Financial Analysis puzzles, solved step by step
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017A vendor offers a game: a fair coin is tossed until it shows heads. If heads comes on the first toss you get Rs 100, on the second toss Rs 200, and the prize doubles with every toss after that. The game costs Rs 1,000 to play, but the vendor has only Rs 10 lakh to pay out. What is the game really worth, and would you play?Equity capital marketsConsulting-style case
Try it first
With the vendor's Rs 10 lakh limit, what is the game worth on average?
Show the worked solution
About Rs 761, so at Rs 1,000 the game is not worth playing. Each possible toss adds Rs 50 of expected value: a prize that doubles times a chance that halves. Prizes fit under the Rs 10 lakh cap for 14 tosses, which is Rs 700, and all longer runs are paid Rs 10 lakh, which adds only about Rs 61. The infinite value of the textbook game rests entirely on prizes no vendor can pay.
Why does every toss add exactly Rs 50?
Imagine a lottery stall where a ticket wins Rs 200 one time in four, or Rs 400 one time in eight. Each prize is worth the same on average, Rs 50, because the prize doubles exactly as the chance halves. In this game, heads first appearing on toss k has probability one over 2 to the k, and pays 100 times 2 to the k minus 1, so every possible toss contributes Rs 50 of expected value. With no limit there are infinitely many tosses, so the expected value is infinite. That is the famous St Petersburg paradox, and it is why nobody sensible pays a fortune to play.
The relationshipk the toss on which the first head appears 1/2^k the chance the first head comes on toss k 100 x 2^(k-1) the promised prize 10^6 the vendor's limit, Rs 10 lakh What it says in wordsFourteen tosses each add Rs 50; every longer run pays the capped Rs 10 lakh, and the chance of reaching toss 15 is one in 16,384.What does the Rs 10 lakh limit do to the value?
The prize on toss 14 is Rs 100 times 2 to the 13, which is Rs 8,19,200; on toss 15 it would be Rs 16,38,400, more than the vendor holds. So only 14 tosses pay in full, worth Rs 700, and every longer run pays Rs 10 lakh, which happens one time in 16,384 and adds about Rs 61. The game is worth about Rs 761. Paying Rs 1,000 means losing about Rs 239 a game on average.
Each of the first 14 tosses adds Rs 50 of expected value, reaching Rs 700, and the capped tail adds only about Rs 61, so the game is worth about Rs 761. Without the cap the value would keep rising Rs 50 a toss, but the capped game never reaches the Rs 1,000 price. How rich would the vendor need to be for Rs 1,000 to be fair?
Value grows very slowly with the vendor's wealth, because each doubling of the bankroll adds just one more Rs 50 toss. A vendor holding Rs 1,000 crore would make the game worth only about Rs 1,425, and the game reaches Rs 1,000 only if the vendor can pay about Rs 2.6 crore. Say the limit too: even a fair price ignores how much risk you can stomach, since almost every game pays Rs 100 or Rs 200. That is why economists use this game to show people value money by its usefulness to them, not by its face amount.
Where candidates lose it
The common loss is reciting "infinite expected value" and stopping, or worse, saying you would pay any price. The interviewer added the vendor's limit precisely to see whether you notice that infinite value depends on payouts that cannot happen.
The second loss is getting the cap wrong by assuming the tail adds a lot. Work out the last toss that pays in full, count Rs 50 per toss, and add the small capped tail; it takes thirty seconds.
What the interviewer asks next
- What would you pay if the vendor could pay out Rs 100 crore?
- The vendor offers to play the game ten times in a row for Rs 7,000. Does that change your answer?
- How does this game relate to valuing a company with a tiny chance of an enormous outcome?
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?Wolverine 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.
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 relationshipP(F | flag) the chance an invoice is fraud given that it was flagged 0.01 the base rate of fraud 0.05 the 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
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?AQR 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.
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 relationshipp the probability that the deal completes 500 the cash offer, received if the deal closes 350 the 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.
087You lend to 10 borrowers. Each has a 2% chance of defaulting over the year, independently of the others. What is the chance that at least one defaults, and what is the expected number of defaults?Rating agenciesBank credit
Try it first
What is the chance that at least one of the ten defaults?
Show the worked solution
The chance of at least one default is 18.3%, and the expected number of defaults is 0.2. Work from the complement: each loan survives with probability 98%, so all ten survive with probability 0.98^10 = 81.7%, and at least one default is the rest. The expected number is simply 10 x 2% = 0.2, which holds whether or not the loans are independent. Independence matters for the first answer, not the second.
Why work from the chance that nothing happens?
Ask what the chance is that at least one of ten friends is late for dinner, each being late one time in fifty. Counting the ways someone could be late is messy: one late, two late, any combination. Counting the single way nobody is late is easy: everyone on time. 'At least one' is one minus 'none', and 'none' for independent events is just the single probabilities multiplied. Each loan survives with probability 0.98, so all ten survive with probability 0.98^10.
To do 0.98^10 in your head, use the shortcut that (1 minus x) to the power n is close to 1 minus nx plus a correction of n(n minus 1)/2 times x squared: 1 minus 0.20 plus 45 x 0.0004 = 0.818. The exact figure is 0.8171, so the chance of at least one default is 18.29%. A second check is the Poisson shortcut, 1 minus e to the minus 0.2, which gives 18.1%.
With ten independent loans each carrying a 2% default chance, no default happens 81.7% of the time, so at least one default happens 18.3% of the time, while expected defaults are 0.2 whether the loans are independent or perfectly linked. Why is the expected number so much simpler?
Expected values add, always. Each loan contributes 0.02 expected defaults, and ten loans contribute 0.2. The expected number of defaults is 0.2 regardless of how the loans are connected, because the expectation of a sum is the sum of the expectations even when the events are correlated. The distribution behind it is lopsided: 81.7% of the time nothing happens, 16.7% of the time exactly one loan fails, and two or more fail 1.6% of the time.
What does correlation change?
Take the extreme. If all ten borrowers are suppliers to one factory and fail together or survive together, there is a 2% chance that all ten default and a 98% chance that none does. Expected defaults are still 0.2. But the chance of at least one default falls to 2%, and the chance of losing the entire book jumps from practically zero to 2%. Correlation does not change the average loss; it moves probability from many small losses into rare large ones, which is the loss a lender cannot survive. That is why credit portfolio work spends its effort on concentration and correlation rather than on the average default rate, and why a portfolio of ten loans to one industry is riskier than its expected loss suggests.
Where candidates lose it
The fast wrong answer is 20%: ten loans times 2%. Adding probabilities only works for events that cannot happen together, and two loans can both default. The error is small here, 20% against 18.3%, but it grows quickly: with 100 such loans the same method gives 200%, which is impossible.
The second loss is giving 18.3% and stopping. The question says independently for a reason. Saying what correlation would do, same expected defaults, fatter tail, is the part a credit interviewer is listening for.
What the interviewer asks next
- How many such loans do you need before at least one default is more likely than not?
- If each loan is Rs 10 crore and recovers 40% on default, what is the expected loss on the book?
- Two of the ten borrowers are in the same group of companies. Does the chance of at least one default go up or down?
