Debt Capital Markets puzzles, solved step by step

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  1. 1004% of issuers in a sector default within a year. An early warning model flags 75% of the issuers that will default and 10% of those that will not. An issuer is flagged. What is the probability that it defaults?Probability and expected valueCoreBelvedere TradingChicago · 2022

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    About 23.8%. Picture 1,000 issuers: 40 will default and 960 will not. The model flags 75% of the 40, which is 30, and 10% of the 960, which is 96. Of the 126 flagged issuers, 30 default, so the chance is 30 / 126 = 23.8%. The flag multiplies the risk about six times, but most flags are still false alarms because defaults are rare.

    Why is the answer not 75%?

    A smoke alarm that goes off for every real fire and also for one in ten batches of toast will ring mostly for toast, because toast is far more common than fire. The early warning model is the same. The chance of default given a flag depends on how common defaults are to begin with, and when the base rate is low, false flags from the large healthy group swamp the true flags from the small defaulting group. 75% answers a different question: how often a defaulter gets flagged.

    Out of 1,000 issuers, most flags land on companies that are fine1,000 issuers40 will default4%960 will not96%30 flagged75%10 missed25%96 flagged10%864 cleared90%An issuer is flagged. It is one of 30 + 96 = 126 flagged issuers.Only 30 of them default:30 / 126 = 23.8%most flags are false alarms
    Of 1,000 issuers, the 40 that will default produce 30 flags and the 960 that will not produce 96 false flags, so only 30 of the 126 flagged issuers default, a probability of 23.8%.
    The relationship
    P(D∣F)=P(F∣D)P(D)P(F∣D)P(D)+P(F∣Dˉ)P(Dˉ)=0.75×0.040.75×0.04+0.10×0.96=30126=23.8%P(D \mid F) = \frac{P(F \mid D)P(D)}{P(F \mid D)P(D) + P(F \mid \bar D)P(\bar D)} = \frac{0.75 \times 0.04}{0.75 \times 0.04 + 0.10 \times 0.96} = \frac{30}{126} = 23.8\%
    P(D)the base rate of default, 4%
    P(F | D)the chance a defaulter is flagged, 75%
    P(F | not D)the chance a healthy issuer is flagged, 10%
    What it says in wordsOf all the flags, the share that come from real defaulters is the chance a flag means default.

    How should a credit desk use a 23.8% flag?

    As a reason to look harder, not a verdict. A flag lifts the default probability from 4% to 23.8%, roughly six times, which is worth a review but not a sale on its own. A second, independent signal compounds the evidence: if a separate test with the same accuracy also flagged the issuer, the probability would rise to about 70%. The honest limitation is that two warning signals about the same company are rarely independent, so the real lift from a second flag is usually smaller.

    Where candidates lose it

    The common error is answering 75%, swapping the probability of a flag given default for the probability of default given a flag. It is the most frequent mistake in Bayes questions, and interviewers set it up deliberately.

    The second loss is doing the formula silently and producing 23.8% with no picture. Say the 1,000 issuers out loud, 30 true flags and 96 false ones, and the interviewer can follow every step.

    What the interviewer asks next

    • What false alarm rate would the model need for a flag to mean a 50% chance of default?
    • The sector's default rate doubles to 8%. What does a flag mean now?
    • Two independent models both flag the issuer. What is the probability of default?

    Asked at Belvedere Trading, Capital Markets, Chicago, 2022 (Wall Street Oasis): The technical portion of the interview consisted of probability questions including one questions relating to Bayes' theorem