Investment Banking puzzles, solved step by step

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

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073A company buys a Rs 100 crore asset. It depreciates it over 10 years in its books and over 5 years for tax, at a 25% tax rate. What deferred tax balance exists at the end of year 1 and year 5, and what happens to it afterwards?Accounting riddlesHardBulge bracket IBMiddle market IB
Try it first
Before you work it: what sits on the balance sheet at the end of year 5?
Show the worked solution
A deferred tax liability of Rs 2.5 crore at year 1, Rs 12.5 crore at year 5, and it unwinds to zero by year 10. Book depreciation is Rs 10 crore a year; tax depreciation is Rs 20 crore for five years then nothing. In years 1 to 5 taxable profit is Rs 10 crore below book profit, so the company pays Rs 2.5 crore less tax than it charges, and the difference accrues as a liability. From year 6 the position reverses and the liability drains at Rs 2.5 crore a year.
Why is there a liability when the company has paid less tax?
Think of a shopkeeper allowed to pay this year's electricity bill next year. Cash looks better today, but the bill has not vanished; it sits as something owed. Faster tax depreciation lets the company deduct the asset's cost sooner, so it pays less tax now and more later, and the accounts record the later tax as a liability the moment the saving is taken. The income statement charges tax on book profit, Rs 2.5 crore a year more than the cash actually paid in years 1 to 5, and that extra charge is what builds the balance.
The asset's book value falls by Rs 10 crore a year while its tax value falls by Rs 20 crore, so the gap between them reaches Rs 50 crore at year 5 and 25% of that, Rs 12.5 crore, is the deferred tax liability, which then unwinds by Rs 2.5 crore a year as book depreciation continues with no tax deduction left.
How do you get the balance without a schedule?
Compare the two values of the asset. The deferred tax liability is the tax rate times the gap between the asset's book value and its tax value, because that gap is profit the tax authority has not yet taxed. At year 1 the book value is Rs 90 crore and the tax value Rs 80 crore, a gap of 10, and 25% of 10 is Rs 2.5 crore. At year 5 the book value is Rs 50 crore and the tax value is zero, a gap of 50, so the liability is Rs 12.5 crore. At year 10 both are zero and so is the liability.
Year end Book value Tax value Gap Deferred tax liability at 25%
1 90 80 10 2.5
3 70 40 30 7.5
5 50 0 50 12.5
7 30 0 30 7.5
10 0 0 0 0.0
Rs crore. The liability is always a quarter of the gap between the book value and the tax value of the asset, which is why it peaks at Rs 12.5 crore when the tax value hits zero at year 5 and disappears when the book value catches up at year 10.
The relationship
$DTL = t \times (\text{book value} - \text{tax value}) \qquad 0.25 \times (50 - 0) = 12.5$
DTL the deferred tax liability, Rs crore
t the tax rate, 25%
book value cost less book depreciation, Rs 50 crore at year 5
tax value cost less tax depreciation, zero at year 5
What it says in wordsThe liability is the tax rate applied to the profit the tax authority has not yet taxed.
Why does a banker care about a timing difference?
Because it is cash. In years 1 to 5 the company keeps Rs 2.5 crore a year more cash than its income statement suggests, which shows up as an increase in deferred tax liabilities in operating cash flow, and in years 6 to 10 the same amount drains out. A model that uses book tax on EBIT misses both legs, and a buyer of a company with a large and growing deferred tax liability should ask whether it is still growing because the company keeps buying assets, or about to reverse. Say the limitation: the balance reverses only if the company stops adding new assets, and tax rules on depreciation differ by country and change, so confirm the current rates before building this into a model.
Where candidates lose it
The common loss is calling the balance a deferred tax asset, because the company has paid less tax and that feels like a benefit. Paying less now means paying more later, which is a liability.
The second loss is saying the two methods cancel out and so nothing appears. They cancel only over the full ten years; at every year end in between, the gap is real and sits on the balance sheet.
What the interviewer asks next
Now the asset is sold at the end of year 5 for Rs 60 crore. What happens to the deferred tax liability?
What would create a deferred tax asset instead of a liability?
The tax rate rises to 30% at the start of year 3. What happens to the balance, and where does the change hit?
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?
076An office building earns net operating income of Rs 10 crore a year and trades at an 8% cap rate. If cap rates fall to 7%, what happens to its value?Valuation riddlesCoreElite boutique IBBulge bracket IB
Try it first
Before you compute: how much does the value move?
Show the worked solution
The value rises about 14%, from Rs 125 crore to Rs 142.9 crore. A cap rate is the income yield a buyer demands, so value is net operating income divided by the cap rate. Rs 10 crore over 8% is Rs 125 crore; the same Rs 10 crore over 7% is Rs 142.9 crore. The ratio of the two values is 8 over 7, a rise of 14.3%. Nothing happened to the building; the market now pays more for each rupee of its rent.
Why does a lower cap rate mean a higher value?
A fixed deposit paying Rs 10,000 a year is worth Rs 1.25 lakh if savers demand 8%, and Rs 1.43 lakh if they will accept 7%. The cheque has not changed; the price people pay for it has. A cap rate is the yield a buyer requires on a building's income, so value is income divided by the cap rate, and a fall in the required yield raises the price of the same income. Rs 10 crore over 0.08 is Rs 125 crore; over 0.07 it is Rs 142.9 crore. The cap rateCapitalisation rate: net operating income divided by the property value, the income yield a buyer requires. works exactly like a bond yield, and value moves the other way.
With net operating income fixed at Rs 10 crore, a fall in the cap rate from 8% to 7% lifts the building's value from Rs 125 crore to Rs 142.9 crore, a rise of 14.3%, because value is income divided by the cap rate and 8 over 7 is 1.143.
How do you get the percentage without computing either value?
The income cancels. The ratio of new to old value is the old cap rate over the new cap rate, 8 over 7, so the rise is one seventh, 14.3%. The move is not symmetric: the same one point rise, from 8% to 9%, would cut value by 8 over 9 minus 1, about 11.1%. Small cap rates make the swings bigger: from 5% to 4% is a 25% jump in value, from 10% to 9% only 11.1%. Say that pattern after the number, because the follow-up is usually about a different starting rate.
The relationship
$V = \frac{NOI}{c} \qquad \frac{V_{7\%}}{V_{8\%}} = \frac{0.08}{0.07} = 1.143 \Rightarrow +14.3\%$
V the building's value
NOI net operating income, Rs 10 crore a year
c the cap rate, 8% falling to 7%
What it says in wordsValue is income over the cap rate, so the value changes by the ratio of the old cap rate to the new one.
Cap rate Value of Rs 10 crore of NOI, Rs crore Change from the 8% case
6% 166.7 +33.3%
7% 142.9 +14.3%
8% 125.0 +0.0%
9% 111.1 -11.1%
10% 100.0 -20.0%
Each one point step in the cap rate moves the value by a different percentage, larger as the cap rate gets lower, because the value is a fixed income divided by a shrinking rate.
What would a banker add about why cap rates move?
Cap rates track the cost of money and the perceived risk of the rent. A fall from 8% to 7% usually means lower interest rates, more buyers chasing the asset, or more confidence in the tenant, and none of those is something the owner did. That is why a real estate team separates value created by raising income from value handed over by the market, and why an owner who bought at 8% and sells at 7% has earned a 14% gain on cap rate compression alone. Mention the limit: the formula assumes a stable income, so a building with leases about to expire does not deserve the same cap rate as one with ten years of rent locked in.
Where candidates lose it
The common loss is reading a falling cap rate as bad news for the value, because falling sounds like a fall. A cap rate is a yield, and a lower required yield means a higher price for the same income.
The second loss is saying 12.5%, from 1 over 8, or 1%, from the one point move. The value scales by the ratio of the rates, 8 over 7, and saying that ratio aloud is the answer.
What the interviewer asks next
NOI also rises 5% as the cap rate falls to 7%. What is the value now?
Why is a one point move in the cap rate a bigger deal at 5% than at 10%?
If the building was bought with 60% debt at the 8% cap rate, what did the cap rate move do to the equity?
077In your head, and out loud: what is 48 x 52, and what is 97 x 103?Mental maths and countingWarm upBulge bracket IBMiddle market IB
Try it first
Pick the pair.
Show the worked solution
2,496 and 9,991. Both pairs sit the same distance either side of a round number, so use the difference of squares: (a minus b)(a plus b) is a squared minus b squared. 48 x 52 is 50 squared less 2 squared, 2,500 minus 4, 2,496. 97 x 103 is 100 squared less 3 squared, 10,000 minus 9, 9,991. Say the middle number and the gap out loud, then subtract.
Why is the product always a little less than the middle number squared?
Take a square courtyard 50 paces on each side, 2,500 square paces. Cut a strip 2 paces wide off one side and lay it along the bottom. The courtyard is now 48 by 52, but the strip you moved was only 48 long, so a 2 by 2 corner is left bare. Two numbers either side of a round one multiply to the round number squared minus the gap squared, because the strip you move never quite fills the corner. 48 x 52 is 2,500 less 4, which is 2,496; the bare corner is the 4.
A 50 by 50 square with a strip moved from its side to its base becomes 48 by 52 with a 2 by 2 corner missing, so 48 x 52 is 2,500 less 4, 2,496, and the same picture at 100 by 100 with a 3 by 3 corner missing gives 97 x 103 as 10,000 less 9, 9,991.
How do you spot when the trick applies?
Add the two numbers and halve: if the result is round, the trick is on. 48 plus 52 is 100, half is 50, gap 2. 97 plus 103 is 200, half is 100, gap 3. Any pair that straddles a round number is a difference of squares: 73 x 67 is 70 squared less 3 squared, 4,900 minus 9, 4,891. If the pair does not straddle a round number, fall back on splitting one factor: 48 x 53 is 48 x 52 plus 48, 2,496 plus 48, 2,544. The reflex to look for the structure first is the skill the question tests.
The relationship
$(a-b)(a+b) = a^2 - b^2 \qquad 48 \times 52 = 50^2 - 2^2 = 2,496 \qquad 97 \times 103 = 100^2 - 3^2 = 9,991$
a the round number in the middle, 50 or 100
b the gap on each side, 2 or 3
What it says in wordsThe product of two numbers equally spaced around a middle is the middle squared minus the spacing squared.
Why does a banking interview bother with this?
Because the same reflex speeds up every number you say in a meeting. A 4% discount on a Rs 1,040 crore valuation, or a share price of Rs 97 times 103 crore shares, is a difference of squares in disguise, and getting it in two seconds without a calculator buys you credibility. Say your working out loud as you go: middle, gap, square, subtract. The interviewer is listening for the method as much as the answer, and a candidate who mumbles the right number silently earns less than one who narrates the route.
Where candidates lose it
The common loss is adding the small square instead of subtracting it, giving 2,504 and 10,009. The missing corner picture settles the sign: the product is always a little less than the middle squared.
The second loss is reaching for long multiplication and taking twenty seconds for 48 x 52. Check the sum of the two numbers first; if its half is round, the whole thing is one subtraction.
What the interviewer asks next
What is 995 x 1,005?
What is 48 x 53, and how does the trick help even though the pair is not symmetric?
Use the same idea to find 49 squared in your head.
078Estimate how many ATMs India needs. Walk me through the logic, then tell me whether your number is sensible.Estimation and market sizingCoreBulge bracket IBConsulting style brainteasers
Try it first
Which input turns a count of cash users into a count of machines?
Show the worked solution
About 2.7 lakh ATMs, on these assumptions. Take 100 crore adults, 80% with a bank account and half of those using ATMs: 40 crore users. Two withdrawals a month each is 80 crore a month, or 2.67 crore a day. If one machine handles about 100 withdrawals a day, you need roughly 2.67 lakh machines. Every input is an assumption to say out loud, not a fact.
Where does population stop being the useful number?
Think of a petrol pump. Nobody decides how many pumps a town needs by counting residents; they count cars, how often each one fills up, and how many fills one nozzle can manage in a day. ATMs work the same way. Population only tells you demand; capacity per machine is what turns demand into a number of machines. So the estimate has two halves: build daily withdrawals from people, then divide by what one machine does.
The demand half, with round numbers. Start with about 100 crore adults, rounded so the arithmetic stays easy. Say 80% hold a bank account with a debit card, and half of those actually take cash from a machine in a typical month: 40 crore users. Two withdrawals a month each gives 80 crore withdrawals, which over 30 days is about 2.67 crore a day.
Starting from 100 crore adults, the banked and ATM-using shares and two withdrawals a month give 2.67 crore withdrawals a day, and dividing by 100 uses per machine gives about 2.67 lakh ATMs, because one machine is busy only about 3.3 hours of its day.
How many withdrawals can one machine really handle?
A withdrawal takes about two minutes from card in to cash out, so a machine busy all 24 hours could do 720 a day. It never is. Demand bunches around mornings, lunch, evenings and salary days, and sits near zero at three in the morning. Size the machine for its realistic daily use, not its theoretical maximum, because idle night hours are capacity nobody can use. About 100 a day, a little over three hours of real use, is a fair working figure, and the one you should defend if pushed.
The relationship
$N = \frac{A \times b \times u \times w}{30 \times c} = \frac{100\text{ cr} \times 0.8 \times 0.5 \times 2}{30 \times 100} \approx 2.67\text{ lakh}$
A adults, 100 crore
b share with a bank account, 80%
u share of those who use ATMs, 50%
w withdrawals a month per user, 2
c withdrawals one machine handles a day, 100
What it says in wordsMachines needed equal daily withdrawals divided by what one machine serves in a day.
How do you sense-check 2.7 lakh?
Turn the answer into a ratio a person can judge. 2.67 lakh machines for 100 crore adults is about 27 ATMs per lakh adults, or one machine for every 3,750 adults. In a city that feels like a machine every few streets; in a village it feels sparse, which matches deployment leaning toward cities. RBI publishes the actual installed count, so say you would check your figure against the latest data. The second check is direction: digital payments cut cash withdrawals, so the honest answer drifts down over time, not up. If withdrawals fall from two a month to 1.5, the requirement falls to about 2.0 lakh.
Where candidates lose it
The common failure is starting from population and multiplying by an ATMs-per-person figure pulled from the air. That is a guess with arithmetic around it, and the interviewer has no step to follow or challenge.
The second loss is using a machine's theoretical capacity. Seven hundred and twenty uses a day assumes a queue at three in the morning, and it cuts the answer roughly seven times. Say why you picked 100 before anyone asks.
What the interviewer asks next
How would wider UPI use change your estimate over five years?
How would you size the cash a bank must load into these machines each week?
Which single assumption would you research first, and why?
080You roll a fair six-sided die repeatedly. On average, how many rolls will it take before every face from 1 to 6 has appeared at least once?Expected value and gamesHardBulge bracket IBConsulting style brainteasers
Try it first
Gut call before you work it: how many rolls on average?
Show the worked solution
About 14.7 rolls. Break the hunt into stages. The first roll always gives a new face. After that a new face comes up with chance 5/6, then 4/6, and so on down to 1/6. The average wait for an event of chance p is 1/p, so the stages take 1, 1.2, 1.5, 2, 3 and 6 rolls, which add to 6 x (1 + 1/2 + ... + 1/6) = 14.7.
Why does breaking it into stages make it easy?
Collecting cricket cards from cereal packets feels quick at first: almost every packet has a new card. The last card is the painful one, because nearly every packet repeats something you already own. The die is the same. Each stage is a simple waiting game with a fixed chance of success, and the average wait for a chance p is 1/p rolls. With k faces already found, a new one turns up with chance (6 minus k)/6, so that stage takes 6/(6 minus k) rolls on average.
Why is the wait 1/p? If a new face comes up one roll in three, you expect to wait three rolls, the same way a bus that comes one minute in ten keeps you waiting about ten. That one fact, applied six times, is the whole puzzle.
The six stages take 1, 1.2, 1.5, 2, 3 and 6 rolls on average, adding to 14.7, and the final face alone accounts for 6 rolls because only one roll in six finds it.
How do you add it up quickly in the room?
The relationship
$E[N] = \frac{6}{6} + \frac{6}{5} + \frac{6}{4} + \frac{6}{3} + \frac{6}{2} + \frac{6}{1} = 6 \sum_{k=1}^{6} \frac{1}{k} \approx 14.7$
E[N] the expected number of rolls to see all six faces
6/(6 minus k) the average wait for a new face once k faces are found
What it says in wordsAdd the average wait for each new face; the waits grow because new faces get rarer.
Pair the fractions to keep the arithmetic out loud: 1 plus 6 is 7, 1.2 plus 3 is 4.2, 1.5 plus 2 is 3.5, total 14.7. The last face costs six rolls on its own, more than the first four stages put together. That shape, cheap early progress and an expensive tail, is worth one sentence because it shows you see the structure, not just the sum.
One limitation: 14.7 is an average. The spread is wide, and plenty of runs take more than 20 rolls, so if the interviewer asks for a number you would bet on for a single run, say the average is not a likely outcome.
Where candidates lose it
The instinct is six, or a vague 10, because candidates picture one roll per face. Saying a number without the stage structure gives the interviewer nothing to follow, and an unexplained guess counts as a miss even if it is close.
The second loss is getting the stages right and fumbling the sum. Pair the terms that make round numbers and say 14.7, not fourteen point something.
What the interviewer asks next
How many rolls on average to see all faces of a 20-sided die?
What is the expected number of rolls until you see a 6?
Roughly how many rolls would you need to be 90% sure of seeing every face?
081Your incoming analyst class has 30 people. What is the probability that at least two of them share a birthday? Roughly how large must the class be for the odds to pass 50%?ProbabilityCoreBulge bracket IBSales and trading
Try it first
Before working it: what is the chance of a shared birthday in a class of 30?
Show the worked solution
About 71% for 30 people, and the odds pass 50% at 23. Work the chance that nobody matches: the second person avoids the first with 364/365, the third avoids both with 363/365, and so on. Multiply 30 of these and you get about 29.4%, so a shared birthday has a 70.6% chance. Repeating the product shows it first passes half at 23 people.
Why does the answer feel far too high?
Your instinct pictures yourself in the room: what are the odds someone shares my birthday? That is about 30 in 365, under 10%. But the question allows any two people to match. In a group of 30, every person can be compared with every other, and there are 30 x 29 / 2 = 435 such pairs. The number of pairs, not the number of people, drives the answer, and pairs grow roughly with the square of the group. Four hundred chances to hit a one-in-365 coincidence is a lot of chances.
The chance of a shared birthday crosses 50% at 23 people and reaches 70.6% at 30, because 23 people already form 253 pairs and 30 form 435, each one a separate chance of a match.
How do you compute it without a list of 435 cases?
Count the opposite. Matching can happen in many overlapping ways, but nobody matching happens in exactly one: each new person must miss every birthday already taken. When at least one is messy, compute none and subtract from one.
The relationship
$P(\text{match}) = 1 - \frac{365}{365} \cdot \frac{364}{365} \cdots \frac{336}{365} \approx 1 - 0.294 = 0.706$
365/365 the first person can have any birthday
364/365 the second avoids the first
336/365 the thirtieth avoids the 29 taken
What it says in wordsThe chance of at least one match is one minus the chance every person lands on a fresh day.
For a quick estimate in your head, use pairs: the chance of no match is close to e raised to minus pairs over 365. With 253 pairs that is e to the minus 0.69, almost exactly a half, which is why 23 is the crossing point. The method assumes birthdays are spread evenly across the year; real birthdays bunch a little, which only nudges the odds higher. By 60 people the chance is 99.4%.
Where candidates lose it
The fast wrong answer is 30 over 365, under 10%. It quietly fixes one person and asks whether anyone matches them, which is a much smaller event than any two people matching.
The second loss is trying to add up the ways a match can happen, which double counts groups with two or three matches. Say complement first, then multiply.
What the interviewer asks next
What is the chance that someone in the class shares your birthday specifically?
How large must the class be for a 99% chance of a shared birthday?
Where does the same pairs logic show up in hashing or in risk of correlated failures?
082A company's revenue grew from Rs 100 crore to Rs 250 crore over five years. What was its compound annual growth rate?Growth and compoundingWarm upMiddle market IBPrivate equity
Try it first
Answer before you calculate.
Show the worked solution
About 20.1% a year. Revenue multiplied by 2.5 over five years, so the annual rate is 2.5 to the power one fifth, minus one. A quick check: 1.2 to the fifth power is about 2.49, so the rate is just above 20%. Dividing the 150% total growth by five gives 30%, which overstates it, because 30% compounded for five years would take 100 to about 371.
Why is 30% wrong when 150% over five years looks like 30% a year?
A child's height or pocket money grows on what is already there. If your allowance rises 10% a year, year three's rise is 10% of a bigger allowance than year one's. Compound growth earns on its own past growth, so a constant rate produces bigger rupee steps every year, and dividing the total by the years overstates the rate. Thirty per cent a year would take 100 to 130, 169, 220, 286 and finally 371.
Dividing 150% total growth by five years gives 30%, but 30% compounded reaches 371; the rate that actually takes revenue from 100 to 250 in five years is 20.1%.
How do you get 20.1% without a calculator?
The relationship
$\text{CAGR} = \Big(\frac{\text{end}}{\text{start}}\Big)^{1/n} - 1 = 2.5^{1/5} - 1 \approx 20.1\%$
end / start the total multiple, 250 / 100 = 2.5
n the number of years, 5
What it says in wordsFind the single yearly multiplier that, applied n times, turns start into end.
Guess and check from a round rate. Try 20%: 1.2 squared is 1.44, cubed is 1.73, then 2.07, then 2.49, just shy of 2.5, so the answer is a hair above 20%. A second route is the rule of 72: at 20% money doubles in about 3.6 years, and 2.5x in five years is a little more than one doubling plus a bit, consistent with roughly 20%.
Say the limitation. CAGR describes only the start and end points. Revenue could have jumped to 250 in year one and sat flat, or fallen and recovered; the CAGR is the same 20.1%. If the path matters, ask for the yearly figures.
Where candidates lose it
Saying 30% is the whole trap. It sounds reasonable and the interviewer asks it fast precisely to see whether you notice that growth compounds.
The second loss is knowing the formula but freezing on the fifth root. Guess 20%, multiply up five times out loud, and adjust. Interviewers prefer that to silence.
What the interviewer asks next
What CAGR doubles revenue in five years?
If revenue then grows 10% a year for five more years, what is the ten-year CAGR?
Why might a company quote the simple average growth rate instead of the CAGR?
083A stock trades at Rs 100 and pays no dividend. A one-year European call struck at Rs 100 costs Rs 10, and the one-year interest rate is 5%. What should a put with the same strike and expiry cost?Rates, risk and optionsHardDebt capital marketsSales and trading
Try it first
Pick the put price before you work it.
Show the worked solution
About Rs 5.24. A call plus cash worth the strike's present value pays the larger of the share price and 100 at expiry, and so does a put plus the share. Identical payoffs must cost the same today: 10 + 100/1.05 = P + 100. The present value of 100 is 95.24, so the put is 5.24. No pricing model or volatility estimate is needed.
Why can you price the put without knowing volatility?
If two combo meals contain exactly the same food, a restaurant cannot charge different prices for long: everyone would buy the cheaper one. Options work the same way. Two portfolios that pay the same in every outcome must cost the same today, so if you know the price of one, the other follows without any view on where the share goes. Volatility matters for what a call is worth, but the call's price already contains it. The put inherits it through the parity relationship.
Build the two portfolios. A: buy the call and put aside 95.24 at 5%, which grows to exactly 100. B: buy the put and buy one share. If the share ends at 130, A exercises the call for 30 and adds its 100 cash, B holds a share worth 130 and lets the put expire: both have 130. If the share ends at 70, A lets the call expire and holds 100 cash, B exercises the put to sell the share for 100: both have 100.
A call plus cash of 95.24 and a put plus one share both pay the larger of the share price and 100 at expiry, so their prices today must match, which pins the put at 5.24.
What does the formula say, and what if the put traded at Rs 7?
The relationship
$C + \frac{K}{1+r} = P + S \quad\Rightarrow\quad P = 10 + \frac{100}{1.05} - 100 \approx 5.24$
C call price, Rs 10
P put price, the unknown
S share price today, Rs 100
K/(1+r) the strike discounted one year at 5%, 95.24
What it says in wordsA call plus the discounted strike is worth the same as a put plus the share.
If the put traded at 7, portfolio B would cost 107 against 105.24 for A, so you would sell B and buy A, pocketing 1.76 today with payoffs that cancel at expiry. Selling B means writing the put and short selling the share. Traders arbitraging that gap is what holds parity in place, which is why quoted put and call prices sit close to it in practice.
Say the assumptions: European options, so no early exercise; no dividend before expiry; and annual compounding. With continuous compounding the strike's present value is 100 x e to the minus 0.05, about 95.12, and the put comes out near 5.12. Give the convention, not just the number.
Where candidates lose it
The instinctive answer is Rs 10, on the idea that at-the-money calls and puts must cost the same. They do not, because the call holder defers paying the strike for a year, and that deferral is worth something when rates are positive.
The second loss is remembering parity but flipping a sign and answering 4.76. Build the two portfolios out loud before writing the formula and the signs take care of themselves.
What the interviewer asks next
If the stock paid a Rs 2 dividend before expiry, how would the put price change?
Why does parity hold exactly for European options but only as a bound for American ones?
What position replicates a long put using only the call, the share and borrowing?
084Four people must cross a narrow bridge at night with one torch. They take 1, 2, 5 and 10 minutes to cross. At most two can be on the bridge at once, a pair walks at the slower person's pace, and the torch must be carried on every crossing. What is the fastest total time?Logic and brainteasersHardConsulting style brainteasersSales and trading
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What is the fastest time you can get everyone across?
Show the worked solution
17 minutes. The 1 and 2 cross (2), the 1 returns (1), the 5 and 10 cross together (10), the 2 returns (2), and the 1 and 2 cross again (2). The obvious plan, the fastest person escorting each of the others, takes 19. The saving comes from pairing the two slowest so the 5 minute walk overlaps the 10 minute one.
Why does the escort plan feel right but lose two minutes?
Picture sharing taxis to the airport. Putting the two people with the latest flights in the same cab costs nothing extra for either; putting each in a cab with someone in a hurry makes every ride as slow as its slowest passenger. A pair always moves at the slower pace, so the 5 minute walker is free if he travels with the 10, and costs a full five minutes if he travels with anyone else. The escort plan pays for the 10, the 5 and the 2 separately.
Escorting everyone with the fastest walker takes 19 minutes because the 10 and the 5 each pay for their own crossing, while sending the 10 and the 5 together takes 17, since the 5 minute walk is absorbed inside the 10.
How do you know 17 cannot be beaten?
Count what every plan must pay. Five crossings are needed: three trips over and two back with the torch. The 10 must cross at least once, costing 10. If the 5 crosses separately from the 10, that is another 5 on top. Pairing them leaves two forward trips and two returns to do with the 1 and the 2, and the cheapest way to do that is 2 + 1 + 2 + 2. The trade is paying the 2 minute walker an extra return in order to save the 5 minute walker's crossing, and 2 is less than 5 minus 1.
The relationship
$\text{pair the slowest if } 2b < a + c: \quad 2 \times 2 = 4 < 1 + 5 = 6$
a the fastest walker, 1 minute
b the second fastest, 2 minutes
c the second slowest, 5 minutes
What it says in wordsSending the two slowest together wins when two trips by the second fastest cost less than the fastest walker plus the second slowest.
The rule is worth stating because it is not always pairing. If the walkers took 1, 4, 5 and 10 minutes, twice 4 is 8, more than 1 plus 5, so escorting wins: 21 minutes against 23 for pairing. Showing the interviewer you know when your trick stops working is better than the trick.
Where candidates lose it
Most candidates find 19 with the escort plan, test one or two variations and declare it optimal, because using the fastest walker for every return looks obviously efficient. The interviewer is waiting to see whether you notice the slow walkers can share a crossing.
The second loss is reaching 17 by trial and error and being unable to say why. Have the one-line reason ready: the 5 hides inside the 10, at the price of one extra 2 minute return.
What the interviewer asks next
What if the times are 1, 4, 5 and 10?
Add a fifth person who takes 12 minutes. What is the fastest time now?
Where does the same logic of batching slow jobs together show up on a deal team?

Year end	Book value	Tax value	Gap	Deferred tax liability at 25%
1	90	80	10	2.5
3	70	40	30	7.5
5	50	0	50	12.5
7	30	0	30	7.5
10	0	0	0	0.0

DTL	the deferred tax liability, Rs crore
t	the tax rate, 25%
book value	cost less book depreciation, Rs 50 crore at year 5
tax value	cost less tax depreciation, zero at year 5

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

V	the building's value
NOI	net operating income, Rs 10 crore a year
c	the cap rate, 8% falling to 7%

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Why is there a liability when the company has paid less tax?

How do you get the balance without a schedule?

Why does a banker care about a timing difference?

Where candidates lose it

What the interviewer asks next

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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

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Why does a lower cap rate mean a higher value?

How do you get the percentage without computing either value?

What would a banker add about why cap rates move?

Where candidates lose it

What the interviewer asks next

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Why is the product always a little less than the middle number squared?

How do you spot when the trick applies?

Why does a banking interview bother with this?

Where candidates lose it

What the interviewer asks next

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Where does population stop being the useful number?

How many withdrawals can one machine really handle?

How do you sense-check 2.7 lakh?

Where candidates lose it

What the interviewer asks next

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Why does breaking it into stages make it easy?

How do you add it up quickly in the room?

Where candidates lose it

What the interviewer asks next

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Why does the answer feel far too high?

How do you compute it without a list of 435 cases?

Where candidates lose it

What the interviewer asks next

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Why is 30% wrong when 150% over five years looks like 30% a year?

How do you get 20.1% without a calculator?

Where candidates lose it

What the interviewer asks next

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Why can you price the put without knowing volatility?

What does the formula say, and what if the put traded at Rs 7?

Where candidates lose it

What the interviewer asks next

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Why does the escort plan feel right but lose two minutes?

How do you know 17 cannot be beaten?

Where candidates lose it

What the interviewer asks next