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ywilfred
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Probability of Bond default 25 Aug 2009, 08:16
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Bonds rated 'B' have a 25% chance of default in five years. Bonds rated 'CCC' have a 40% chance of default in 5 years. A portfolio consists of 30% 'B' rated bonds 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?



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hoping_for_stern
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Probability of Bond default Updated on: 25 Aug 2009, 09:49
Originally posted by hoping_for_stern on 25 Aug 2009, 09:04.
Last edited by hoping_for_stern on 25 Aug 2009, 09:49, edited 1 time in total.
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Guess I need to read the questions more thoroughly :oops:

A)Probability of 'B' bonds in 5 years = 1/4.
B) Probability of 'B' in random sample = 3/10

Probability of 'B' in random sample of 5 years includes dependent events (A) and (B) / hence multiplication
1/4*3/10 = 3/40 = 7.5%

so here is where I start to doubt myself - since i am in initial phase of prepartion,any answer that gets derived very easily,
makes me doubt if I am doing it correct or not .

What is the OA ?
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Probability of Bond default 25 Aug 2009, 09:09
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post answer options pls
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Probability of Bond default 25 Aug 2009, 09:33
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ywilfred
Bonds rated 'B' have a 25% chance of default in five years. Bonds rated 'CCC' have a 40% chance of default in 5 years. A portfolio consists of 30% 'B' rated bonds 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?
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The portfolio contains 30% B-rated bonds, and 70% CCC-rated bonds. We're not interested in the entire portfolio, however; we know in advance that the bond we're selecting defaulted. 40% of CCC-rated bonds default, so (0.7)*(0.4) = 28% of the portfolio is defaulted CCC bonds. 25% of B-bonds default, so (0.25)(0.3) = 7.5% of the portfolio is defaulted B-bonds. Thus, if the bond we selected randomly was in default, we must have selected it from the 35.5% of bonds which defaulted. The probability it was B-rated was thus 7.5/35.5 = 15/71.
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Probability of Bond default 25 Aug 2009, 09:43
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Default rate for B in the portfolio = 1/4*3/10 = 3/40 = .075
Default rate for CCC in the portfolio = 4/10*7/10 = .28
Probability that a B rated bond is selected among the defaulted bonds = .075/(.28 +.075)
= .075/(.355)

Please post OA
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Probability of Bond default 25 Aug 2009, 11:28
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hoping_for_stern

- You are on the right track! You just solved for the probability of the randomly selected bond being B bond and defaulting. The questions is "given that it defaulted" what is the probability of a B bond.

Changes the approach a bit.

You need to find also the probability of a CCC bond defaulting (.7)(.4) = 28%

Probability of B/Probability bond defaulted = 7.5/(28+7.5) = slightly more than 20%
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Probability of Bond default 25 Aug 2009, 21:23
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IanStewart
ywilfred
Bonds rated 'B' have a 25% chance of default in five years. Bonds rated 'CCC' have a 40% chance of default in 5 years. A portfolio consists of 30% 'B' rated bonds 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?

The portfolio contains 30% B-rated bonds, and 70% CCC-rated bonds. We're not interested in the entire portfolio, however; we know in advance that the bond we're selecting defaulted. 40% of CCC-rated bonds default, so (0.7)*(0.4) = 28% of the portfolio is defaulted CCC bonds. 25% of B-bonds default, so (0.25)(0.3) = 7.5% of the portfolio is defaulted B-bonds. Thus, if the bond we selected randomly was in default, we must have selected it from the 35.5% of bonds which defaulted. The probability it was B-rated was thus 7.5/35.5 = 15/71.
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Good explanation IanStewart and good question, thanks for poster
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ywilfred
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Probability of Bond default 25 Aug 2009, 23:54
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Ian stewart did it nicely. The answer is 0.211.



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