Financial Analysis puzzles, solved step by step

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  1. 0391% of a company's invoices are fraudulent. A screening rule flags 90% of fraudulent invoices and 5% of clean ones. If an invoice is flagged, how likely is it to be fraud?Probability and expected valueCoreWolverine TradingChicago · 2017

    Try it first

    A flagged invoice: what is the chance it is really fraud?

    Show the worked solution

    About 15.4%. Picture 10,000 invoices. 100 are fraudulent and the screen flags 90 of them. 9,900 are clean and the screen flags 5% of them, 495. The flagged pile holds 585 invoices, of which 90 are fraud, so a flag means fraud only 90 times in 585. The rare base rate swamps the screen's accuracy.

    Why is a 90% accurate screen right only 15% of the time?

    Think of a smoke alarm that never misses a fire but also goes off whenever someone makes toast. In a house where fires are rare and toast is daily, almost every alarm is toast. When the thing you are hunting is rare, even a small false positive rate on the large innocent pile produces more false alarms than true hits. Here 5% of 9,900 clean invoices is 495 false flags, against only 90 true ones.

    Count 10,000 invoices through the screen, then look only at the flagged pileAll invoices10,000Fraud, 1%100Clean, 99%9,900Flagged, 90%90Missed10Flagged, 5%495Passed9,405Flagged pile: 585495 clean90 fraud90 / 58515.4%of flags arereal fraud
    Of 10,000 invoices, the screen flags 90 of the 100 frauds and 495 of the 9,900 clean invoices, so the flagged pile of 585 is only 15.4% fraud even though the screen catches 90% of frauds.
    The relationship
    P(F∣flag)=0.90×0.010.90×0.01+0.05×0.99=0.0090.0585=15.4%P(F\mid \text{flag}) = \frac{0.90 \times 0.01}{0.90 \times 0.01 + 0.05 \times 0.99} = \frac{0.009}{0.0585} = 15.4\%
    P(F | flag)the chance an invoice is fraud given that it was flagged
    0.01the base rate of fraud
    0.05the false positive rate on clean invoices
    What it says in wordsTrue flags divided by all flags, where all flags include the false ones from the much larger clean pile.

    What would make the screen useful, and how would an auditor use it?

    Cutting the false positive rate does far more than raising the catch rate. At a 0.5% false positive rate the flagged pile would be 90 frauds and about 49.5 clean invoices, and a flag would mean fraud 65% of the time. Raising the catch rate from 90% to 100% would only move the answer from 15.4% to about 16.8%. In practice a screen like this is a triage tool: it shrinks 10,000 invoices to 585 for a human to review, and that review is where the 495 false alarms are cleared. Say the limit too: the 1% base rate is itself an estimate, and the answer is only as good as it is.

    Where candidates lose it

    The trap answer is 90%, which swaps the chance of a flag given fraud for the chance of fraud given a flag. Interviewers ask this question to see whether you notice the swap.

    Work in counts, not formulas. Saying "imagine 10,000 invoices" makes every number concrete, and the 495 false alarms jump out before you have written a single probability.

    What the interviewer asks next

    • If an invoice is flagged twice by two independent screens, what is the chance it is fraud?
    • What false positive rate would make a flag mean fraud at least half the time?
    • How would the answer change if fraud were 10% of invoices?

    Asked at Wolverine Trading, Prop Trading, Chicago, 2017 (Wall Street Oasis): Phone interviews were pretty standard brainteasers and fit questions. There was a Bayes question

  2. 062A target company's shares trade at Rs 450. A buyer has offered Rs 500 a share in cash. If the deal fails, you expect the shares to fall to Rs 350. Ignoring time value, what probability of completion does the market price imply?Probability and expected valueCoreAQR Capital ManagementGreenwich · 2021

    Try it first

    What probability of completion does Rs 450 imply?

    Show the worked solution

    About 67%. If the price is the probability-weighted average of the two outcomes, p x 500 + (1 - p) x 350 = 450, so p = (450 - 350) / (500 - 350) = 100 / 150 = 2/3. The price sits two thirds of the way from the failure value to the offer. The answer is only as good as the Rs 350 failure estimate, which nobody can observe directly.

    Why does a price between two outcomes reveal a probability?

    Picture a resale ticket for a cricket match that may be rained off. If the match is played the ticket is worth Rs 1,000; if it is washed out you get a Rs 400 refund. If tickets change hands at Rs 800, buyers are betting on play two times in three. When a price can end at one of two known values, where it sits between them is the market's probability, read off by distance from the bad outcome. A merger target is the same ticket: it ends at the offer price or falls back to where it would trade alone.

    Where the price sits between the two outcomes is the probability300350400450500550100 to lose50 to gainDeal fails: 350Offer: 500Market: 450p = 100 / 15066.7%chance of completionIgnore time value66.7%Six months at 8% a year: price x 1.04 = 46878.7%Fallback is Rs 380, not 35058.3%
    Rs 450 sits 100 above the Rs 350 failure value and 50 below the Rs 500 offer, two thirds of the way along, so the market implies about a 67% chance of completion; allowing for time value raises that to 78.7%, and a higher Rs 380 fallback lowers it to 58.3%.
    The relationship
    p×500+(1−p)×350=450  ⇒  p=450−350500−350=100150≈66.7%p \times 500 + (1 - p) \times 350 = 450 \;\Rightarrow\; p = \frac{450 - 350}{500 - 350} = \frac{100}{150} \approx 66.7\%
    pthe probability that the deal completes
    500the cash offer, received if the deal closes
    350the expected share price if the deal fails
    What it says in wordsThe implied probability is the distance from the failure value to today's price, divided by the full distance from failure to offer.

    What changes once you allow for time?

    Deals take months to close, and an arbitrageur who ties up Rs 450 wants paying for the wait. Say closing is six months away and the required return is 8% a year, 4% for the half year. Then the expected payoff must be 450 x 1.04 = Rs 468, and p = (468 - 350) / 150 = 78.7%. Ignoring time value understates the implied probability, because part of the gap to the offer is simply the return for waiting.

    What would you check before trusting the number?

    The failure value first, because it is an estimate and the answer swings on it. If the shares would fall only to Rs 380, say because the market has risen since the bid, the implied probability drops to 58.3%. Next the shape of the bet: Rs 50 to gain against Rs 100 to lose, so an arbitrage desk needs real confidence in the regulatory approvals, the buyer's financing and the shareholder vote. The limit to say aloud is that a probability read from prices also carries a premium for bearing deal risk, so it is not a pure forecast of completion.

    Where candidates lose it

    The fast wrong answer is 90%, reading the price as a fraction of the offer. That ignores the failure value entirely, and the failure value is half the information in the question.

    The quieter slip is measuring from the wrong end and saying one third. The price sits close to the offer, so completion is the likelier outcome; a quick sense check of the direction catches it.

    What the interviewer asks next

    • Closing is a year away and arbitrageurs want 10% a year. What probability is implied now?
    • The buyer raises the offer to Rs 520 and the shares jump to Rs 480. What happened to the implied probability?
    • Why might a stock-for-stock deal need a hedge that a cash deal does not?

    Asked at AQR Capital Management, Quantitative Research, Greenwich, 2021 (Wall Street Oasis): Questions about merger arbitrage strategies. Hedging. Python programming. Data analysis and regression.