Hello, In this question you were given probabilities of default for two bonds of 0.05 and 0.15, and asked to work out the probability of both defaulting. The question doesn't really indicate which tail of the distribution the default probability is in (e.g. is the event of default considered in terms of amount of loss, which would indicate the upper tail, or the return on investment which would indicate the lower?). The solutions have gone with lower tail. Was there anything in this question that points to use of lower tail (and hence inputting F(0.05), F(0.15) into the copula)? Thanks.
Hi Alibaba! I agree with you, it is not immediately obvious. Based on previous questions that I've seen on bonds defaulting, it's often the 'time until default', T, that is the random variable. Seeing the references in the question to 'a year' makes me think that we are modelling that here too. So P(T(A) ≤ 1) = 0.05 And P(T(B) ≤ 1) = 0.15 Then we are using the copulas to work out the joint CDF, ie: P(T(A) ≤ 1, T(B) ≤ 1) Does this help? Anna
Hi Anna, Yes that helps a lot! Thank you that's really clear, I'll look out for that in similar questions so. Thanks again
Hello, May I ask why is the lower tail copula used? We are using a lower tail because one year is considered to be short and in the lower tail of distribution function of T?
Yes, we are saying that ... if there is a recession in the next year ... then if bond A defaults, there is a high (conditional) probability of bond B defaulting. And yes, assuming that the next year represents the lower tail of the distribution of time until default.