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Quant B - Question 2
 
You are here: Forum Home  >  Forums  >  2008 FRM Screencast Tutorial Q&A  >  Thread
David Harper, CFA, FRM, CIPM
Posted: 05 April 2008 08:51 PM   [ Ignore ]  
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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%.

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Lee
Posted: 13 April 2008 09:07 AM   [ Ignore ]   [ # 1 ]  
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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

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Lichuan-Lee Kang

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David Harper, CFA, FRM, CIPM
Posted: 13 April 2008 09:20 AM   [ Ignore ]   [ # 2 ]  
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Hi Lee,

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

David

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shuihongwong
Posted: 14 August 2008 02:57 PM   [ Ignore ]   [ # 3 ]  
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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

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David Harper, CFA, FRM, CIPM
Posted: 15 August 2008 04:48 PM   [ Ignore ]   [ # 4 ]  
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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

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