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FRM 2007 PII q45 - PD calculation (Mental fart, please help)


New Member
Company ABC was incorporated on January 1, 2004. It has an expected annual default
rate of 10%. Assuming a constant quarterly default rate, what is the probability that
company ABC will not have defaulted by April 1, 2004?

a. 2.40%
b. 2.50%
c. 97.40%
d. 97.50%


For an annual default rate dA and assuming a constant default rate, the quarterly default
rate dQ satisfies:

(1 – dA) = (1 – dQ)^4

Since dA is 10%, dQ = 2.6%, and so the probability of the company surviving to the
second quarter is: 100% - 2.6% = 97.40%.

I must be having a colossal mental fart at the moment because I don't see why the following method won't yield the correct answer. I mean, the answer provided totally makes sense, but...

If (no yearly default) = (no quarterly default)^4
then what is wrong with saying
(yearly default) = (quarterly default)^4? (in which case, .1 = dQ^4 so dQ = 56% and this definitely is way to high, but still)

I mean, I see from the answer how the previous question is not correct, but, from an intuitive sense, why can't that be done? I'll stop here because I don't want to confuse things further. Any thoughts?

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi FoQ,

Your dQ = 56% would be correct in the (strange) world where 10% annual default required four consecutive defaults in all four quarters.
i.e., if dQ = 56%, then what is the probability that default occurs four times in a row? 56%^4 = 10%

so the issue seems to be about the asymmetry between default and survival: survival requires consecutive survival, but default only needs to occur once.

for this reason, cumulative PD needs to operate on survival:
P(cumulative PD) = 1 - (survive1)*(survive2) = 1 - p*p*p ...
put another way, a cumulative default needs only one defualt and therefore incudles all outcomes that are not cumulative survivial