#### ahnnecabiles

##### New Member

I have questions regarding the probability of default. First, regarding your screencast on the cumulative probability of default, why don't we use the 2-year spot rates for the treasury and corporate instead to compute for the 2-yr cumulative probability of default, i.e. 1-{1+(2-yr treasury)/1+(2-yr corporate)} = the 2-yr cumulative probability of default? Isn't it that the difference between the 2-yr treasury and 2-yr corporate includes the expected probability that the bond will default on the the two year period (its 2-yr cumulative probability)? Why do we have to take the long step of getting the forward rate to compute for the 2-yr cumulative probability?

Second, what is the difference between the cumulative probability of default and the conditional probability of default? According to Hull, the cumulative probability of default (the probability of the asset defaulting at the end of year 2 on the condition that it did not default on year 1) is just the sum of 2 marginal probabilities of default (i.e. marginal probability of default for year 1 and the marginal probability of default for year 2). Thus, if an asset has a 2-yr cumulative probability of default of .57 percent (as in his example in his book, Chapter 20 - credit risk), and has a marginal default probability of .20 percent in year 1, then it has .37 percent marginal default probability for year 2 (.57 - .20 = .37). Or, we will just add the two marginal probabilities to come up with the probability that it will default at the end of year 2. This is different from what Saunders says that it has to be 1-(probability of repayment year 1)(probability of repayment year 2). Whereas, conditional probability of default is the same as default intensity, in which the probability of default at time t is equal to 1-e^default intensity x t. I always thought that cumulative default probability and conditional default probability (the probability of default on the nth year on the condition that it did not default on the previous year/s) are the same. Please help.

Lastly David (sorry for the long queries but only your explanation on these could make me sleep :long, if we are asked, there are 10 independent bonds with the same marginal 1-yr edf of 5%, what is the probability that exactly one of the bonds will default at the end of the year? According to the solution (FRM Handbook p. 426), the probability is equal to 10 x .05 x (1-.05)^9. Why? Isn't it that it is about the probability of the union of x and y, such that p(x or y) = P(x) + P(y) where P(x) is the pd of one bond, such that the sum of their probabilities (10 x .05) is the probability that any one of them will default?

Hope my questions are clear, thanks so much. Will be waiting for your reply.

Thanks and more power.