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Based on historical evidence, a B-rated counter party is approximately 16 times more likely to default over a 1-year time period than a BBB-rated counter party. Over a 10-year time period, a B-rated counter party is how many more times likely to default than a BBB-rated counter party?

Answer provided is 5 times. How is that computed?

I don't know the criteria for BBB-rated and B-rated Bonds.

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    If not mistaken, this "counterparty" is commonly called a "bond issuer". – Flux Oct 21 '20 at 9:40
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I think this is a very confusingly worded probability problem.

Consider the chance that the BBB issuer defaults to be x, making the chance that the B issuer defaults to be 16x.

If we look at the chances that an issuer has not defaulted, a.k.a. a survival rate, that would be 1-x and 1-16x, respectively.

Then, if we want to see if the issuers 'survive' multiple years, that would be (1-x)^n, and (1-16x)^n, again respectively, where n would be the number of years.

And then, after n years, we can again subtract from 1 to turn that back into a default rate, 1-(1-x)^n and 1-(1-16x)^n, respectively.

And if you try some numbers like x = 0.02 or a 2% failure rate per year for the BBB issuer, at 10 years:

BBB: 1-(1-0.02)^10 = 0.18 or 18%

B: 1-(1-0.02*16)^10 = 0.98 or 98%, which is awfully close to the 5 times more likely to default as indicated in the question.

It is really a little tricky in that the chances of B defaulting in 10 years is almost certain, and what happens is that as BBB accumulates more chances to default over the years, that naturally brings it closer to 100% certainty. Just stating it as '16 times more likely to default in 1 year but only 5 times more likely to default in 10 years' is an extremely obtuse way of stating it.

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