# Bayes - hard

Re: Bayes’ formula new Posted by: suny (IP Logged) [hide posts from this user] Date: May 23, 2008 04:59PM Thanks. Can you help to solve the following question please? Bonds rated B have a 25% chance of default in five years. Bonds rated CCC have a 40% chance of default in five years. A portfolio consists of 30% B and 70% CCC-rated bonds. If a randomly selected bond defaults in a five-year period, what is the probability that it was a B-rated bond CAN ANYONE DO THIS ONE CLEARLY?

21.13%? P(B|default)=P(default|B)*P(B)/P(default)=0.25*0.3/[(0.3*0.25)+(0.7*0.4)]

no way! bayes used to be hard until i found the tree method…just make a tree first branch 70% going up and 30% going down from the 70% branch, you have 40% chance of default from the 30% branch you have 25% chance of default… .7 *.4 = .28 .3 *.25= .075 so the chanc of randomly selected bond defaulting can be said to equal .28 + .075 = .355 Chance of it being a B bond is .075/.355= 21.13 if they asked for chances of ccc bond, it would have been .28/.355 much easier than memorizing a formula

P(Rated B|Default) = [P(Default AND Rated B)] / [P(Default)] Draw a tree diagram and find the total probability of a bond being in default: it will be 0.355 = 0.3*0.25/0.355 = 0.2113

The tree is the secret. See the tree, you’ll see the light:)