Equity Research puzzles, solved step by step

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All topicsProbability and brainteasers12Expected value and decisions8Market sizing and estimation12Returns and compounding9Valuation riddles12Three statement riddles10EPS and share count9Cost of capital and rates8Growth, mix and unit economics8Mental maths6Data and reasoning traps6
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  1. 013You roll a fair die and receive Rs 100 times the face shown. Before you are paid you may reroll once, but then you must keep the second roll. What is the right to reroll worth?Expected value and decisionsCoreHedge fund long/shortMulti-manager pod

    Try it first

    Which first rolls should you keep?

    Show the worked solution

    The reroll right is worth Rs 75. A single roll is worth Rs 350 on average. With the reroll, keep 4, 5 and 6, which average Rs 500, and reroll 1, 2 and 3 for an expected Rs 350. Half the time you get Rs 500 and half the time Rs 350, so the game is worth Rs 425, Rs 75 more than without the right.

    What is the rule for keeping or rerolling?

    Think of a job offer in hand while you wait on another interview. You take the offer if it beats what you expect the other process to give you, not if it beats your dream job. Keep any result worth more than the expected value of the alternative; here the alternative is a fresh roll worth Rs 350. So 4, 5 and 6 are kept, and 1, 2 and 3 are rerolled.

    Keep any face worth more than a fresh roll, Rs 350 on averageRs 1001Rs 2002Rs 3003Rs 4004Rs 5005Rs 6006Fresh roll: Rs 350reroll: lime shows the gainkeepKeep 4, 5, 6: averageRs 500, half the timeReroll 1, 2, 3: Rs 350,half the timeGame worth Rs 425Reroll right worthRs 75
    The value of a fresh roll, Rs 350, cuts between faces 3 and 4, so you keep 4, 5 and 6 and reroll 1, 2 and 3, which makes the game worth Rs 425 and the reroll right worth Rs 75.

    How do you value the right itself?

    Value the game with the right, then subtract the game without it. With the right: half the time you hold 4, 5 or 6, averaging Rs 500; half the time you reroll and expect Rs 350. That is Rs 425. Without it the game is worth Rs 350. The right is worth the difference, Rs 75, and all of it comes from the three bad faces being replaced.

    The relationship
    V=12×500+12×350=425425−350=75V = \tfrac{1}{2}\times 500 + \tfrac{1}{2}\times 350 = 425 \qquad 425 - 350 = 75
    500average of the kept faces 4, 5 and 6, in rupees
    350expected value of a fresh roll, in rupees
    What it says in wordsThe game with the reroll right is worth the average of the kept outcomes and the expected reroll, each half the time.

    This is an option, and a research interviewer asks it to see whether you value flexibility correctly. The right has value only because you can refuse it when the first roll is good. Its value is the gain in the bad states, Rs 250, 150 and 50 on faces 1, 2 and 3, averaged over six faces: Rs 75. The same logic prices an option to expand a project or to delay an investment.

    Where candidates lose it

    The common loss is keeping only 5 and 6, or only 6, because a 4 feels ordinary. The cut-off is the expected reroll, Rs 350, and a sure Rs 400 beats it.

    The second is answering Rs 425 when asked what the right is worth. That is the game with the right; the right is the difference, Rs 75.

    What the interviewer asks next

    • What if you may reroll twice?
    • What would you pay to play if you had to announce keep or reroll before seeing the first roll?
    • How does the answer change for a 20-sided die paying Rs 100 a face?
  2. 063A company is waiting for a regulator's decision on its main drug. The stock trades at Rs 190. You estimate it is worth Rs 320 if the drug is approved and Rs 150 if it is rejected. What probability of approval is the market pricing, and what is the stock worth to someone who puts the chance of approval at 30%?Expected value and decisionsCoreHedge fund long/shortBuy-side equity research

    Try it first

    What approval probability is the Rs 190 price implying?

    Show the worked solution

    The market is pricing about a 23.5% chance of approval; at 30% the stock is worth about Rs 201. Set the price equal to the probability-weighted outcomes: 190 = p x 320 + (1 - p) x 150, so p = 40/170 = 23.5%. At 30%, the value is 150 + 0.3 x 170 = Rs 201, about 5.8% above the price, ignoring the time value of money and any premium for bearing a binary risk.

    How does a price turn into a probability?

    Think of a ticket that pays Rs 320 if it rains tomorrow and Rs 150 if it does not. Nobody would sell it below Rs 150 or buy it above Rs 320, and where it trades in between tells you how likely the crowd thinks rain is. Where the price sits between the two outcomes is the market's probability: Rs 190 is 40 of the 170 rupees from the downside to the upside, 23.5% of the way.

    The relationship
    p=P−VdownVup−Vdown=190−150320−150=23.5%p = \frac{P - V_{down}}{V_{up} - V_{down}} = \frac{190 - 150}{320 - 150} = 23.5\%
    Ptoday's share price, Rs 190
    V_upthe value if the drug is approved, Rs 320
    V_downthe value if it is rejected, Rs 150
    pthe probability of approval the price implies
    What it says in wordsThe implied probability is how far the price has climbed from the bad outcome, as a share of the gap between the outcomes.
    Where the price sits between the two outcomes is the market's probabilityRejected: Rs 150Approved: Rs 320Price Rs 190: 23.5% of the way from 150 to 320Your 30% view: Rs 150 + 0.3 x 170 = Rs 2010%100%From today's Rs 190, what each outcome doesRejected-21%Approved+68%
    The Rs 190 price sits 23.5% of the way from the Rs 150 rejection value to the Rs 320 approval value, which is the market's implied probability, while a 30% view places the value at Rs 201.

    What does the gap between 23.5% and 30% actually buy you?

    The expected edge is the difference in probability times the spread: 6.5 points x Rs 170, about Rs 11 a share, or 5.8% of the price. But the outcome is binary: the stock either falls 21% to Rs 150 or rises 68% to Rs 320, and on a 30% view it falls seven times in ten. That is why desks size binary positions small and spend their effort asking how confident the 30% itself is, not just whether it beats 23.5%.

    What does the simple formula leave out?

    Three things worth naming. The decision may be months away, so the outcomes should be discounted. Investors want paying for binary risk, so part of the gap between your view and the price is a premium, not a mistake. And the implied probability is only as good as the two end values: if the rejection value is Rs 160 instead of Rs 150, the implied probability drops from 23.5% to 18.8%. Most of the real work is in the end values, not the division.

    Where candidates lose it

    The common slip is dividing the price by the upside, 190 over 320, and saying 59%. That treats the stock as worthless if the drug fails, when it keeps Rs 150 of value. Probability is measured along the distance between the outcomes, not from zero.

    The second loss is reading Rs 201 against Rs 190 as a clear opportunity. Say that the edge is small against a 21% downside that happens seven times in ten on your own numbers, and you sound like someone who has sized a position.

    What the interviewer asks next

    • If the decision is a year away and your discount rate is 12%, how does the implied probability change?
    • The stock jumps to Rs 230 with no news. What probability is the market now pricing?
    • How would you size a position when your edge is 6.5 points of probability?
  3. 077The bill at a restaurant comes to exactly pi rupees, and you can only pay in whole paise. How do you pay a perfectly fair amount?Expected value and decisionsCoreMillennium ManagementSheung Wan · 2025

    Try it first

    Which approach makes your payment exactly fair?

    Show the worked solution

    Randomise: pay Rs 3.15 with probability about 0.159 and Rs 3.14 otherwise. Pi is 3.14159, which is 15.9% of the way from 3.14 to 3.15. Weighting the two payable amounts by those odds gives an expected payment of exactly pi. Neither side is favoured on average, which is the only sense in which an amount you cannot pay can be paid fairly.

    What does fair mean when the exact amount cannot be paid?

    Five children and four mangoes: nobody can get four fifths of a mango each without a knife, so a fair parent draws lots and one child misses out, with every child facing the same chance. When the exact amount is impossible, fair means fair on average: the expected payment equals what you owe. Rounding to 3.14 short-changes the restaurant every time; rounding to 3.15 overpays every time. Only a random choice between the two neighbours can hit pi exactly.

    Pick the two amounts either side of pi, then weight them so the balance sits on pi84.1%15.9%Rs 3.14Rs 3.15pi = 3.14159, 15.9% of the way from 3.14 to 3.15pay Rs 3.14pay Rs 3.15Coin method: the target odds 0.1593 are 0.0010100011... in binary. Flip a coin for digits 0 and 1.Stop at the first flip that differs from that digit: below the target, pay 3.15; above, pay 3.14.
    Pi sits 15.9% of the way from Rs 3.14 to Rs 3.15, so putting 84.1% of the probability on 3.14 and 15.9% on 3.15 balances the payment exactly at pi, and a fair coin can deliver those odds by comparing random binary digits with 0.0010100011 and so on.

    How do you find the right odds?

    Call the chance of paying 3.15 by the letter p. The expected payment is 3.14 plus p times one paisa, and you want that to equal 3.14159. So p is the fraction of the gap you still owe: 0.00159 over 0.01, which is 0.1593. Say it as a balance point: the further pi sits towards 3.15, the more often you pay 3.15.

    The relationship
    3.14+p×0.01=π⇒p=π−3.140.01≈0.15933.14 + p \times 0.01 = \pi \quad\Rightarrow\quad p = \frac{\pi - 3.14}{0.01} \approx 0.1593
    pthe probability of paying Rs 3.15
    0.01one paisa, the gap between the two payable amounts
    pi - 3.14the part of a paisa you still owe after paying 3.14
    What it says in wordsPay the higher amount with a probability equal to the share of the paisa you still owe.

    How do you actually produce odds of 0.159 with a coin?

    This is the follow-up that separates people. Write p in binary: 0.0010100011 and so on. Flip a fair coin to generate your own random binary digits, one at a time, and compare each with p's digit in the same place. At the first flip that differs, you know whether your random number is below p or above it, and you pay 3.15 or 3.14 accordingly. Each flip has a one in two chance of settling it, so on average two flips decide, even though p itself is irrational. The same idea, called randomised roundingRounding a fractional quantity up or down at random, with odds set so that the expected result equals the exact fraction., is used to split odd lots of shares fairly across client accounts.

    Where candidates lose it

    Most candidates say pay 3.14 or 3.15 and argue about which side should bear the fraction of a paisa. That treats the question as etiquette. The interviewer wants the expected value framing: fair on average is the only fairness available.

    The second loss comes on the follow-up. Candidates who reach p = 0.159 then say roll a thousand-sided die, which only approximates it. The binary coin comparison hits the odds exactly and needs about two flips.

    What the interviewer asks next

    • You eat at the same restaurant every day. Is there a non-random way to be fair over time?
    • How would you generate a probability of exactly one third with a fair coin?
    • The waiter is risk averse. Does paying pi in expectation still feel fair to him?

    Asked at Millennium Management, Quantitative Research, Sheung Wan, 2025 (Wall Street Oasis): How to pay the restaurant fairly if I owe pi dollars. Need to pay with usual dollars and cents.