Investment Banking puzzles, solved step by step

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

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042A company has revenue of Rs 730 crore and collects from customers in 45 days. Next year revenue grows 20% and collection slows to 60 days. How much cash does the receivables line absorb?Accounting riddlesCoreBulge bracket IBMiddle market IB
Try it first
How much cash does the receivables line absorb next year?
Show the worked solution
About Rs 54 crore. Receivables today are Rs 730 crore times 45 over 365, which is Rs 90 crore. Next year revenue is Rs 876 crore and collection takes 60 days, so receivables are Rs 144 crore. The increase of Rs 54 crore is cash the business has earned but not collected. Rs 18 crore of it comes from growth and Rs 36 crore from slower collection, twice as much.
Why does growth eat cash at all?
Think of a tailor who lets regular customers pay a month and a half after collecting their clothes. If the business grows, more money is out with customers at any moment, even though nobody is paying late. Receivables are revenue earned but not yet collected, so when revenue grows at the same collection period the balance grows with it, and the increase is cash the business has to fund. Here 20% more revenue at 45 days lifts receivables from Rs 90 crore to Rs 108 crore.
Receivables rise from Rs 90 crore to Rs 144 crore: Rs 18 crore of the increase comes from 20% growth at the old 45 days and Rs 36 crore from collecting in 60 days instead, so slower collection absorbs twice as much cash as growth.
How do you split the Rs 54 crore?
Take the two changes one at a time. Growth first: Rs 876 crore at the old 45 days is Rs 108 crore, so growth absorbs Rs 18 crore. Then slower collection: going from 45 to 60 days on Rs 876 crore adds 15 days of sales, Rs 36 crore. Fifteen extra days of collection on the larger business costs twice as much cash as 20% growth. The line to watch is days sales outstandingReceivables divided by revenue, times 365: the average number of days customers take to pay..
The relationship
$\Delta AR = \left(876 \times \tfrac{45}{365} - 90\right) + 876 \times \tfrac{60-45}{365} = 18 + 36 = 54$
876 x 45/365 next year's receivables if collection had not slowed
90 today's receivables, 730 x 45/365
876 x 15/365 the extra 15 days of sales tied up by slower collection
What it says in wordsThe change in receivables is the growth effect at the old collection period plus the slower collection on the new revenue.
Does the order of the split matter?
A little. Slower collection first, on today's revenue, costs Rs 30 crore, and growth then adds Rs 24 crore; the Rs 6 crore difference is growth and slower collection compounding each other. The total is Rs 54 crore whichever way you split it, and slower collection is the bigger cause either way. Say which order you used. Then say what it means: a business growing fast while collecting slowly can report rising profit and falling cash at the same time, which is the pattern a lender looks for first.
Where candidates lose it
Candidates answer Rs 144 crore, the new balance, rather than Rs 54 crore, the change. Cash is absorbed by the increase in receivables, not by their level.
The second loss is blaming it all on growth. Splitting the change shows that slower collection costs twice as much, and that is the insight the question is built to draw out.
What the interviewer asks next
Collection improves to 30 days instead. How much cash does the receivables line release?
Payables also stretch from 30 to 45 days of costs. How does that change the answer?
Why might a company's collection period lengthen as it grows?
051Estimate the number of two-wheelers sold in India in a year.Estimation and market sizingCoreBulge bracket IBConsulting style brainteasers
Try it first
Pick a ballpark before you build it.
Show the worked solution
About 2 crore a year. India has about 140 crore people in roughly 30 crore households. If about half of households own a two-wheeler, 15 crore are in use. A vehicle lasts about 10 years, so 1.5 crore are replaced each year, and a fleet growing 3% adds about 45 lakh more, close to 2 crore. A top-down check, 18 crore vehicles on the road over a 9-year life, lands at 2.0 crore too. The published industry figure is the thing to confirm, not to recall.
Where do you start, and what do you refuse to guess?
Think of guessing how many birthday cakes a town buys in a year. Nobody knows the number, but everybody can picture the pieces: people, birthdays per person, the share with a cake. Market sizing works from a number you can picture to the number you cannot, and the only figure taken on trust is the population. About 140 crore people at 4.5 per household is 31 crore households, call it 30. The one real judgement is the ownership rate, the penetrationThe share of possible buyers who already own the product. Half of households here.: higher in towns, lower in villages, so about half overall.
Working up from 30 crore households, half owning a two-wheeler that lasts 10 years, gives 1.95 crore sales a year once 3% fleet growth is added, and working down from 18 crore vehicles on the road over a 9-year life gives 2.0 crore, so the two routes agree within 4%.
How does the bottom-up build go?
Thirty crore households at 50% is 15 crore two-wheelers in use. Sales are what it takes to keep that fleet going plus what it takes to grow it. Yearly sales of a durable are the fleet divided by its life, plus the fleet's growth, so the life assumption matters as much as the ownership rate. At a 10-year life, 1.5 crore are replaced each year; a fleet growing 3% a year needs another 45 lakh; together about 1.95 crore. Say each assumption as you use it, so the interviewer can push on the one they disagree with.
The relationship
$S = \frac{H \times p}{L} + H \times p \times g = \frac{30 \times 0.5}{10} + 15 \times 0.03 \approx 1.95\text{ crore}$
H households, about 30 crore
p the share of households with a two-wheeler, 50%
L the vehicle's life, 10 years
g growth in the fleet, 3% a year
What it says in wordsSales replace the vehicles that wear out and add the ones that a growing fleet needs.
Why build a second route, and what if it disagrees?
Come at it from the fleet instead. About 13 two-wheelers per 100 people is 18 crore on the road, and dividing by a 9-year average life gives 2.0 crore a year, within 4% of the first route. A demand estimate is credible only once an independent route lands near it; agreement does not prove either number, but disagreement tells you which assumption to revisit. Stress the two judgements: 40% ownership and a 12-year life give 1.4 crore, 60% and 8 years give 2.8 crore. So the honest answer is a range of about 1.5 to 3 crore with 2 crore most likely, and the published sales figure from the industry body is the check to look up afterwards.
Where candidates lose it
Candidates reach for a remembered figure, right or wrong, and have nothing to say when asked how they got it. The interviewer is marking the structure and the stated assumptions, not the number.
The second loss is forgetting replacement. A fleet of 15 crore does not mean 15 crore sales; sales are the fleet divided by its life plus growth, which is why the life assumption deserves its own sentence.
What the interviewer asks next
How would the estimate change if you were asked for cars instead?
What share of those sales would you expect to be first-time buyers rather than replacements?
How would you estimate the number of two-wheeler dealerships in India?
053You roll a fair die and are paid its face value in lakh rupees. After seeing the first roll you may roll once more and take the second result instead, whatever it turns out to be. What is the game worth, and what is your stopping rule?Expected value and gamesCoreBulge bracket IBConsulting style brainteasers
Try it first
On which first rolls do you roll again?
Show the worked solution
The game is worth Rs 4.25 lakh, and you roll again on a 1, 2 or 3. A single roll is worth 3.5 lakh on average, so the chance to roll again is worth exactly that. Keep any first roll above 3.5, which means 4, 5 or 6. Half the time you keep, averaging 5 lakh; half the time you roll again for 3.5. Half of 5 plus half of 3.5 is 4.25 lakh. The option to re-roll adds 0.75 lakh over a single roll.
Why is the rule to compare against 3.5?
Think of boarding a train with one visible seat by the door and the rest of the carriage out of sight. You walk on only if the seat you can see is worse than the average seat down the carriage, because once you walk you take whatever is there. Work backwards: the last decision has no choices left, so its value is the plain average, 3.5 lakh, and any earlier roll is kept only if it beats that. A 4, 5 or 6 beats 3.5; a 1, 2 or 3 does not; no face equals it, so the rule has no ties. The number you are comparing with is the expected valueThe average outcome if the same gamble were repeated many times: each result weighted by its probability. of rolling again.
A first roll of 1, 2 or 3 falls below the 3.5 that a second roll is worth, so you roll again, while a 4, 5 or 6 is kept, and the kept faces contribute 2.5 and the re-rolls 1.75 for a game worth 4.25 lakh.
How does the 4.25 come together?
Three faces out of six are kept and contribute (4 + 5 + 6) over 6, which is 2.5 lakh; three are re-rolled and contribute three sixths of 3.5, which is 1.75; the total is 4.25 lakh. Check the rule by moving it. Keep the 3 as well: (3 + 4 + 5 + 6) over 6 plus two sixths of 3.5 is 4.17. Re-roll the 4 too: (5 + 6) over 6 plus four sixths of 3.5 is 4.17. Both sit below 4.25, about Rs 8,333 a game, so 3.5 is the only bar that gives the full value.
The relationship
$V = \frac{4 + 5 + 6}{6} + \frac{3}{6} \times 3.5 = 2.5 + 1.75 = 4.25$
(4 + 5 + 6)/6 the kept faces, each one in six
3/6 the chance the first roll is a 1, 2 or 3 and you roll again
3.5 what the second roll is worth on average
What it says in wordsAdd what you keep, weighted by its chance, to what you get from rolling again, weighted by its chance.
What happens with a third roll, and why do interviewers ask?
Allow two re-rolls. The game from the second roll onward is now worth 4.25, so on the first roll you keep only a 5 or 6: (5 + 6) over 6 plus four sixths of 4.25 is 4.67 lakh. Every extra chance to try again raises both the value of the game and the bar for stopping, and that is the whole logic of an option: the right to swap a known result for an uncertain one is worth paying for only when the known result is poor. The same reasoning prices walking away from a signed deal or waiting for a better bid. The limit: this assumes you care only about the average. Someone who badly needs 3 lakh might keep a 3, and that is a preference, not an error.
Where candidates lose it
The common miss is a rule based on feel, such as re-rolling anything below 5 because 5 and 6 feel like wins. That throws away a 4 that beats the 3.5 you get by rolling again, and costs about Rs 8,333 of value per game.
The second loss is valuing the game at 3.5 lakh because it is still just a die. The option to re-roll is worth 0.75 lakh here, and pricing an option at zero is the error the interviewer is listening for.
What the interviewer asks next
You may re-roll twice. What is the game worth and what is the rule on the first roll?
The re-roll costs Rs 50,000. Do you still take it, and when?
How does this connect to the decision to walk away from a signed deal?
064Estimate the annual revenue of a single petrol pump on a busy national highway.Estimation and market sizingCoreBulge bracket IBConsulting style brainteasers
Try it first
Before building it: which assumption deserves the most of your time?
Show the worked solution
About Rs 82 crore a year, built as vehicles, stop rate, litres and price. Say 15,000 vehicles pass a day and 5% stop, which is 750 fills. At 30 litres a fill that is 22,500.0 litres, and at Rs 100 a litre about Rs 22.5 lakh a day. Times 365 gives roughly Rs 82 crore. The stop rate is the shakiest input: 3% gives Rs 49 crore and 8% gives Rs 131 crore.
How do you turn a vague question into a chain you can multiply?
Think of a chai stall outside a railway station. You would not guess its takings in one go; you would ask how many people walk past, how many stop, what they buy and what it costs. A petrol pump is the same chain. Revenue is vehicles passing, times the share that stop, times litres per fill, times the price per litre, and the whole job is to say a defensible number for each link out loud. A busy national highway might carry 15,000 vehicles a day past one spot. If one in twenty pulls in, that is 750 fills. A car takes 25 to 35 litres and a truck far more, so 30 litres is a fair blended figure once you note that most vehicles are cars. Use Rs 100 a litre as a round illustrative price and say that the live pump price should be checked, because it moves.
Starting from 15,000 vehicles a day, a 5% stop rate, 30 litres a fill and Rs 100 a litre, the pump sells 22,500.0 litres and takes about Rs 22.5 lakh a day, roughly Rs 82 crore a year, and moving the stop rate between 3% and 8% swings that between Rs 49 crore and Rs 131 crore.
Which assumption should you shake, and which can you leave alone?
Not every link is equally uncertain. Shake the input you know least, because the answer moves one for one with every link and the widest range sits in the stop rate. Litres per fill is boxed in by the size of a fuel tank: 25 to 35 litres moves the annual figure only from Rs 68 crore to Rs 96 crore. The stop rate could honestly be 3% on a stretch with pumps every few kilometres or 8% at the only station for an hour, and that range takes the answer from Rs 49 crore to Rs 131 crore. Saying which input is soft is worth more to the interviewer than the point estimate.
Input Low Base High Revenue swing, Rs crore a year
Share of vehicles that stop 3% 5% 8% 49 to 131
Litres per fill 25 30 35 68 to 96
Price per litre Printed on the pump Rs 100 Known, confirm the current figure
With everything else at the base case, the stop rate swings the annual revenue across a range of Rs 82 crore while litres per fill swings it only Rs 27 crore, so the stop rate is where the estimate lives or dies.
The relationship
$\text{Revenue} = 15{,}000 \times 5\% \times 30 \times 100 \times 365 \approx \text{Rs }82\text{ crore}$
15,000 vehicles passing the pump in a day
5% the share that stop to fill
30 litres in an average fill
100 an illustrative price per litre, in rupees
365 days a year, a highway pump rarely closes
What it says in wordsAnnual revenue is the number of fills a day, times the litres and the price, times the days in the year.
What does a banker add after the number?
Say what the number is not. Revenue is not profit: a dealer keeps a margin of a few rupees a litre, so Rs 80 crore of sales is a small fraction of that in gross margin, and the rest passes through to the oil company and the exchequer. Then offer one sanity check: Rs 80 crore a year is about Rs 22 lakh a day, which at Rs 3,000 a fill is roughly 750 customers, about one every two minutes across an 18-hour day. If that sounds like a busy forecourt, the estimate hangs together.
Where candidates lose it
The common loss is a single guessed number with no chain behind it, or a chain where every link is stated with equal confidence. The interviewer is marking the structure and the honesty about which link is soft, not the final figure.
The second loss is stopping at revenue. One sentence on margin, and one sanity check from the other direction, separates a market-sizing answer from a guess.
What the interviewer asks next
How would you estimate the pump's annual profit rather than its revenue?
What changes if the pump sits inside a city rather than on a highway?
The owner wants to add a convenience store. How would you size that revenue?
066On an island everyone either always tells the truth or always lies. A says: "At least one of the two of us is a liar." B says nothing. What are A and B?Logic and brainteasersCoreConsulting style brainteasersSales and trading
Try it first
Before you work it: what can you say about B, who said nothing?
Show the worked solution
A always tells the truth and B always lies. Suppose A is a liar. Then A's statement is false, so neither of them is a liar, which means A tells the truth: a contradiction. So A is truthful, the statement is true, at least one of the two is a liar, and since A is not, B is. B never needed to speak.
Why does assuming A is a liar fall apart?
Picture someone who says, with a straight face, "I am lying right now." If that is a lie, they are telling the truth, and if it is the truth, they are lying; the sentence eats itself. A's statement is a milder cousin. A liar can never truthfully announce that there is a liar among us, because the announcement would then be true. So test the liar case first: if A lies, the statement is false, which means neither person is a liar, which means A is truthful. The assumption contradicts itself and is thrown out.
Supposing A lies makes A's statement false, which would mean neither is a liar and so A is truthful, a contradiction, while supposing A is truthful makes the statement true and points at B, so A tells the truth and silent B is the liar.
How does B get classified without saying a word?
Once A is truthful, A's sentence becomes a fact about the pair: at least one of them lies. A has been cleared, so the only person left who can be the liar is B, and B's silence changes nothing because the information came from A. The question is built this way to see whether you notice that evidence about a person can come from someone else. Say the two steps in order, A first and then B, and name the contradiction out loud rather than skipping to the answer.
Assumption about A Is the statement true? What follows Verdict
A lies No Neither is a liar, so A is truthful Contradiction, rejected
A tells the truth Yes At least one lies, and A does not B is the liar
Only the second row survives the test, so A is truthful and B, who said nothing, is the liar, because the statement A made is about both of them.
Why do interviewers like this shape of puzzle?
It rewards the method you need for any logic test: assume a case, chase it to a contradiction, and move on. The same discipline is how you check a model: assume the circular reference resolves, trace it, and if it cannot, the assumption was wrong. Candidates who fail it usually try to argue from B's silence, which carries no information, instead of from A's sentence, which carries all of it.
Where candidates lose it
The common loss is saying that nothing can be known about B, because B gave no statement. The statement that convicts B is A's, and it only works because a liar cannot utter it.
The second loss is reversing the pair and calling A the liar, from the feeling that whoever accuses must be the guilty one. Test the case rather than trusting the feeling: the liar assumption breaks in two steps.
What the interviewer asks next
A instead says: "Both of us are liars." What are A and B now?
A says: "B would say I am a liar." What can you conclude?
What if A says: "At least one of us tells the truth"?
068How many zeros are at the end of 100 factorial, written out in full?Mental maths and countingCoreBulge bracket IBMiddle market IB
Try it first
Before you count: how many trailing zeros?
Show the worked solution
24 trailing zeros. Each zero at the end is a factor of 10, which is a 2 paired with a 5. Among the numbers 1 to 100 there are 97 factors of 2 but far fewer 5s, so the 5s decide it. The 20 multiples of 5 contribute one 5 each, and the four multiples of 25 contribute a second, giving 20 + 4 = 24. There is no multiple of 125 below 100, so there is no third layer.
Why is the answer about fives and not about tens?
Think of making pairs of shoes from a pile of lefts and rights. If you have 97 left shoes and 24 right shoes, you can make 24 pairs; the extra lefts do not help. A trailing zero is a pair of one 2 and one 5 multiplied together, and in 1 to 100 the 2s vastly outnumber the 5s, so the number of 5s sets the number of zeros. Counting multiples of 10 misses the zeros built from a 5 in one number and a 2 in another, such as 5 x 2 = 10 or 15 x 4 = 60.
The 20 multiples of 5 from 5 to 100 each contribute one factor of 5, and the four multiples of 25 contribute a second one, so 100 factorial contains 24 fives, each paired with one of its 97 twos to make 24 trailing zeros.
Where do the extra four come from?
A number like 50 is 2 x 5 x 5: it carries two fives, not one. Every multiple of 25 hides a second five, and 25, 50, 75 and 100 are the four of them below 100, so the count is 20 from the multiples of 5 plus 4 from the multiples of 25. The next layer would be multiples of 125, which carry a third five, but 125 is beyond 100, so the count stops at 24. The pattern continues for larger factorials: divide by 5, then 25, then 125, and add the whole-number parts.
The relationship
$Z = \left\lfloor \tfrac{100}{5} \right\rfloor + \left\lfloor \tfrac{100}{25} \right\rfloor + \left\lfloor \tfrac{100}{125} \right\rfloor = 20 + 4 + 0 = 24$
Z the number of trailing zeros
floor the whole-number part of the division
5, 25, 125 each power of 5 adds a layer of fives
What it says in wordsAdd up the whole-number parts of 100 divided by 5, by 25 and by 125, because each layer counts one more five per number.
How do you check it without writing out 100 factorial?
Test the method on a factorial you can write. 10 factorial is 3,628,800, with two trailing zeros, and the rule gives 10 over 5 = 2, plus 10 over 25 = 0, which matches. Showing the rule on a small case you can verify is the fastest way to earn trust for the big case you cannot. Then say the general version: the number of trailing zeros of n factorial is the whole-number sum of n over 5, n over 25, n over 125 and so on, and for 1,000 factorial that is 200 + 40 + 8 + 1 = 249.
Where candidates lose it
The fast wrong answers are 10, from counting multiples of 10, and 20, from counting multiples of 5 once. The second is close enough that candidates say it with confidence and lose the point on the four multiples of 25.
The other loss is explaining it as a memorised formula. Say why fives are the scarce factor first; the formula then sounds like reasoning rather than recall.
What the interviewer asks next
How many trailing zeros does 1,000 factorial have?
How many factors of 2 are in 100 factorial, and why does that number not matter here?
What is the smallest n for which n factorial ends in exactly 30 zeros, and is every count reachable?
069Two companies both trade at 8x EV/EBITDA. One spends 20% of its EBITDA on capex each year and the other spends 60%. Which is cheaper, and which multiple shows it?Valuation riddlesCoreElite boutique IBBulge bracket IB
Try it first
Before you work it: which company is cheaper?
Show the worked solution
The company spending 20% is cheaper, and EV over EBITDA less capex shows it. Give both Rs 100 crore of EBITDA, so each is worth Rs 800 crore. The first keeps Rs 80 crore after capex and trades at 10x that cash figure; the second keeps Rs 40 crore and trades at 20x. EV/EBITDA cannot see capital intensity, so when capex differs this much, compare value against cash earnings instead.
Why does the same 8x mean different things?
Two taxi drivers each earn Rs 1 lakh a month before costs. One drives a new car that needs a service a year; the other drives an old one that eats Rs 60,000 a month in repairs. You would not pay the same for both businesses. EBITDA is earnings before the spending needed to keep the assets working, so two companies with equal EBITDA can hand their owners very different amounts of cash. At 8x, both are valued at Rs 800 crore on Rs 100 crore of EBITDA. The first spends Rs 20 crore of that on capex, the second Rs 60 crore, and that difference never appears in the headline multiple.
Both companies carry Rs 100 crore of EBITDA and an Rs 800 crore enterprise value, but after capex the first keeps Rs 80 crore and the second Rs 40 crore, so on EV over EBITDA less capex they trade at 10x and 20x and the first is the cheaper company.
Which multiple fixes it?
Take the capex off before dividing. EV over EBITDA less capex puts the two companies on cash they can actually keep, and the gap opens up to 10x against 20x. EV over EBIT does a rougher version of the same job, because depreciation is yesterday's capex spread over time, but it lags when a company is spending more than it depreciates. The cleanest version is EV over unlevered free cash flow, which also takes tax and working capital off. Name the limitation too: capex can be lumpy, so use a normal year or an average, and growth capex that adds capacity deserves a different reading from maintenance capex that merely keeps the lights on.
Rs crore a year Company A Company B
EBITDA 100 100
Capex 20 60
EBITDA less capex 80 40
Enterprise value 800 800
EV / EBITDA 8.0x 8.0x
EV / (EBITDA less capex) 10.0x 20.0x
On identical EBITDA and value, company A keeps Rs 80 crore after capex and company B Rs 40 crore, so company B costs twice as much per rupee of cash earnings.
The relationship
$\frac{EV}{EBITDA - capex} = \frac{800}{100 - 20} = 10x \qquad \frac{800}{100 - 60} = 20x$
EV enterprise value, Rs 800 crore for both
EBITDA Rs 100 crore for both
capex capital spending, Rs 20 crore and Rs 60 crore
What it says in wordsDivide value by the cash left after capex, and the capital-hungry company looks twice as expensive.
When would the market be right to pay 8x for both?
Say the case against your own answer. If company B's Rs 60 crore of capex is building capacity that will lift EBITDA sharply, the market may be paying for growth that company A will never deliver. The question has no growth figures, so the honest answer is that on today's cash the 20% company is cheaper, and that the interviewer would need to tell you about growth and the split between maintenance and growth capex before you could defend an equal multiple.
Where candidates lose it
The common loss is answering that neither is cheaper because the multiples match. The interviewer has deliberately picked a multiple that is blind to capex to see whether you know what EBITDA leaves out.
The second loss is naming the fix without the limitation. Capex is lumpy and some of it is growth, so say which capex you would use and why.
What the interviewer asks next
Both companies also have the same EV/EBIT. What would that tell you about their depreciation?
How would you separate maintenance capex from growth capex if the company does not disclose the split?
Which industries tend to show the biggest gap between EV/EBITDA and EV/(EBITDA less capex)?
070A fund earns 10% a year before fees and charges 2% a year. Over 30 years, what share of the gross ending wealth does the fee consume?Growth and compoundingCoreMiddle market IBPrivate equity
Try it first
Before you compute: roughly what share of the final gross pot goes to fees?
Show the worked solution
About 42% of the gross ending wealth. Rs 1 compounding at 10% for 30 years becomes Rs 17.45. With a 2% fee the money compounds at 8% and becomes Rs 10.06. The gap is Rs 7.39, and 7.39 over 17.45 is 42%. A fee that sounds like a fifth of the return takes more than two fifths of the final pot, because every rupee of fee also forfeits the growth it would have earned.
Why does a 2% fee take far more than 2%?
Imagine a farmer who hands over two in every hundred sacks of grain each harvest. The loss is not two sacks; it is two sacks plus everything those sacks would have grown into had they been sown. A fee is taken from a growing pot, and each slice removed stops compounding, so the cost builds year on year. After one year the fee has cost about 2% of the pot. After 10 years it has cost 17%, after 20 years 31%, and after 30 years 42%. The number is still climbing at year 30; it only stops when the money is withdrawn.
Rs 1 compounding at 10% reaches Rs 17.45 in 30 years while at 8% after fees it reaches Rs 10.06, and the shaded gap of Rs 7.39 is 42% of the gross ending wealth, having been 17% at year 10 and 31% at year 20.
How do you get the number in your head?
Use the rule of 72 for the two growth rates. At 10% money doubles roughly every 7.2 years, so in 30 years it doubles a little over four times, about 17x. At 8% it doubles every 9 years, so just over three times, about 10x. The ratio of the two end points, 10 over 17, is roughly 0.58, so the fee has taken about 42% of the gross pot. The exact figures are 17.45x and 10.06x, and the exact share is 42.3%, but the rule of 72 gets you within a point in ten seconds.
The relationship
$\text{fee share} = 1 - \frac{(1.08)^{30}}{(1.10)^{30}} = 1 - \frac{10.06}{17.45} = 42.3\%$
1.10 gross growth of Rs 1 each year at 10%
1.08 net growth each year after a 2% fee
30 years the money compounds
What it says in wordsCompare the net end point with the gross end point; the shortfall is the share the fee consumed.
What does an interviewer want after the number?
The link to the desk. Carried interest, management fees and transaction costs all work the same way: a small annual drag compounds into a large share of the terminal value, which is why net-of-fee returns, not gross, are what an investor should compare. Say the limitation too. The 10% gross return is an illustration, not a forecast, and a fee is worth paying if the manager's gross return beats the alternative by more than the fee. The question is about arithmetic, not about whether any fund is worth its fee.
Where candidates lose it
The fast wrong answer is 60%, from multiplying 2% by 30 years, or 20%, from saying the fee is a fifth of the return. Both treat the fee as a flat deduction rather than a slice of a compounding pot.
The other loss is a number without a check. The rule of 72 gives roughly 17x against 10x in seconds, and saying it shows you can sanity check compounding without a calculator.
What the interviewer asks next
What share of the gross pot does a 1% fee take over 30 years?
If the fee is charged on gains only, like a 20% carry, how does the arithmetic change?
Over how many years would the 2% fee consume half of the gross pot?
071Final-year free cash flow is Rs 100 crore, long-run growth is 4% and WACC is 10%. By what percentage does the terminal value fall if WACC rises to 11%?DCF and cost of capitalCoreBulge bracket IBMiddle market IB
Try it first
Before you compute: how much does the terminal value fall?
Show the worked solution
It falls by about 14.3%, one seventh. Terminal value is next year's cash flow over the spread between WACC and growth. At 10% the spread is 6 points, so the value is Rs 104 crore over 0.06, about Rs 1,733 crore. At 11% the spread is 7 points, giving Rs 1,486 crore. The new value is 6 over 7 of the old, a fall of 14.3%. The spread, not the WACC, is what the value is sensitive to.
Why does a one point move in WACC cost a seventh of the value?
Think of a shop whose rent is Rs 10 a month and whose takings are Rs 16: the owner keeps Rs 6. If rent goes up by one rupee the owner's margin shrinks by a sixth, far more than the rent went up in percentage terms. The terminal value divides by the gap between WACC and growth, and when that gap is thin, a small move in either rate is a large move in the gap. Here the gap goes from 6 points to 7, so the value falls to six sevenths of itself.
With growth fixed at 4%, the terminal value of Rs 100 crore of final-year cash flow falls from Rs 1,733 crore at a 10% WACC to Rs 1,486 crore at 11%, a drop of 14.3%, because the spread in the denominator grows from 6 points to 7.
What is the shortcut, and when does it break?
You do not need either terminal value to answer. The ratio of new to old value is the ratio of the old spread to the new spread, 6 over 7, so the percentage change is the spread change divided by the new spread. The same shortcut runs the other way: a fall in WACC to 9% narrows the spread to 5 points, and the value rises by 6 over 5 minus 1, 20%. Notice the asymmetry. A one point fall lifts value 20% while a one point rise cuts it 14.3%, which is why DCF sensitivity tables look lopsided.
The relationship
$TV = \frac{FCF \times (1+g)}{WACC - g} \qquad \frac{TV_{11\%}}{TV_{10\%}} = \frac{0.10 - 0.04}{0.11 - 0.04} = \frac{6}{7} \Rightarrow -14.3\%$
FCF final-year free cash flow, Rs 100 crore
g long-run growth, 4%
WACC the discount rate, 10% rising to 11%
What it says in wordsThe cash flow cancels; the terminal value changes by the ratio of the old spread to the new spread.
WACC Spread over 4% growth Terminal value, Rs crore Change from the 10% case
8% 4 points 2,600 +50.0%
9% 5 points 2,080 +20.0%
10% 6 points 1,733 +0.0%
11% 7 points 1,486 -14.3%
12% 8 points 1,300 -25.0%
Each one point step in WACC moves the terminal value by a different percentage, larger on the way down in WACC than on the way up, because the value is a cash flow divided by a thin and changing spread.
Why does this matter beyond the arithmetic?
The terminal value is usually most of a DCF, often two thirds or more of the enterprise value. If a one point move in WACC moves the terminal value by a seventh, the whole valuation is a judgement about two inputs that nobody knows to within a point. That is why a banker shows a sensitivity table rather than a single number, and why a growth assumption of 4% against a WACC of 10% should be cross-checked against the implied exit multiple before anyone trusts it.
Where candidates lose it
The common loss is answering 10%, from the WACC moving from 10 to 11, or 1%, from the one point move itself. Both forget that the value divides by the spread, not by the WACC.
The other loss is computing both terminal values with the Rs 104 crore numerator and losing a minute. Say the spreads, 6 and 7, and the answer is one division.
What the interviewer asks next
What happens to the terminal value if growth rises from 4% to 5% with WACC fixed at 10%?
What exit EV/EBITDA multiple does a Rs 1,733 crore terminal value imply if final-year EBITDA is Rs 150 crore?
Why would a WACC of 10% with 4% growth be hard to defend for a mature company?
074A corridor has 100 closed lockers. Student 1 opens every locker. Student 2 toggles every second locker, student 3 every third, and so on up to student 100, who toggles only locker 100. Which lockers end up open?Logic and brainteasersCoreConsulting style brainteasersSales and trading
Try it first
Before you reason it out: how many lockers end up open?
Show the worked solution
The 10 perfect squares: lockers 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Locker n is toggled once by every student whose number divides n, so it ends open if n has an odd number of divisors. Divisors come in pairs, d and n over d, so the count is even unless one divisor pairs with itself, which happens only when n is a perfect square. Those ten lockers are open; the other 90 are shut.
What decides whether a locker ends open or closed?
Think of a light switch flicked by a series of people walking past. Only the number of flicks matters: an odd count leaves the light on, an even count leaves it off. Locker n is flicked by student d exactly when d divides n, so its final state depends only on how many divisors n has. Locker 12 is touched by students 1, 2, 3, 4, 6 and 12, six flicks, and ends closed. Locker 16 is touched by 1, 2, 4, 8 and 16, five flicks, and ends open.
Of the 100 lockers only the ten perfect squares stay open, because locker 12 is toggled by its six divisors and ends closed while locker 16 is toggled by its five divisors, 4 pairing with itself, and ends open.
Why do only perfect squares have an odd number of divisors?
Divisors arrive in pairs. If d divides n then so does n over d: for 12 the pairs are 1 and 12, 2 and 6, 3 and 4. Pairs give an even count, and the only way to break a pair is for a divisor to be its own partner, d equal to n over d, which means n is d squared. So 16 has the pairs 1 and 16, 2 and 8, and then 4 standing alone. Every perfect square has exactly one such unpaired divisor, and no other number has any, which is why the open lockers are exactly the squares.
The relationship
$\text{toggles}(n) = \#\{d : d \mid n\} \qquad \text{odd} \iff n = m^2$
toggles(n) the number of times locker n changes state
d | n d divides n, so student d touches locker n
m^2 a perfect square, the only case where a divisor pairs with itself
What it says in wordsA locker is toggled once per divisor, and the count is odd only for perfect squares.
How should you present it in the room?
Do not simulate 100 students. Work one small locker aloud, say the pairing rule, name the exception, and list the ten squares: that is a ninety second answer that shows structure rather than stamina. Then offer the check: the number of open lockers is the whole-number part of the square root of 100, which is 10, and for 1,000 lockers it would be 31. Interviewers use this puzzle to see whether you look for the rule behind a process instead of running the process, which is the same instinct a model builder needs when a spreadsheet has a thousand rows and one formula.
Where candidates lose it
The usual loss is trying to simulate the first twenty lockers by hand under time pressure and never reaching the rule. The puzzle rewards stepping back and asking what one locker's fate depends on.
The other loss is spotting that the squares stay open and being unable to say why. Have the pairing argument ready: divisors pair up, and only a square root pairs with itself.
What the interviewer asks next
How many lockers end open if there are 1,000 lockers and 1,000 students?
Which lockers are toggled exactly twice, and what kind of numbers are they?
Student 1 now skips the corridor entirely. Which lockers end open?

876 x 45/365	next year's receivables if collection had not slowed
90	today's receivables, 730 x 45/365
876 x 15/365	the extra 15 days of sales tied up by slower collection

H	households, about 30 crore
p	the share of households with a two-wheeler, 50%
L	the vehicle's life, 10 years
g	growth in the fleet, 3% a year

(4 + 5 + 6)/6	the kept faces, each one in six
3/6	the chance the first roll is a 1, 2 or 3 and you roll again
3.5	what the second roll is worth on average

Input	Low	Base	High	Revenue swing, Rs crore a year
Share of vehicles that stop	3%	5%	8%	49 to 131
Litres per fill	25	30	35	68 to 96
Price per litre	Printed on the pump	Rs 100		Known, confirm the current figure

Try it first

Why does growth eat cash at all?

How do you split the Rs 54 crore?

Does the order of the split matter?

Where candidates lose it

What the interviewer asks next

Try it first

Where do you start, and what do you refuse to guess?

How does the bottom-up build go?

Why build a second route, and what if it disagrees?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the rule to compare against 3.5?

How does the 4.25 come together?

What happens with a third roll, and why do interviewers ask?

Where candidates lose it

What the interviewer asks next

Try it first

How do you turn a vague question into a chain you can multiply?

Which assumption should you shake, and which can you leave alone?

What does a banker add after the number?

Where candidates lose it

What the interviewer asks next

Try it first

Why does assuming A is a liar fall apart?

How does B get classified without saying a word?

Why do interviewers like this shape of puzzle?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer about fives and not about tens?

Where do the extra four come from?

How do you check it without writing out 100 factorial?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the same 8x mean different things?

Which multiple fixes it?

When would the market be right to pay 8x for both?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a 2% fee take far more than 2%?

How do you get the number in your head?

What does an interviewer want after the number?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a one point move in WACC cost a seventh of the value?

What is the shortcut, and when does it break?

Why does this matter beyond the arithmetic?

Where candidates lose it

What the interviewer asks next

Try it first

What decides whether a locker ends open or closed?

Why do only perfect squares have an odd number of divisors?

How should you present it in the room?

Where candidates lose it

What the interviewer asks next