Equity Research puzzles, solved step by step

Puzzles: 100
Traced to a firm: 19
Topics: 11
Hard: 30

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All topics Probability and brainteasers12 Expected value and decisions8 Market sizing and estimation12 Returns and compounding9 Valuation riddles12 Three statement riddles10 EPS and share count9 Cost of capital and rates8 Growth, mix and unit economics8 Mental maths6 Data and reasoning traps6

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Any Warm up Core Hard

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001You roll a fair die until each even number, 2, 4 and 6, has appeared at least once. You are told the game ended on a 2. What is the probability that the first roll was a 1, and why is it not 1/5?Probability and brainteasersHardSquarepoint CapitalLondon · 2026
Try it first
Commit before you work it: given the game ended on a 2, what is the chance the first roll was a 1?
Show the worked solution
The answer is 1/6. After a first roll of 1, 3 or 5 the game ends on 2 with chance 1/3; after a 2 it cannot; after a 4 or 6 it ends on 2 with chance 1/2. Overall the game ends on 2 with chance 1/3, so a first roll of 1 carries (1/6 x 1/3) / (1/3) = 1/6. The 4 and 6 absorb the share the 2 lost.
Why does 1/5 feel right, and where does it go wrong?
Think of a cricket team told that the match was won off the last ball. That news does not just rule out one scenario; it makes the close games far more likely than the easy ones. Conditioning works the same way here. Being told the game ended on 2 is evidence, and evidence reweights every starting point by how well it explains what you saw. The 1/5 answer treats the news as if it only deleted the 2 and left the other five faces equally likely.
So ask, for each first roll, how likely it is that 2 ends up the last even number. After a 1, 3 or 5, all three evens are still missing and each is equally likely to come last, so 1/3. After a 2, the 2 is already seen and cannot come last, so 0. After a 4, only 2 and 6 are missing, and 2 comes last exactly when 6 shows first, so 1/2. The same holds after a 6.
The chance the game ends on 2 is 1/3 after a first roll of 1, 3 or 5, zero after a 2, and 1/2 after a 4 or 6, so once you learn it ended on 2 the first roll carries weight 1/6 for each odd face and 1/4 each for the 4 and the 6.
How do you turn those chances into the answer?
Apply Bayes ruleThe chance of a cause given what you saw equals its prior chance times how likely it made the observation, divided by the total chance of the observation.. Each face starts at 1/6. Multiply by its chance of ending on 2 and add them up: three faces at 1/18, one at 0, two at 1/12, which totals 1/3. The 1 contributes 1/18 of that 1/3, which is 1/6, exactly its starting weight. The 4 and the 6 rise from 1/6 to 1/4 each, and the 2 falls to zero.
The relationship
$P(1 \mid \text{end on }2) = \frac{\tfrac{1}{6}\cdot\tfrac{1}{3}}{\tfrac{3}{6}\cdot\tfrac{1}{3} + \tfrac{1}{6}\cdot 0 + \tfrac{2}{6}\cdot\tfrac{1}{2}} = \frac{1/18}{1/3} = \frac{1}{6}$
1/6 the chance of each first roll before you know anything
1/3 the chance of ending on 2 after an odd first roll
1/2 the chance of ending on 2 after a first roll of 4 or 6
What it says in wordsWeight each first roll by how likely it makes ending on 2, then divide by the total chance of ending on 2.
The research version of this is reading a data point. A company that beats estimates is more likely to be one that guided low, not just one that is doing well. Before you update, ask which starting stories make the thing you observed more likely, and shift weight towards them.
Where candidates lose it
The trap is saying 1/5 fast, because it sounds like careful conditioning: remove the impossible case and spread the rest evenly. The interviewer asked why it is not 1/5 precisely because that answer throws away how strongly each start predicts the ending.
The second loss is getting 1/6 by luck and being unable to explain it. Say the three conditional chances, 1/3, 0 and 1/2, out loud; that list is the whole argument.
What the interviewer asks next
What is the probability the first roll was a 4, given the game ended on 2?
What is the expected number of rolls until all three evens have appeared?
If the game instead ends when any two evens have appeared, how does the answer change?
Asked at Squarepoint Capital, Quant Research Intern Interview, London, 2026 (Wall Street Oasis): why is the probability of seeing a 1 on our first roll, given that we end on a 2, not 1/5
002A private company is worth somewhere between Rs 0 and Rs 100 crore to its owner, every value equally likely, and the owner knows the exact figure. Under your management it would be worth 1.5 times whatever it is worth to the owner. The owner accepts any bid at or above the company's value to them. How much should you bid?Expected value and decisionsHardBuy-side equity researchHedge fund long/short
Try it first
Pick your bid before you work it.
Show the worked solution
Bid nothing: every positive bid loses money on average. If you bid b, the owner accepts only when the value is below b, so accepted deals average b/2. Worth 1.5 times that to you, they return 0.75b for a price of b, a loss of a quarter of the bid each time. At Rs 60 crore you would win 60% of the time and lose Rs 15 crore on every win.
Why does the average value of Rs 50 crore mislead?
Imagine buying a used car from a private seller who has driven it for five years. If the seller happily accepts your first offer, the most useful thing you have learnt is what the seller knows about the car. The same logic runs here. The owner says yes only when your bid is above what the company is worth to them, so acceptance itself is bad news about the value. Across all companies the average is Rs 50 crore, but you never get to buy the average company; you buy the ones worth less than your bid.
A bid of Rs 60 crore is accepted only when the owner's value is below Rs 60 crore, so the accepted cases average Rs 30 crore, worth Rs 45 crore to you, and every accepted deal loses Rs 15 crore, a quarter of the bid.
How do you show that no bid works?
Take a general bid b. The owner accepts with chance b/100, and given acceptance the value is spread evenly from 0 to b, averaging b/2. To you that is worth 1.5 x b/2 = 0.75b. You pay b and receive 0.75b, so each accepted deal loses 0.25b, whatever b is. Multiply the loss by the chance of acceptance and the expected result is minus b squared over 400, which is zero only at b = 0. At a bid of Rs 60 crore that is minus Rs 9 crore.
The relationship
$E[\text{profit}] = \frac{b}{100}\left(1.5\cdot\frac{b}{2} - b\right) = -\frac{b^2}{400}$
b your bid, Rs crore
b/100 the chance the owner accepts
b/2 the average value of a company whose owner accepts
What it says in wordsThe chance of a deal times what each deal makes, and each deal loses a quarter of the bid.
This is the winner's curseThe tendency of the winning bid in an auction or negotiation to be the one that most overestimated the value, because the other side or other bidders knew better., and it is why a buy-side analyst asks who is on the other side of a trade. When the seller knows more than you do, the times you get filled are skewed towards the times you were wrong. The answer changes only if your edge is large enough: with a multiple of 2 instead of 1.5, accepted deals exactly break even.
Where candidates lose it
The fast answer takes the Rs 50 crore average, multiplies by 1.5 and bids anything under Rs 75 crore. It ignores that the owner chooses whether to sell, so the companies you actually buy are not a random sample.
The second loss is getting the zero answer and not generalising it. Say that the loss is a fixed quarter of any bid, so no bid escapes it, and name the multiple at which the answer flips.
What the interviewer asks next
What multiple of the owner's value would you need before any positive bid breaks even?
How does the answer change if the owner does not know the value either?
Where do you see the winner's curse in IPO allotments or block trades?

003Estimate India's annual cement demand in million tonnes. Do it two ways: once from consumption per person, and once from what gets built, housing, infrastructure and commercial construction. Then reconcile the two answers.Market sizing and estimationHardIndian brokerage researchSell-side equity research

Try it first

Your two routes give different answers. What is the best next move?

Show the worked solution

About 325 to 420 million tonnes, on these illustrative inputs. The per capita route, 1,400 million people at 0.30 tonnes each, gives 420. Adding up housing, repairs, infrastructure and commercial building gives 325. The 95 million tonne gap points at the two softest inputs: the per capita anchor and the number of homes built each year. Check either against published industry data.

How does the per capita route work?

It is the way a household guesses its monthly rice: people times how much each eats. Take a population of about 1,400 million and an assumed consumption of 0.30 tonnes, 300 kg, a head. That gives 420 million tonnes. The route is quick but hangs on a single number you cannot see, the per capita figure, so it is only as good as your anchor. Say that you would check the anchor against published data rather than quoting one from memory.

How do you build the end use route, and why does it disagree?

Now count what gets built. Assume 10 million new homes a year at 600 sq ft and a builder's thumb rule of about 20 kg of cement per sq ft: 120 million tonnes. Repairs and extensions: 250 million existing homes, 6% doing a job a year, about 2 tonnes each: 30. Infrastructure: assume Rs 15 lakh crore of spending a year, cement at 5% of project cost and Rs 6,000 a tonne: 125. Commercial and industrial: 2,000 million sq ft at 25 kg: 50. Total 325.

On these illustrative inputs the per capita route gives 420 million tonnes and the end use route gives 325, of which new housing is 120 and infrastructure 125, leaving a gap of 95 million tonnes that tells you which assumptions to test.

When two routes disagree, the gap tells you which assumption to test, not which answer to average. Close the 95 million tonne gap from each side in turn. A per capita figure of 0.23 tonnes instead of 0.30 would close it alone. So would roughly 18 million new homes instead of 10, which is a big move, so the home count is less likely to be the whole story. Self-built rural homes are the category most often missed, which is where you would dig.

End use	Build-up	Million tonnes
New housing	10 m homes x 600 sq ft x 20 kg	120
Repairs and extensions	250 m homes x 6% x 2 t	30
Infrastructure	Rs 15 lakh crore x 5% / Rs 6,000 a t	125
Commercial and industrial	2,000 m sq ft x 25 kg	50
Total	325

Every input is an assumption made for the exercise, stated so the interviewer can challenge it one line at a time.

Where candidates lose it

The common loss is presenting one route and one number with false precision. The interviewer asked for two routes because the reconciliation is the test: can you say which input you trust least and how far it would have to move.

The second is quoting a national consumption figure from memory as fact. Build from assumptions you state, and say which published source you would check.

What the interviewer asks next

How would the answer move if housing starts fell 20% in a downturn?
Which end use would you model first for a cement company with most of its plants in one region?
How would you turn this demand estimate into a utilisation rate for the industry?

004An index rises 10% one day and falls 10% the next, alternating for ten days. Where does it end? A leveraged product returns exactly twice the index's move each day. Where does that end?Returns and compoundingHardHedge fund long/shortLong-only asset management
Try it first
The index ends ten days down about 4.9%. Where does the 2x daily product end?
Show the worked solution
The index ends at 0.951, down 4.9%, and the 2x product ends at 0.815, down 18.5%. Each up-down pair multiplies the index by 1.1 x 0.9 = 0.99 and the product by 1.2 x 0.8 = 0.96. Five pairs give 0.99 to the fifth and 0.96 to the fifth. The product loses nearly four times as much, not twice.
Why does a flat-looking path lose money at all?
A shop marks a shirt up 10% and then runs a 10% sale: the tag ends at 99% of where it began, because the discount is taken on the higher price. A gain and an equal percentage loss never cancel; the pair always leaves you with one minus the square of the move. For 10% that is 1 minus 0.01, so each pair costs 1%. Five pairs cost a little under 5%.
Over ten alternating days the index ends at 0.951 while the 2x daily product ends at 0.815, below the 0.902 that simply doubling the index's loss would give, because each pair costs the product 4% against 1% for the index.
Why is the 2x product four times worse and not twice?
Doubling the daily move doubles the swing, and the pair loss is the square of the swing. Twice the move means four times the loss per pair: 0.2 squared is 0.04 against 0.1 squared at 0.01. Compounded over five pairs the product lands at 0.815, a 18.5% loss, about 3.8 times the index's 4.9%. Someone who expected twice the index would have looked for 0.902.
The relationship
$(1+r)(1-r) = 1 - r^2 \qquad 0.99^5 = 0.951 \qquad 0.96^5 = 0.815$
r the daily move, 0.10 for the index and 0.20 for the 2x product
1 - r^2 what one up-down pair leaves you with
What it says in wordsEach up and down pair shrinks the value by the square of the move, so doubling the move quadruples the shrinkage.
The general name is volatility dragThe gap between the average of a set of returns and the compound return they produce, roughly half the variance of the returns.. It is why a daily leveraged product can fall over a month in which its index ended flat, and why it is built for short holding periods. The limitation is honest: in a steady trend with little back and forth, daily compounding can leave the product ahead of twice the index.
Where candidates lose it
The trap is doubling the index's result and answering down 9.8%. It treats a product that resets its leverage every day as if it held a fixed position for ten days.
The second loss is getting the numbers without the reason. Say that the pair loss is the square of the move, and the four times falls out of that in one line.
What the interviewer asks next
What if the index rises 10% every day for ten days? Is the 2x product ahead of or behind twice the index's return?
What about a minus 2x daily product on the same alternating path?
How would you estimate the monthly drag on a 3x product from the index's daily volatility?
005A DCF has flat free cash flow of 100 a year for five years, a WACC of 10% and a terminal growth rate of 5% after year five. What share of the value comes from the terminal value, and how much does the value fall if WACC rises to 11%?Valuation riddlesHardSell-side equity researchBuy-side equity research
Try it first
Before the arithmetic: roughly how much does value fall when WACC goes from 10% to 11%?
Show the worked solution
The terminal value is about 77% of the value, and a one point rise in WACC cuts the total by about 16%. At 10% the five years are worth 379 and the terminal value 1,304 in today's money, total 1,683. At 11% they are worth 370 and 1,039, total 1,408. The explicit years barely move; the terminal value drops by a fifth.
Where does most of the value sit?
Think of valuing a flat you plan to rent out for five years and then keep forever. The five years of rent are real, but the flat itself, the part you keep, is most of what you are paying for. A DCF is the same. The five explicit years are worth 379; everything after year five, the terminal value, is worth 1,304 in today's money, 77.5% of the total. The terminal value at year five is 100 x 1.05 / (0.10 - 0.05) = 2,100, discounted back five years.
At a 10% WACC the terminal value is 77.5% of a total value of 1,683; at 11% the total falls to 1,408, down 16.3%, and almost all of the fall comes from the terminal value rather than the five explicit years.
Why does one point of WACC move the value so much?
The terminal value divides by the gap between WACC and growth. Moving WACC from 10% to 11% widens that gap from 5% to 6%, cutting the undiscounted terminal value from 2,100 to 1,750, a sixth, before the extra discounting takes more. The explicit years fall only from 379.1 to 369.6. Together the value drops from 1,683 to 1,408, down 16.3%.
The relationship
$TV_5 = \frac{FCF\,(1+g)}{WACC - g} = \frac{100 \times 1.05}{0.10 - 0.05} = 2{,}100$
FCF free cash flow in year five, 100
g terminal growth, 5%
WACC - g the gap that sets the terminal multiple, 5% here
What it says in wordsThe terminal value is next year's cash flow divided by the gap between the discount rate and growth, so a small gap makes it very sensitive.
This is why a research note shows a sensitivity table of WACC against terminal growth rather than a single number. It is also why you check the implied exit multiple: 2,100 is 21 times year-five cash flow, and if peers trade nowhere near that, the inputs need a second look. The limitation is that the table shows sensitivity; it does not tell you which rate is right.
Where candidates lose it
The common loss is treating WACC as a small adjustment and guessing a fall of a few per cent. The terminal value's denominator is the gap between WACC and growth, and that gap moves by a fifth.
The second is presenting a DCF value without saying how much of it sits beyond the forecast years. Give the share first; it tells the interviewer you know where the model's risk lives.
What the interviewer asks next
What terminal growth rate at 11% WACC would restore the original value?
What exit multiple is implied by the terminal value, and how would you sanity check it?
Why does a high-growth company usually have an even larger terminal value share?
007A company has revenue of 100. Variable costs are 50% of revenue, fixed costs are 40, interest is 5 and the tax rate is 25%. Revenue rises 10%. By how much do EBIT and net income grow?EPS and share countHardSell-side equity researchBuy-side equity research
Try it first
Revenue is up 10%. What happens to net income?
Show the worked solution
EBIT grows 50% and net income 100%. Revenue of 110 leaves 55 after variable costs; less fixed costs of 40, EBIT is 15 against 10. Less interest of 5, pre-tax profit is 10 against 5, and after 25% tax net income is 7.5 against 3.75. Fixed costs magnify the change five times and interest doubles it again.
Why does a 10% revenue change become a 50% EBIT change?
Think of a tea stall with a fixed monthly rent. Once the rent is covered, every extra cup sold is almost pure profit, so a busy month feels far better than the extra sales alone suggest. Fixed costs do not grow with revenue, so the whole extra contribution lands in EBIT, and a thin EBIT makes that addition a large percentage. Here revenue up 10 adds 5 of contribution, and 5 on an EBIT of 10 is 50%.
Revenue rises from 100 to 110 while fixed costs stay at 40, so EBIT rises from 10 to 15; interest stays at 5, so net income rises from 3.75 to 7.5, a 100% increase from a 10% revenue gain.
Why does net income grow twice as fast as EBIT?
Interest is a second fixed charge, sitting below EBIT. With interest of 5 taking half of an EBIT of 10, the next 5 of EBIT doubles pre-tax profit, and tax at a flat rate keeps that doubling intact. The shortcut: operating leverageHow much operating profit moves for a given change in revenue, driven by the share of costs that are fixed. is contribution over EBIT, 50 over 10, which is 5; financial leverage is EBIT over pre-tax profit, 10 over 5, which is 2. Together 5 x 2 = 10, and 10 x 10% = 100%.
The relationship
$\%\Delta NI = \underbrace{\frac{50}{10}}_{\text{operating}} \times \underbrace{\frac{10}{5}}_{\text{financial}} \times 10\%\ = 100\%$
50 contribution: revenue less variable costs
10 EBIT before the change
5 pre-tax profit before the change
What it says in wordsOperating leverage times financial leverage times the revenue change gives the change in net income.
The same arithmetic runs in reverse, which is the point a research analyst should add. A 10% revenue fall would halve EBIT and wipe out net income entirely. Leverage magnifies both directions, so a highly geared, high fixed cost company is the one whose earnings estimates move most on a small change in the top line. The limit of the shortcut: it holds only while costs behave as fixed, and over a few years most costs flex.
Where candidates lose it
The fast answer is 10% for everything, because it assumes every line scales with revenue. The question is built to see whether you notice which costs do not move.
The second loss is getting 50% for EBIT and stopping, forgetting that interest is a second fixed layer. Walk the income statement all the way to net income out loud.
What the interviewer asks next
What happens to net income if revenue falls 10% instead?
At what revenue does net income reach zero?
How would you spot a company with high operating leverage from its annual report?
008A company's unlevered cost of capital is 12% and it can borrow at 8%. Ignore taxes. What happens to its cost of equity and its WACC when it moves from no debt to debt equal to equity?Cost of capital and ratesHardBuy-side equity researchHedge fund long/short
Try it first
Debt at 8% replaces half the 12% capital. What is the new WACC?
Show the worked solution
The cost of equity rises from 12% to 16% and WACC stays at 12%. With debt equal to equity, shareholders carry the same business risk on half the capital, so their required return rises by the spread between the unlevered rate and the debt rate, 4 points, times debt over equity. Half at 16% and half at 8% is still 12%. Without taxes, cheap debt only moves risk around.
Why can cheap debt not lower the cost of capital on its own?
Think of two friends buying a food truck. If one lends at a fixed rate and the other takes whatever is left after paying the loan, the truck's takings are no less risky; the owner just now carries all of the ups and downs on a smaller stake. Borrowing does not change the business, so it cannot change the total return the business must earn for all its funders; it only shifts risk from lenders to shareholders.
Without taxes the cost of equity rises in a straight line from 12% at no debt to 20% at debt twice equity, while WACC stays flat at 12%; the dashed red line is the mistaken WACC that holds equity at 12% and falls to 10% at debt equal to equity.
How do you get the 16%?
The Modigliani and MillerThe 1958 result, from Franco Modigliani and Merton Miller, that in a world without taxes or distress costs the value of a firm does not depend on how it is financed. relation gives it directly. The cost of equity is the unlevered rate plus the gap between the unlevered rate and the debt rate, scaled by debt over equity. Here that is 12% plus (12% minus 8%) times 1, which is 16%. Check with WACC: half at 16% plus half at 8% is 12%, exactly the unlevered rate.
The relationship
$r_E = r_U + (r_U - r_D)\frac{D}{E} = 12\% + 4\% \times 1 = 16\%$
r_U the unlevered cost of capital, the return the business itself must earn
r_D the cost of debt
D/E debt over equity at market values
What it says in wordsShareholders demand the business's return plus a premium for each rupee of debt standing ahead of them.
Now add back what the puzzle removed. With tax, interest is deductible, so debt does lower WACC a little; at high debt, the cost of debt itself rises and distress costs appear. That is why an analyst who sees WACC fall sharply as a model adds debt should check whether the cost of equity was left unchanged. In practice this shows up as re-levering beta: the equity beta must rise when leverage rises.
Where candidates lose it
The trap is averaging 12% and 8% and announcing a WACC of 10%. It holds the cost of equity fixed while the equity becomes riskier, and it quietly creates value from nothing.
The second loss is getting 12% and not being able to say why. The one line to say is that financing slices the same cash flows differently; it does not change them.
What the interviewer asks next
Add a 25% tax rate. What is the WACC at debt equal to equity now?
If the debt cost rises to 10% at this leverage, what happens to the cost of equity?
How do you re-lever a peer's beta for a company with more debt?
026You start with Rs 2. Each round you win Rs 1 with probability 0.6 or lose Rs 1 with probability 0.4. You stop when you reach Rs 5 or go broke at Rs 0. What is the chance you reach Rs 5?Probability and brainteasersHardTwo SigmaNew York · 2023
Try it first
Before any algebra: roughly where does the answer sit?
Show the worked solution
135 over 211, about 64%. Let P(i) be the chance of reaching Rs 5 from Rs i. Each P(i) is 0.6 times P(i+1) plus 0.4 times P(i-1), with P(0) = 0 and P(5) = 1. The solution is P(i) = (1 - (2/3)^i) / (1 - (2/3)^5), and at i = 2 that is (5/9) divided by (211/243), or 135/211. A fair coin would give 40%.
How do you set the problem up without drawing every path?
Think of a cricket team chasing a small target with few wickets in hand. What matters is not the path of each ball but where the game stands now: runs needed and wickets left. The only thing that matters at any moment is how much money you hold, so write one unknown for each amount and one equation linking it to its neighbours. From Rs i you move to Rs i+1 with chance 0.6 and to Rs i-1 with chance 0.4, so P(i) = 0.6 P(i+1) + 0.4 P(i-1). The two ends are fixed: P(0) = 0 because you are broke, and P(5) = 1 because you have won.
The relationship
$P(i) = \frac{1-(q/p)^{i}}{1-(q/p)^{N}} \qquad P(2) = \frac{1-(2/3)^2}{1-(2/3)^5} = \frac{135}{211}$
p, q the chances of winning and losing one round, 0.6 and 0.4
N the target, Rs 5
i your starting capital, Rs 2
What it says in wordsThe ratio of losing to winning odds, raised to your capital and to the target, gives the chance of hitting the target first.
With a fair coin the chance of reaching Rs 5 rises in a straight line with starting capital, 40% from Rs 2. A 60/40 edge lifts every point well above that line, to 64.0% from Rs 2 and 92.4% from Rs 4.
How do you check 64% without trusting the formula?
Solve the chain by hand, starting from the bottom. Write P(1) = a. Then P(2) = (a - 0.4 x 0) / 0.6 = a/0.6 = 1.667a, and each later step follows the same rule until P(5) = 211/81 x a. Setting P(5) = 1 gives a = 81/211, or 38.4%, and P(2) = 135/211. Two routes landing on the same fraction is the check an interviewer wants to hear. It also tells you the shape: the edge matters most in the middle, where the fair line and the edge curve are furthest apart.
Say the desk version in one line. A small, repeatable edge does not make ruin impossible when you start thin: from Rs 2 you still go broke about 36 times in 100. More capital, not a bigger edge, is what pushes the ruin chance towards zero.
Where candidates lose it
The common loss is answering 40%, the fair coin result, because the candidate remembers that the chance of reaching the target is capital over target. That rule only holds when winning and losing are equally likely.
The second loss is trying to list paths. The game can run for any number of rounds, so path counting never ends. Name the state, write one equation per state, and use the two boundaries.
What the interviewer asks next
What is the chance of reaching Rs 5 if the win probability drops to 0.5?
If the target were Rs 1,000 instead of Rs 5, roughly what is your chance of ever going broke from Rs 2?
How many rounds do you expect the game to last from Rs 2?
Asked at Two Sigma, Research, New York, 2023 (Wall Street Oasis): Biased gamblers ruin problems; Markov Chain problems; sampling uniformly from triangle
027You can repeat a bet that wins 60% of the time and pays even money, as often as you like. What fraction of your capital should you stake each time to grow your money fastest, and what happens if you stake double that fraction?Expected value and decisionsHardHedge fund long/shortMulti-manager pod
Try it first
Stake double the growth-maximising fraction. What happens to your money over many bets?
Show the worked solution
Stake 20% of capital each time; at 40% your money slowly shrinks. The Kelly fraction for an even-money bet is the win chance minus the loss chance, 0.6 minus 0.4. At 20% the typical path grows about 2.0% a bet. At 40% each bet still has a positive expected gain, but growth is about -0.24% a bet, below zero, because big losses compound harder than big wins.
Why is the bet with the highest expected value not the one that grows fastest?
Imagine a shopkeeper who puts half the till into stock every morning. A good day lifts the till by half; a bad day cuts it in half. One of each leaves 1.5 x 0.5 = 0.75 of where she started, even though the good and bad days were the same size. Wealth compounds, so what matters over many bets is the average of the logarithm of each outcome, not the average outcome. Expected value per bet rises in a straight line with the stake; growth rises, peaks and then falls.
The relationship
$g(f) = 0.6\,\ln(1+f) + 0.4\,\ln(1-f) \qquad f^{*} = p - q = 0.2$
f the fraction of capital staked on each bet
g(f) growth of capital per bet on the typical path, in log terms
p - q the edge: win chance less loss chance
What it says in wordsGrowth per bet is the chance-weighted log of what each outcome does to your capital, and it peaks when you stake your edge.
Growth per bet peaks at 2.01% when 20% of capital is staked, falls to zero at about 38.9%, and turns negative at 40%, so overbetting a real edge can make you poorer over many bets.
What do the numbers look like at each stake?
Stake Expected gain per bet Growth per bet, typical path
10% +2.0% +1.50%
20% +4.0% +2.01%
30% +6.0% +1.47%
40% +8.0% -0.24%
50% +10.0% -3.40%
Expected gain keeps rising with the stake while growth peaks at 20% and turns negative near 40%.
Read the two columns against each other. Every row has a positive expected gain, yet the 40% and 50% rows lose money on the path you will actually live through. Half Kelly at 10% keeps about 75% of the maximum growth with far smaller swings, which is why many desks size below full Kelly. Say the limitation too: the formula assumes you know the 60% exactly. Real edges are estimates, and overestimating one pushes you towards the overbetting side of the curve.
Where candidates lose it
Candidates maximise expected value and conclude you should stake everything, because every bet is favourable. That answer goes broke on the first loss. The interviewer is testing whether you know that repeated bets compound, so the log of wealth is what you should maximise.
The second loss is saying 60%, the win probability, as the stake. The Kelly fraction for even money is the edge, 0.6 minus 0.4, not the win chance.
What the interviewer asks next
The bet now pays 2 to 1 with a 40% win chance. What is the Kelly fraction?
Why might a portfolio manager size positions at half Kelly?
How does position sizing on a stock idea resemble this bet, and where does the analogy break?
028Estimate the annual premium pool for two-wheeler insurance in India. Build it from the fleet on the road, the share of vehicles that stay insured once the upfront cover runs out, and the average premium for third-party and own-damage cover.Market sizing and estimationHardIndian brokerage researchSell-side equity research
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Which single assumption moves this estimate the most?
Show the worked solution
About Rs 19,000 crore a year on these assumptions. Take an assumed fleet of 25 crore two-wheelers, 1.8 crore in each of the last five years and 1.6 crore in each older year. All young vehicles are insured; after year five the insured share falls from 60% to 20%. That leaves 14.6 crore insured vehicles paying Rs 800 to Rs 1,500 a year, a pool near Rs 18,700 crore.
Where do you start, the vehicles or the policies?
Think of a gym. Counting everyone who ever signed up tells you little; the revenue comes from the members who still renew. Start from the fleet, but the number that decides the pool is how many vehicles are still insured, not how many are on the road. In India third-party cover is compulsory by law and new two-wheelers are sold with a multi-year third-party policy, so the young fleet is close to fully insured. Confirm the current rules before quoting them. Once that upfront cover runs out, many owners of older, cheaper bikes let it lapse.
Vehicle age Fleet, crore Share insured Insured, crore Premium, Rs a year Pool, Rs crore
1 to 5 years 9.0 100% 9.0 1,500 13,500
6 to 10 years 8.0 45% 3.6 1,000 3,600
11 to 15 years 8.0 25% 2.0 800 1,600
Total 25.0 58% 14.6 18,700
Every input here is an assumption for the estimate, not a reported figure. Premium per insured vehicle falls with age because the own-damage part is priced on the vehicle's value.
Every vehicle in its first five years is insured, but after year five the insured share falls from 60% to 20%, so only 14.6 crore of 25 crore vehicles pay a premium and the pool is about Rs 18,700 crore rather than Rs 27,900 crore.
How do you show the interviewer which assumption matters?
Run one sensitivity out loud. If ten more vehicles in every hundred older than five years renewed their cover, the pool would rise by about Rs 1,440 crore, roughly 8% of the total. That is the lever an insurer or a regulator can pull: enforcement of the third-party requirement at the roadside. The premium per vehicle matters less because it is set in narrow bands for third-party cover. The limitation to say: premiums, fleet and lapse rates here are assumptions for the method, and an analyst would replace each with a sourced number before writing it into a note.
Where candidates lose it
The costly mistake is multiplying the whole fleet by an average premium. That quietly assumes every old bike is insured and overstates the pool by about 49% on these numbers. The interviewer is waiting to see whether you ask how many of those vehicles actually carry cover.
The second loss is presenting assumed inputs as facts. Say each one as an assumption, give a round number, and move on; the structure is what is being marked.
What the interviewer asks next
How would the pool change if the upfront third-party period were shortened from five years to one?
Which part of the pool, third-party or own-damage, is more exposed to price competition between insurers?
How would you check your fleet assumption against registration data?

1/6	the chance of each first roll before you know anything
1/3	the chance of ending on 2 after an odd first roll
1/2	the chance of ending on 2 after a first roll of 4 or 6

b	your bid, Rs crore
b/100	the chance the owner accepts
b/2	the average value of a company whose owner accepts

FCF	free cash flow in year five, 100
g	terminal growth, 5%
WACC - g	the gap that sets the terminal multiple, 5% here

50	contribution: revenue less variable costs
10	EBIT before the change
5	pre-tax profit before the change

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Where do you start, the vehicles or the policies?

How do you show the interviewer which assumption matters?

Where candidates lose it

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