste a table from Excel into your message - it's not very pretty but it is easier to read (there are other, prettier, ways to display matrices here).

				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​

Does anyone know how these default probabilities were calculated and why it is not just squarring the matrix?

No, I agree with your calculations. Can you provide a link to the paper?

 

[pbuk]

The transition matrices shown appear to be transposed from what I would expect. Normally, it acts on a column vector of prior states, so the ith column of the matrix lists the probabilities of transitions from the ith state. Hence each column of the matrix should add to 1, but in the matrices shown each row adds to 1.

Both conventions are equally valid: you describe a left stochastic matrix but the right stochastic matrix presentation in the OP is more common IME.

Edit:

see Wikipedia, this Cambridge University course and this MIT open course.

/Edit.

Likewise, the default probabilities in any given year should also add to 1, but the Elton and Gruber numbers add to about 30%.

The vector we are looking for is the probability that a bond initially in state $i$ is in the state 'default' after 2 years. There is no reason this sum should be one (if all bonds default within 2 years the sum will be 7). The sum actually represents the number of bonds that are in default after 2 years if we start with one bond with each rating (I am talking about the sum of the OP's vector, I have no idea what the numbers in the paper represent).

 

[haruspex]

Both conventions are equally valid: you describe a left stochastic matrix but the right stochastic matrix presentation in the OP is more common IME.

Edit:

/Edit.The vector we are looking for is the probability that a bond initially in state $i$ is in the state 'default' after 2 years. There is no reason this sum should be one (if all bonds default within 2 years the sum will be 7). The sum actually represents the number of bonds that are in default after 2 years if we start with one bond with each rating (I am talking about the sum of the OP's vector, I have no idea what the numbers in the paper represent).

ok, I completely misunderstood how 'default' was being used here. I thought it referred to the a priori vector, but I see now it is just one of the classes.