Bayes' Formula

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. A) 0.211 B) 0.625 C) 0.429 D) 0.250 --------------------------- SPOILER --------------------------- Ok, the question I have is regarding the given answer. So if you want to figure it out yourself stop reading and don’t look below. The answer uses this formula as the way to find the answer: P(B-rated | Default) = P(Default | B-rated) / P(Default) First, why do you find P(B-rated | Default) and not P(Default | B-rated) ? Second, why does the formula exclude the multiplication part of the Bayes’ formula? Why isn’t it: P(B-rated | Default) = [P(Default | B-rated) / P(Default)] * P(B-rated) ??? Thanks guys.

draw a tree Default .25 Tot Prob=.075 B .3 No Default .75 Tot Prob= .225 CCC .7 Default .4 Tot Prob = .28 No Default .6 tot Prob =.42 Sum of .075, .225, .28 and .42 = 1 (Check that things are ok, thus far) Now total probability of default = .075 + .28 = .355 Out of this, a B Default = .075 So Probability it is a B bond, given a Default has occurred = .075 / .355 = .211 (A) Might not look pretty when it publishes on the site, but hope you get the idea. I do not recommend memorizing the formula, and instead prefer using the above approach.

thanx cp. this has helped me grasp the bayes concept as well. didnt get it from looking at the text before.

Hi kant, You want to know the probability that the bond is a B rated bond, given that the bond defaults. You don’t want to know what the probability is that the bond defaults, given that it is a B rated bond.

excellent explanation cpk. ALWAYS use the tree approach. much more intuitive then trying to deal with the ugly equation.


Think of Bayes’ this way: “The probability of what we want” divided by “the sum of all the ways it can happen”. So for this problem “what we want is a B rated bond that defaults” (30% * 25%) = 7.5% “sum of all ways it can happen” (B-rated bond that defaults) + (C rated Bond that defaults) = (30% * 25%) + (70% * 40%) = 7.5% + 28% So, (7.5%) / (7.5% + 28%) = 21.1%

Honestly I dont like looking at Bayes formula. Its a great way to get yourself screwed up in my opinion. Too confusing. Just use logic and math skills here. Ok so lets say there are 100 Bonds. Thus B bonds = 30 and C bonds = 70. Now out of the 5 years 25% of the B bonds will fail. That is 7.5 will fail (.25*30) 40% of the C bonds will fail, or 28. (.4*70). Now since its asking for the probability of a B bond failure GIVEN A BOND FAILS… this is key here. We know the bond will fail. Thus we have 35.5 bonds will fail over the 5 years. (28+7.5) Now its simple form here: 7.5/35.5 = .211 A