CFA Level IQuantitative MethodsHard
An analyst knows that 5% of bonds in a large portfolio are internally flagged as financially 'distressed.' Historically, 40% of distressed bonds default within one year, while only 2% of non-distressed bonds default within one year. A randomly selected bond defaults within the year. Using Bayes' formula, what is the probability that this bond was originally flagged as distressed?
- A20.00%
- B5.00%
- C40.00%
- D51.28%
Show answer & explanationAnswer & explanation
Correct answer: D. 51.28%
P(Default) = P(D)×P(Default|D) + P(not D)×P(Default|not D) = (0.05×0.40) + (0.95×0.02) = 0.02 + 0.019 = 0.039. By Bayes' rule, P(D|Default) = P(D)×P(Default|D)/P(Default) = 0.02/0.039 ≈ 51.28%.
Why the other options are wrong
- A. Arbitrary intermediate value; doesn't reflect correct Bayesian updating.
- B. This is simply the unconditional prior probability of being distressed, ignoring the default information.
- C. This is just P(Default|Distressed), the prior conditional probability, not the updated posterior.
Bayes' Formula
A rule for updating the probability of an event based on new information, combining prior probabilities with conditional likelihoods to produce a posterior probability.
- Updated Probability = [P(new info|event)×P(event)] / P(new info)
- Denominator computed via the total probability rule across all mutually exclusive scenarios
- Widely used in credit risk, diagnostic testing, and Bayesian statistics
Memory trick: Start with your prior belief, update it once new evidence knocks