Question:

A priori, assume the odds of a recession next year are 25% (and 75% that there will be no recession, so only two outcomes). If the economy does NOT go into a recession, the likelihood that bond XYZ will default is only 1.0% (therefore, there is a 99% probability of no default). If on the other hand the economy DOES descend into recession, the likelihood bond XYZ will default increases to fully 9.0%. At the end of the year, we observe the bond defaulted.

What is the POSTERIOR probability that the economy went into recession? (Using Bayes Theorem)

Answer:

Let R = probability of recession. R’ = probability of not recession. Unconditional P(R) = 25%
Let D = probability of bond default. D’ = probability of not default.

We are looking for P (R | D)
“what is probability of recession (R) given that (conditional on) we observe a default (D)”

P(R|D) = P(D|R)P(R)/[P(D)]

P(D|R)P(R) = (9%)(25%) = 2.25%. Note this is the JOINT probability the economy enter recession AND the bond will default.
P(D) = P(D|R)P(R) + P(D|R’)P(R’) = (9%)(25%) + (1%)(75%) = 2.25% + 0.75% = 3%. This is the UNCONDITIONAL probability of default.

P(R|D) = 2.25% / 3% = 75%.

David:

If you look at your tutorial 12min59second or PDF page 14. you gave the P(G|U) = .... / P(U|G) + P(U|G’). Don’t you think denominator should be P(UG) + P(UG’) ? Or like what your answer is: P(U|G)P(G) + P(U|G’)P(G’) ??

Based on the total probability rule, P(U) = P(UG) + P(UG’) if G and G’ are mutually exclusive and exhaustive.

Please correct me if I am wrong.
thanks
Lee

Hi Lee,

Yes, agreed, the 2nd formula on p. 14 is incorrect. Denominator should be just as you say. Apologies for error.

David

David,
Just wonder why the question is asking P(R given D) but not P(R given not D)? Could you please clarify? Thanks.

Philip

Hi Philip,

The question is arbitrary that way, nothing special in that choice.
It could have been, what is P(R | D’) in which case I think the answer looks like it would have been (25%)(91%)/97% = 23.45%

David