Investment Banking puzzles, solved step by step

Puzzles: 100
Traced to a firm: 36
Topics: 12
Hard: 29

Showing 1–10 of 29 · filtered from 100Clear filters

001You have two empty bowls, 50 white balls and 50 black balls. Split all 100 balls between the bowls any way you like. A bowl is then picked at random and one ball drawn from it. How do you maximise the chance of drawing white?ProbabilityHardDeutsche BankMumbai · 2024
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Before you work it: what is the best chance of white you can get?
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Put one white ball alone in bowl A and the other 99 balls in bowl B. Bowl A gives white every time. Bowl B gives white 49 times in 99. Each bowl is picked half the time, so the chance of white is 0.5 x 1 + 0.5 x 49/99, about 74.7%, against 50% for an even split.
Why does the number of balls in a bowl not matter?
Think of two queues at a ticket counter, one with a single person and one with ninety nine. If a coin decides which queue gets served, the lone person gets served half the time. The bowls work the same way. The bowl is chosen with even odds, so a bowl holding one ball carries exactly as much weight as a bowl holding ninety nine. That is the lever: you can make one half of the draw certain at the cost of a single white ball.
One white ball alone makes bowl A a certain white. Bowl B keeps 49 white and 50 black, so white comes up 49.5% of the time there, and the average across the two bowls is 74.7%, well above the 50% an even split gives.
How do you prove no other split beats it?
Bowl A can never beat 100%, and one ball is the cheapest way to get there. Every extra white ball you move into bowl A is a white ball taken out of bowl B, which lowers bowl B's odds without raising bowl A's. Adding black balls to bowl A only hurts it. So the best bowl A is one white ball, and bowl B's best case is whatever is left: 49 white out of 99.
The relationship
$P(\text{white}) = \tfrac{1}{2}\cdot\frac{1}{1} + \tfrac{1}{2}\cdot\frac{49}{99} \approx 0.747$
1/2 the chance each bowl is picked
1/1 the chance of white from bowl A, one white ball
49/99 the chance of white from bowl B, 49 white in 99 balls
What it says in wordsAverage the white odds of the two bowls, because each bowl is equally likely to be picked.
Say the 74.7% figure out loud, then say the general rule: with n white and n black balls, the answer is one half plus one half of (n minus 1) over (2n minus 1), which creeps towards 75% as n grows. Interviewers like hearing that you see the limit.
Where candidates lose it
Most candidates answer 50% and stop, because it feels as if a fixed pool of balls must give a fixed chance. The interviewer is testing whether you notice that the bowl is picked first, with even odds, whatever it holds.
The second trap is getting 74.7% and then being unable to say why it cannot be beaten. Have the one-line proof ready: a white ball moved into bowl A beyond the first one is wasted there and missed in bowl B.
What the interviewer asks next
What if the bowl is picked with probability proportional to how many balls it holds?
What if there are three bowls?
Now I pay you 100 if you draw white. What would you pay to play?
Asked at Deutsche Bank, Equity Capital Markets, Mumbai, 2024 (Wall Street Oasis): The most difficult question was the brain teaser. You have two empty bowls and 100 balls
004A company trades at 20x earnings and 10x EBITDA. Equity value is 200, interest expense is 20 at a 5% interest rate, and depreciation and amortisation is 20. What is the tax rate?Valuation riddlesHardEvercoreMenlo Park · 2025EvercoreSan Francisco · 2026
Try it first
Which two numbers do you need to find before the tax rate falls out?
Show the worked solution
The tax rate is 50%. P/E of 20 on equity of 200 gives net income of 10. Interest of 20 at 5% means debt of 400, so enterprise value is 600 and EBITDA is 60. Take off depreciation of 20 and interest of 20 to reach pre-tax profit of 20. Tax is 20 minus 10, which is 10, so the rate is 50%.
Where do you start when every number looks equally useful?
Start from what the answer is made of. A tax rate is tax divided by pre-tax profit, so the job is to find the two profit lines either side of the tax charge. Everything in the question is a road to one of those two numbers. Say that out loud first: it tells the interviewer you have a plan before you touch the arithmetic.
Net income is one step: equity value over the P/E, 200 over 20, is 10. Pre-tax profit takes longer. The enterprise valueThe value of the whole business to all its funders: equity value plus debt, less cash. multiple is on EBITDA, so you need EV, which needs debt, which the interest line hands you: 20 of interest at 5% means 400 of debt.
Equity of 200 at 20x earnings gives net income of 10; interest of 20 at 5% gives debt of 400, so enterprise value is 600 and EBITDA at 10x is 60. EBITDA of 60 less depreciation of 20 and interest of 20 leaves pre-tax profit of 20, and a tax charge of 10 on that is a 50% rate.
What assumption are you making, and should you say it?
You are assuming the company holds no cash, so enterprise value is simply equity plus debt. Say the assumption before you use it, because with any cash on the balance sheet EV falls, EBITDA falls, and the tax rate changes. A 50% rate is also high for most countries, which is worth one sentence: the interviewer built round numbers, not a real company.
Where candidates lose it
Candidates grab EBITDA first and then stall, because EV/EBITDA looks like the obvious lever but EV is not given. The debt is hiding inside the interest line, and people who do not think of interest divided by rate never find it.
The second loss is silent assumptions. Saying no cash out loud costs two seconds and turns a lucky answer into a reasoned one.
What the interviewer asks next
Now the company holds 100 of cash. What is the tax rate?
What does a 50% effective tax rate tell you about this business?
If the P/E rises to 25x with everything else fixed, what happens to the implied tax rate?
Asked at Evercore, Investment Banking, Menlo Park, 2025 (Wall Street Oasis): P/E of 20x, Ev/EBITDa of 10x, 20 million interest expense, 5% IR, D/A of 20 million, EQ of 200 million -> find tax rate
Asked at Evercore, Mergers and Acquisitions, San Francisco, 2026 (Wall Street Oasis): Given the following, calculate the Tax Rate
012You flip a fair coin until you get two heads in a row. What is the expected number of flips?Expected value and gamesHardBulge bracket IBConsulting style brainteasers
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Your first instinct: how many flips on average?
Show the worked solution
Six flips on average. Track where you stand: at the start, or one head up. From the start, one flip takes you to one head or leaves you at the start; from one head, a head finishes the game and a tail sends you back. Writing the expected flips from each state as an equation and solving gives 6 from the start and 4 from one head.
Why is the obvious answer of 4 wrong?
The guess of 4 comes from the one in four chance that two given flips are both heads. Think instead of climbing a slippery two-step ladder where any slip drops you to the ground, not one rung down. A tail after a head costs you the head you already had, so progress is lost and the wait is longer than the simple odds suggest. The clean way to handle lost progress is to give each position its own equation.
From the start a head moves you to one head and a tail leaves you at the start; from one head, a head finishes and a tail sends you back, so the two equations solve to 6 expected flips from the start and 4 from one head.
How do the equations work?
Call E the expected flips from the start and E(H) the expected flips once you hold one head. Every flip costs one. From the start, half the time you move to one head and half the time you are back where you began. From one head, half the time you finish and half the time a tail sends you to the start. Each equation reads: one flip, plus the average of the waits from wherever that flip leaves you. Substitute the second into the first and E = 1.5 + 0.75E, so E = 6 and E(H) = 4. This way of setting up the problem is called a Markov chainA process where what happens next depends only on the current state, not on how you got there..
The relationship
$E = 1 + \tfrac12 E_H + \tfrac12 E \qquad E_H = 1 + \tfrac12\cdot 0 + \tfrac12 E \quad\Rightarrow\quad E = 6,\; E_H = 4$
E expected flips from the start, with no head in hand
E_H expected flips when the last flip was a head
1 the flip you are about to make
What it says in wordsFrom each position, the expected wait is one flip plus the average wait from wherever that flip lands you.
How do you check 6 a second way?
Wait for the first head, which takes 2 flips on average. Flip once more: half the time it is a head and you are done; half the time it is a tail and you start from scratch. So E = 3 + E/2, which again gives 6. Two routes to the same number is the strongest answer you can give in the room. Simulated 200,000 times, the average comes out at 6.01. Then add the twist interviewers like: waiting for heads then tails takes only 4, because a failed attempt at it, a second head, still leaves you one head up.
Where candidates lose it
Most candidates answer 4, because the chance of two heads is a quarter, and stop. The interviewer is checking whether you notice that a tail after a head resets your progress.
The second loss is trying to write a single equation for the whole game and getting tangled. Name the states first, start and one head, and write one line for each; the algebra is then two lines long.
What the interviewer asks next
What is the expected number of flips to get heads followed by tails?
What about three heads in a row?
If the coin lands heads 60% of the time, what is the expected wait for two heads in a row?
014You have 25 horses and a track that races 5 at a time, with no stopwatch. What is the minimum number of races needed to find the three fastest?Logic and brainteasersHardConsulting style brainteasersSales and trading
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How many races?
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7 races. Race five heats of five, then race the five heat winners: the winner of race 6 is the fastest overall. Now strike out every horse that has three horses known to be faster. Only five can still be second or third: the second and third from the winner's heat, the first two from the runner-up's heat and the winner of the third-placed heat. Race them; the top two are second and third overall.
What does each race actually tell you?
Without a stopwatch, a race tells you the order only among the horses in it. Think of five classrooms each running their own sprint: you know the fastest in each room, but nothing about how one room's second best compares with another room's winner. Every horse must race at least once, so five heats are unavoidable, and they give five separate rankings that a sixth race between the heat winners stitches together. Name the heats A to E in the order their winners finished race 6, so A1 is the fastest horse of all.
Once race 6 orders the heat winners, every horse with three known horses ahead of it is ruled out, which leaves exactly five candidates for second and third, A2, A3, B1, B2 and C1, and they fill race 7.
Which horses can you strike out after race 6?
Any horse with three horses known to be faster cannot finish in the top three. Heats D and E go entirely, because D1 and E1 already finished behind A1, B1 and C1, and everything in those heats is slower still. In heat C only C1 survives: C2 trails C1, B1 and A1. In heat B, B1 and B2 survive, but B3 trails B1, B2 and A1. In heat A, A2 and A3 survive and A4 trails A1, A2 and A3. That leaves exactly five: A2, A3, B1, B2 and C1.
Race 7 puts those five on the track, and its first two finishers are second and third overall. A1 sits out, because it is already known to be the fastest. Could six races ever be enough? No: the five heats alone cannot name the fastest horse, and the race that does name it leaves five horses still unranked for second and third. That is the reasoning to say out loud, because it shows the 7 is a minimum rather than just a method that works.
Where candidates lose it
The common answer is 11 or more: race the heats, then keep racing groups of winners and runners-up until something falls out. It can reach the right horses, but it shows no elimination logic, which is the whole point of the question.
The other slip is stopping at 6 because the winners' race finds the champion. The question asks for three horses; draw the grid, strike out the impossible ones, and the five that remain fit one race.
What the interviewer asks next
How many races do you need to find only the fastest horse?
With 49 horses and a track that takes 7 at a time, how many races find the fastest three?
If the heats were drawn at random each time, would the minimum change?
016A merger promises Rs 100 crore of pre-tax cost synergies at full run-rate: half in year 1 and all of it from year 2. Achieving them costs Rs 150 crore once, in year 1. Tax is 25% and the market capitalises after-tax earnings at 8x. Roughly what are the synergies worth today?Deal mathsHardNomuraNew York · 2026
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Roughly what are the synergies worth?
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About Rs 450 crore. At full run-rate the synergies add Rs 75 crore of after-tax earnings, which the market capitalises at 8x: Rs 600 crore. Two things the headline ignores come off. Year 1 delivers only half, a shortfall of Rs 37.5 crore after tax, and the one-off cost of Rs 150 crore is Rs 112.5 crore after its tax saving.
Why can you not just multiply Rs 100 crore by 8?
A shop owner expects a new supplier to cut her costs by Rs 1 lakh a year. Before celebrating, she has to pay tax on the extra profit, wait for the new contract to start, and pay the one-off cost of switching. A synergy is worth the after-tax earnings it adds, capitalised, less everything it costs to achieve and every rupee that arrives late. The 8x multiple applies to after-tax earnings, so the pre-tax Rs 100 crore first becomes Rs 75 crore at 25% tax, and 75 x 8 is Rs 600 crore.
The pre-tax headline of Rs 800 crore overstates the synergies: after tax the run-rate is worth Rs 600 crore, and taking off the after-tax year-1 shortfall of Rs 37.5 crore and integration cost of Rs 112.5 crore leaves about Rs 450 crore.
What comes off the Rs 600 crore?
Two things. Phasing: the market value assumes the full run-rate, but year 1 delivers only half, so Rs 50 crore of pre-tax savings, Rs 37.5 crore after tax, never arrive. Cost to achieve: Rs 150 crore of severance, systems work and site closures, assumed tax deductible, so it costs Rs 112.5 crore after tax. Net of phasing and the cost to achieve, the synergies are worth about Rs 450 crore, a quarter less than the after-tax figure and 44% less than the pre-tax headline.
The relationship
$V = \underbrace{100 \times 0.75 \times 8}_{600} - \underbrace{50 \times 0.75}_{37.5} - \underbrace{150 \times 0.75}_{112.5} = 450$
100 x 0.75 x 8 run-rate synergies after 25% tax, capitalised at 8x
50 x 0.75 the half of year 1's savings that does not arrive, after tax
150 x 0.75 the one-off cost to achieve, after its tax saving
What it says in wordsCapitalise the after-tax run-rate, then subtract the after-tax value of the savings that arrive late and the cost of getting them.
What does this rough answer leave out?
Three refinements, each worth naming. The year-1 items fall a year from now, so discounting them would trim the deductions slightly. The 8x assumes the market gives synergies the same multiple as the core earnings, while in practice cost synergiesSavings from combining two businesses, such as closing duplicate offices or merging purchasing. earn more credit than revenue synergies and both are discounted for execution risk. The deductibility of integration costs is an assumption to confirm for the jurisdiction. The answer to give is the method and Rs 450 crore, then one line on which assumption you would test first.
The number matters because it caps what a buyer can sensibly pay away. A premium above roughly Rs 450 crore hands the target's shareholders more than the whole value the deal creates, before any risk that the savings slip.
Where candidates lose it
The common slip is Rs 800 crore: a pre-tax saving multiplied by a multiple meant for after-tax earnings. Interviewers hear it constantly, because announcements usually quote synergies before tax.
The second slip is stopping at Rs 600 crore. The question gave you a phasing schedule and a cost to achieve for a reason; both come off, and the cost comes off after tax.
What the interviewer asks next
If the buyer pays a Rs 500 crore premium for the target, is the deal worth doing on these synergies alone?
How would you value Rs 100 crore of revenue synergies at a 20% EBITDA margin?
Why do markets often give cost synergies more credit than revenue synergies?
Asked at Nomura, Investment Banking, New York, 2026 (Wall Street Oasis): Asked about what synergies were in a merger and in-depth questions about how my experiences
018Three cards sit in a hat: one red on both sides, one white on both sides, one red on one side and white on the other. You draw one and see a red face. What is the probability the other side is red?ProbabilityHardBulge bracket IBSales and trading
Try it first
Pick your answer.
Show the worked solution
Two thirds. You are not choosing among cards; you are looking at a face. There are three red faces you could be seeing, all equally likely: two belong to the red-red card and one to the red-white card. In two of the three cases the hidden side is red. The tempting answer of one half treats the two red-capable cards as equally likely and ignores that the red-red card shows red twice as often.
Why is one half so tempting, and where does it go wrong?
Imagine two families at a party: one with two daughters, one with a daughter and a son. If you meet one of the girls at random, she is twice as likely to come from the two-daughter family, because that family brought two girls. The red-red card is twice as likely to be the one showing red, because it has two red faces to show. Treating the two remaining cards as equally likely throws that information away, which is exactly what the question is built to catch.
Of the three red faces you might be looking at, two belong to the card that is red on both sides and one to the red-white card, so the hidden side is red two times in three, not one in two.
How do you prove it with Bayes' rule?
Write it as a conditional probability using Bayes' ruleA formula for updating a probability after seeing evidence: the chance of the evidence given the cause, times the prior chance of the cause, divided by the overall chance of the evidence.. The chance of drawing the red-red card is one third, and if you did, you are certain to see red. The chance of seeing red at all is the share of red faces, three of six, one half. So the chance the card is red-red, given a red face, is one third times one, divided by one half: two thirds.
The relationship
$P(RR \mid \text{red seen}) = \frac{P(\text{red seen} \mid RR)\,P(RR)}{P(\text{red seen})} = \frac{1 \times \tfrac13}{\tfrac12} = \tfrac23$
P(RR) the chance of drawing the red-red card, one third
P(red seen | RR) the chance of seeing red if you hold that card, which is 1
P(red seen) the overall chance of seeing red, 3 red faces out of 6
What it says in wordsWeight each card by how likely it is to produce what you saw, then divide by how likely that sight was overall.
How do you check it without any formula?
Imagine drawing 600 times, each card about 200 times. The red-red card shows red all 200 times, the red-white card shows red about 100 times, and the white card never does. Of the 300 red sightings, 200 have red on the back. Counting outcomes in a large imagined sample is a reliable check whenever a conditional probability feels slippery. The puzzle is a version of Bertrand's box paradox, and the same reasoning sits under the Monty Hall problem.
Where candidates lose it
One half is the answer most candidates give, and they defend it by saying the white card is out, so two cards remain. That is true about cards and irrelevant to faces; the interviewer will ask how many red faces you could be looking at.
The fix is to count the smallest equally likely outcomes. Here that means faces: six of them, three red, and two of those three have a red reverse.
What the interviewer asks next
What if the hat held two red-red cards and one red-white card?
You see a white face. What is the probability the other side is white?
How does this connect to the Monty Hall problem?
019Explain the Black-Scholes intuition by pricing, in your head, a one-year at-the-money call on a Rs 100 stock with 20% volatility and near-zero interest rates.Rates, risk and optionsHardGoldman SachsZurich · 2025
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Roughly what is the call worth?
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About Rs 8. With near-zero rates, an at-the-money call is worth roughly 0.4 times the price times the volatility times the square root of time: 0.4 x 100 x 0.20 x 1 = 8.0. The full Black-Scholes formula gives Rs 7.97. The intuition: the option collects the upside of a bell-shaped spread of outcomes and nothing on the downside, and the average of that upside half is about 0.4 of one standard deviation.
What is Black-Scholes actually doing?
Picture a ticket that pays you whatever a stock gains over the next year and nothing if it falls. You would pay something for it, even though an at-the-moneyAn option whose strike price equals the current stock price, so exercising it today would pay nothing. ticket is worth nothing if cashed in today. Black-Scholes prices that ticket as the average payoff across all the ways the stock could move, with the spread of those moves set by volatility and time. Its inputs are the stock price, strike, time, volatility and interest rate, and with zero rates the formula reduces to S times N(d1) minus K times N(d2), where the two N terms come from the bell curve.
At a stock price of 100 the call pays nothing if exercised today, yet with a year to run at 20% volatility it is worth Rs 7.97, almost exactly the rule of thumb of 0.4 times the Rs 20 one-year standard deviation.
Where does the 0.4 come from?
Over a year at 20% volatility, the stock's likely outcomes form a bell curve around Rs 100 with a standard deviation of about Rs 20. The call keeps the right half and treats the left half as zero. The average of the positive half of a bell curve, counting the negative half as zero, is the standard deviation divided by the square root of 2 pi, about 0.4 of it. So the call is worth about 0.4 x Rs 20, which is Rs 8. Halve the time to six months and the spread shrinks by the square root of 2, so the option is worth about Rs 5.6, not Rs 4.
The relationship
$C \approx \frac{1}{\sqrt{2\pi}}\,S\,\sigma\sqrt{T} = 0.399 \times 100 \times 0.20 \times 1 = 7.98 \qquad C_{BS} = 100\,[N(0.1) - N(-0.1)] = 7.97$
S the stock price, Rs 100, equal to the strike
sigma volatility, 20% a year
T time to expiry in years, here 1
N(.) the cumulative bell-curve probability used in the full formula
What it says in wordsAn at-the-money option is worth about 0.4 of one standard deviation of the stock's price over the life of the option.
What does the shortcut leave out?
It holds for at-the-money options with modest volatility and low rates. Away from the strike, the payoff is no longer the clean right half of the curve, and with meaningful rates the forward price drifts above the spot, which raises a call's value. The rule is a sanity check on a quoted price, not a replacement for the formula. Black-Scholes itself assumes constant volatility and smooth price moves, which is why traders quote options in implied volatility and adjust for the skew they see in the market. Doubling volatility to 40% roughly doubles this call's value, to about Rs 15.9.
Where candidates lose it
The common slip is to say an at-the-money option is worth nothing because exercising it today pays zero. That is intrinsic value; the whole price of this option is time value, the chance of a favourable move before expiry.
The second slip is reciting the formula without any feel for the number. Give the 0.4 rule, reach Rs 8, and then say the exact formula agrees; that is what explaining Black-Scholes sounds like in an interview.
What the interviewer asks next
What is the matching at-the-money put worth here, and why?
If volatility doubles to 40%, roughly what happens to the call's value?
What is the delta of this call, roughly, and what does it tell you?
Asked at Goldman Sachs, 2026 | EMEA | Zurich | Wealth Management | Summer Analyst Interview, Zurich, 2025 (Wall Street Oasis): Moreover, I was asked to explain Black and Scholes.
020Company A trades at 12x earnings and 9x EV/EBITDA. Its closest peer B trades at 15x earnings and 8x EV/EBITDA. Build one set of numbers that explains how A is cheaper on P/E yet dearer on EV/EBITDA.Valuation riddlesHardBarclaysNew York · 2026
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Which difference does most of the work in a good answer?
Show the worked solution
Give B heavy depreciation and more debt, so much less of its EBITDA reaches shareholders. Both earn EBITDA of Rs 100 crore. A has D&A of 2 and interest of 18, keeps net income of 60, and at equity value 720 trades on 12x; its EV of 900 is 9x EBITDA. B has D&A of 25 and interest of 35, keeps only 30, and at equity value 450 trades on 15x, though its EV of 800 is 8x.
How can one company be cheaper on one multiple and dearer on another?
Two shops each take in Rs 1 lakh a month before equipment and loan costs. One works from a simple counter with a small loan; the other leases costly machines it must keep replacing and carries a large loan. Priced on takings before those costs, they look alike; priced on what the owner keeps, they do not. EV/EBITDA values the business before depreciation and before financing, while P/E values what is left for shareholders after both, so a company with heavy D&A or heavy interest can look cheap on one and dear on the other.
Both companies earn EBITDA of Rs 100 crore, but B's heavier D&A and interest leave it net income of Rs 30 crore against A's Rs 60 crore, so B looks cheaper on EV/EBITDA at 8.0x against 9.0x and dearer on P/E at 15.0x against 12.0x.
Which line explains how much of the gap?
Work down from EBITDA. B carries Rs 23 crore more D&A and Rs 17 crore more interest than A, which after 25% tax is Rs 17.25 crore and Rs 12.75 crore of net income: together exactly the Rs 30 crore gap between A's 60 and B's 30. Here depreciation does more of the work than debt, which you can show with EV/EBIT, a multiple that charges D&A but ignores financing: A is 9.2x and B 10.7x, so A is cheaper there too.
Rs crore A B
Enterprise value 900 800
Net debt, at 10% interest 180 350
Equity value 720 450
EBITDA 100 100
D&A 2 25
Interest 18 35
Net income, 25% tax 60 30
EV / EBITDA 9.0x 8.0x
EV / EBIT 9.2x 10.7x
P/E 12.0x 15.0x
Two invented companies built to match the multiples in the question.
So which company is actually cheaper?
It depends on whether B's D&A is a real cost. If it reflects equipment that must be replaced, which capital expenditureCash spent on buying or replacing long-term assets such as machines and buildings. running close to depreciation would confirm, then EBITDA flatters B and A is better value on every measure that counts the cost of staying in business. If B's D&A is mostly amortisation of acquired intangibles with no cash replacement, EBITDA is the fairer lens and B may be the cheap one. The answer the interviewer wants is the diagnosis: check capex against D&A and check leverage before trusting either multiple.
Where candidates lose it
The common slip is to say the market simply disagrees or that one multiple must be wrong. Both are correct; they measure different things, and the job is to name what sits between EBITDA and net income.
The second slip is naming only debt. Leverage can push P/E either way depending on the cost of debt, while heavy D&A reliably depresses earnings relative to EBITDA. Give both, with numbers, and say which one does more of the work.
What the interviewer asks next
How would you check whether B's D&A is a real economic cost?
If B refinanced its debt at 5% instead of 10%, what would its P/E become?
Which multiple would you use to compare a capital-heavy telecoms operator with a software company, and why?
Asked at Barclays, Investment Banking, New York, 2026 (Wall Street Oasis): A company is trading at a lower P/E but a higher EV/EBITDA than peers
026A game doubles your stake on heads and halves it on tails, so every round has positive expected value. After ten rounds on a Rs 1 lakh stake, what is the expected wealth and what is the most likely outcome?Expected value and gamesHardBulge bracket IBConsulting style brainteasers
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After ten rounds, what is the most likely amount in hand?
Show the worked solution
Expected wealth is Rs 9.31 lakh, but the most likely outcome is Rs 1 lakh, exactly where you started. Each round multiplies the stake by 1.25 on average, and 1.25 to the tenth is 9.31. Yet the wealth you actually hold is 2 to the power of heads minus tails, and the likeliest split is five and five, which multiplies to one. The median player goes nowhere; the mean is carried by a handful of lucky runs.
Why is the average so far from the typical result?
Ten friends each put Rs 1 lakh into this game. Most finish near where they began, a few lose most of their money, and one, with a long run of heads, finishes with hundreds of lakh. Add it all up and divide by ten and the average looks wonderful; ask the friend in the middle how it went and the answer is that nothing happened. Gains and losses compound, and a doubling followed by a halving lands exactly back where you started, so the mean grows 25% a round while the median stays flat. The arithmetic average of 2 and 0.5 is 1.25; their geometric average, the square root of 2 times 0.5, is exactly 1, and compounding follows the geometric one.
The mean rises 25% a round to Rs 9.31 lakh after ten rounds while the median stays at Rs 1 lakh, because the likeliest outcome, five heads and five tails, multiplies to one and the games with eight or more heads, 5.5% of the total, supply 68% of the mean.
How do you get both numbers in two lines?
For the mean, use the fact that the expectation of a product of independent rounds is the product of the expectations: 1.25 to the tenth, 9.31. For the typical outcome, count heads. After ten flips the wealth is Rs 1 lakh times 2 to the power of heads minus tails, which is 2 to the power of 2H minus 10. Five heads is the single likeliest count, 24.6% of games, and it gives a multiplier of exactly one; it is also the median, because the outcomes sit symmetrically on either side of it. 37.7% of players finish below Rs 1 lakh and the same share above, and you can say all of that without touching the mean.
The relationship
$E[W_{10}] = \left(\tfrac12\cdot 2 + \tfrac12\cdot\tfrac12\right)^{10} = 1.25^{10} = 9.31 \qquad W_{10} = 2^{\,H-T} = 2^{\,2H-10}$
1.25 the expected multiplier of a single round, in lakh per lakh staked
H, T the number of heads and tails in the ten flips
2H - 10 heads minus tails, the net number of doublings
What it says in wordsThe mean compounds the average multiplier; the wealth you hold compounds the count of heads, and the likeliest count leaves you where you started.
Where does the Rs 9.31 lakh come from, then?
From the tail. A run of ten heads happens once in 1,024 games and turns Rs 1 lakh into Rs 1,024 lakh, which on its own adds a full Rs 1 lakh to the mean. The three best outcomes, eight or more heads, occur in 5.5% of games and contribute Rs 6.31 lakh of the Rs 9.31 lakh average, about 68% of it. Nothing is wrong with the expected value; it is the wrong statistic for a question about what will probably happen to you. The right one is the geometric meanThe growth rate that compounding actually delivers: the nth root of the product of n multipliers. It is never above the arithmetic mean. return, which here is zero.
The lesson a desk draws is about sizing. Bet only half your stake each round and the multipliers become 1.5 and 0.75, whose geometric mean is 1.0607: the typical player now grows about 6.1% a round and finishes ten rounds near Rs 1.80 lakh, even though the expected value per round has fallen from 1.25 to 1.125. Half is the fraction that maximises the typical growth rate in this game. The same arithmetic sits under volatility drag in fund returns: a strategy that gains 50% and then loses a third has a flattering average and nothing to show for it.
Where candidates lose it
Most candidates say the expected value and stop, or say the game must be good because every round is positive on average. The interviewer is waiting for you to notice that a doubling and a halving cancel exactly, so the typical outcome is no change.
The second loss is muddling the median with the mean when pressed. Say the mode and the median are both Rs 1 lakh, give the 24.6% chance of the exact five-five split, and explain that the mean lives in the tail.
What the interviewer asks next
What fraction of your stake should you risk each round to maximise your typical long-run growth, and why?
What is the probability of finishing with more than Rs 1 lakh after ten rounds?
A fund gains 50% one year and loses a third the next. What are its average and its compound returns?
028A company's tax rate rises from 25% to 30%. Name three ways this moves a DCF and put numbers on each for a business with EBIT of Rs 200 crore, funded 30% by debt at 8% before tax and 70% by equity at 12%.DCF and cost of capitalHardGoldman SachsSan Francisco · 2025
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Which way does the valuation move overall?
Show the worked solution
The value falls, by about 4.8% here, because the cut to cash flow outweighs the cheaper debt. One: unlevered free cash flow falls, as NOPAT drops from Rs 150 crore to Rs 140 crore. Two: the after-tax cost of debt falls from 6.0% to 5.6%, taking the WACC from 10.20% to 10.08%. Three: the terminal value, which carries both effects, falls from about Rs 2,516 crore to Rs 2,395 crore at 4% growth.
Why does a tax rate touch both the top and the bottom of the fraction?
A landlord pays more tax on her rent, so the rent she keeps falls; but the interest on her mortgage is deductible, so the loan costs her a little less after tax. Two opposite effects from one change. A DCF is the same fraction: cash flow on top, discount rate below. A higher tax rate shrinks the cash flows directly and shrinks the WACC indirectly through the interest tax shield, and the first effect is almost always larger because it hits every rupee of operating profit while the second touches only the debt slice. Interviewers ask for three ways so that you show both sides and then say which wins.
The tax rise cuts NOPAT from Rs 150 crore to Rs 140 crore, lowers the WACC from 10.20% to 10.08% through the cheaper after-tax debt, and takes the terminal value from Rs 2,516 crore to Rs 2,395 crore, because the cash flow effect of Rs -168 crore outweighs the WACC relief of Rs 46 crore.
How big is each effect on these numbers?
Cash flow. NOPAT is EBIT times one minus the tax rate: Rs 200 crore x 0.75 = Rs 150 crore before and Rs 200 crore x 0.70 = Rs 140 crore after, a fall of Rs 10 crore, or 6.7%, in every forecast year, treating depreciation, capex and working capital as cancelling out. Discount rate. The after-tax cost of debt is 8% x (1 - t): 6.0% before and 5.6% after. Weighted at 30% debt and 70% equity at 12%, the WACC moves from 10.20% to 10.08%, down 12 basis points. Terminal value. At 4% growth, a Gordon growthA terminal value that treats the final cash flow as growing at a constant rate forever: the next cash flow divided by the discount rate less the growth rate. terminal value is Rs 2,516 crore before and Rs 2,395 crore after. Separating the two, the cash flow cut alone takes Rs 168 crore off the terminal value and the lower WACC alone adds Rs 46 crore back, so the net is Rs 121 crore, 4.8% lower.
Line Tax 25% Tax 30% Change
NOPAT, Rs crore 150 140 -6.7%
After-tax cost of debt 6.0% 5.6% -40 bp
WACC 10.20% 10.08% -12 bp
Terminal value at 4% growth, Rs crore 2,516 2,395 -4.8%
Illustrative inputs from the question: EBIT Rs 200 crore, 30% debt at 8%, 70% equity at 12%, long-run growth 4%.
The relationship
$WACC = 0.30 \times 8\%\,(1-t) + 0.70 \times 12\% \qquad TV = \frac{EBIT\,(1-t)\,(1+g)}{WACC - g}$
(1 - t) the share of each rupee left after tax, 0.75 falling to 0.70
0.30, 0.70 the debt and equity weights
g the long-run growth rate, 4% here
What it says in wordsThe tax rate enters the cash flow once and the discount rate once, and the cash flow effect is the larger of the two.
Is there a subtler channel the interviewer might be after?
Yes: the cost of equity. When you relever a beta, the formula carries a (1 - t) term, so a higher tax rate lowers the levered beta a little and the cost of equity with it. With debt to equity of 3 to 7 and a levered beta of 1.0 at 25%, the unlevered beta is 0.757, and relevering at 30% gives 0.984; with an assumed 6% risk-free rate and 6% equity risk premium, the cost of equity slips from 12% to 11.90% and the WACC to 10.01%. Even with that help the value is still about 3.7% lower, so the ranking of the effects does not change. Say the limitation too: the WACC relief assumes the company actually pays tax and keeps its interest deductible, and interest deductions are capped in many jurisdictions, a rule to confirm for the country in question.
Where candidates lose it
The common loss is naming only the cash flow effect, or naming the WACC effect and then concluding that value rises. Three ways means cash flow, the cost of debt inside the WACC, and the terminal value or cost of equity that carries them, and the net is down.
The second loss is putting no numbers on it. The question hands you EBIT, weights and rates so that you can say Rs 150 to Rs 140 crore and 10.20% to 10.08% out loud; a candidate who stays qualitative has not answered.
What the interviewer asks next
If the company had no debt at all, which of the three effects survive?
How would a large tax-loss carryforward change the answer?
Why might a biotech DCF, with years of losses ahead, be almost indifferent to this change?
Asked at Goldman Sachs, Investment Banking, San Francisco, 2025 (Wall Street Oasis): Three ways in which the tax rate affects a DCF?

1/2	the chance each bowl is picked
1/1	the chance of white from bowl A, one white ball
49/99	the chance of white from bowl B, 49 white in 99 balls

E	expected flips from the start, with no head in hand
E_H	expected flips when the last flip was a head
1	the flip you are about to make

100 x 0.75 x 8	run-rate synergies after 25% tax, capitalised at 8x
50 x 0.75	the half of year 1's savings that does not arrive, after tax
150 x 0.75	the one-off cost to achieve, after its tax saving

P(RR)	the chance of drawing the red-red card, one third
P(red seen \| RR)	the chance of seeing red if you hold that card, which is 1
P(red seen)	the overall chance of seeing red, 3 red faces out of 6

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Why does the number of balls in a bowl not matter?

How do you prove no other split beats it?

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Where do you start when every number looks equally useful?

What assumption are you making, and should you say it?

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Why is the obvious answer of 4 wrong?

How do the equations work?

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Which horses can you strike out after race 6?

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Why can you not just multiply Rs 100 crore by 8?

What comes off the Rs 600 crore?

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How do you prove it with Bayes' rule?

How do you check it without any formula?

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What is Black-Scholes actually doing?

Where does the 0.4 come from?

What does the shortcut leave out?

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How can one company be cheaper on one multiple and dearer on another?

Which line explains how much of the gap?

So which company is actually cheaper?

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Why is the average so far from the typical result?

How do you get both numbers in two lines?

Where does the Rs 9.31 lakh come from, then?

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Why does a tax rate touch both the top and the bottom of the fraction?

How big is each effect on these numbers?

Is there a subtler channel the interviewer might be after?

Where candidates lose it

What the interviewer asks next