Investment Banking puzzles, solved step by step

Puzzles: 100
Traced to a firm: 36
Topics: 12
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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?
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?
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?
088Aptitude test style: you convert Rs 5 lakh to dollars at Rs 83.50 per dollar, and the bank first deducts a 1.5% fee on the rupee amount. How many dollars do you receive, to the nearest dollar?Mental maths and countingCoreBulge bracket IBMiddle market IB
Try it first
Pick the answer the test is looking for.
Show the worked solution
5,898 dollars. The fee is 1.5% of Rs 5,00,000, which is Rs 7,500, leaving Rs 4,92,500 to convert. Divided by 83.50 that is about 5,898.2. A quick route: 83.5 x 6,000 is 5,01,000, so Rs 5 lakh buys about 5,988 dollars before the fee, and taking 1.5% off 5,988 gives the same 5,898.
What is the question really testing?
These questions sit in the timed numerical section of an online assessment, and the arithmetic is easy. What they test is whether you apply each adjustment to the right base under time pressure. Think of a restaurant bill with a 10% service charge: it is 10% of the food, not of the food plus tax, and getting the base wrong gives a number that looks almost right. The test designers put the almost-right numbers in the options, so the base you apply the fee to decides whether you pick the correct one.
Deducting the Rs 7,500 fee from Rs 5,00,000 and converting the remaining Rs 4,92,500 at 83.50 gives 5,898 dollars, while forgetting the fee gives 5,988 and dividing by 1.015 gives 5,900, both of which a test will list as options.
Does it matter whether you take the fee before or after converting?
Not for a percentage fee. Taking 1.5% off the rupees and then converting, or converting and then taking 1.5% off the dollars, multiplies by the same two numbers in a different order: 5,988 x 0.985 is also 5,898. Order matters only for a flat fee or a fee charged on a different base; a percentage fee on the same amount commutes with the conversion. Knowing this lets you pick whichever order makes the mental maths easier.
The relationship
$\$ = \frac{5{,}00{,}000 \times (1 - 0.015)}{83.50} = \frac{4{,}92{,}500}{83.50} \approx 5{,}898$
5,00,000 the rupee amount handed over
0.015 the fee, 1.5% of the rupee amount
83.50 rupees per dollar
What it says in wordsDollars received equal the rupees left after the fee divided by the rupee price of one dollar.
Where does 5,900 come from? Dividing by 1.015 answers a different question: how much could you convert if the 1.5% were charged on top of the converted amount. That base is smaller than Rs 5 lakh, so the fee comes out as about Rs 7,389 instead of Rs 7,500. On a test with options two dollars apart, that is a wrong answer.
Where candidates lose it
The trap is speed. Candidates divide 5,00,000 by 83.5, see 5,988 in the options and move on, or remember the fee and divide by 1.015 because it feels like the same thing. Both are listed because both are common.
Read what the fee is charged on, write the base down, and check the order of magnitude: about six thousand dollars, a little under.
What the interviewer asks next
What if the bank also charged a flat Rs 500 per transaction?
What exchange rate are you effectively getting after the fee?
How would you convert the dollars back, and what do the two fees cost you in total?
089Estimate the annual ticket revenue of one screen in a multiplex in a large Indian city.Estimation and market sizingCoreBulge bracket IBConsulting style brainteasers
Try it first
Which input deserves most of your time in this estimate?
Show the worked solution
About Rs 3.1 crore a year, on these assumptions. Take 5 shows a day and 200 seats. Weekday shows run about 25% full over 261 days and weekend shows about 55% over 104 days, giving roughly 122,450 tickets. At an average of Rs 250 a ticket that is about Rs 3.06 crore. Occupancy, not seat count, is the input that swings the answer.
How do you structure it so the interviewer can follow?
Think of a bus route: revenue is trips a day times seats times how full the bus is times the fare. A screen is the same machine with a different product. Revenue is capacity times utilisation times price, and the estimate is only as good as the utilisation guess. Lay the chain out first, then fill each slot with a number and a one-line reason.
Capacity: about 5 shows a day, since a film of two and a half hours plus cleaning takes roughly three hours a slot from late morning to past midnight, and about 200 seats in a mid-sized auditorium. Utilisation: a Tuesday afternoon show might be a fifth full while a Saturday night show is close to sold out, so split the year. With 104 weekend days at 55% and 261 weekdays at 25%, the blended occupancy is 33.5%. Price: Rs 250 as an average across morning discounts and premium evening seats.
Weekday and weekend shows together sell about 122,450 tickets a year, roughly Rs 3.06 crore at Rs 250 each, and a 10 point change in occupancy moves that by about Rs 0.91 crore, three times what a 10% change in seats or price does.
Why is occupancy the swing input and not seats?
Each input moves revenue in proportion, so what matters is how wrong you could plausibly be. Seats are visible on a booking app's seat map and you will not be more than 10% off. Occupancy could reasonably be anywhere from 20% to 45% blended, and every 10 points is about Rs 0.91 crore, close to 30% of the answer. When you present an estimate, name the input you are least sure of and show what a fair range for it does to the total.
The relationship
$\text{Revenue} = \text{shows} \times \text{seats} \times \Big(\sum_{\text{day types}} \text{days} \times \text{occupancy}\Big) \times \text{avg price}$
shows x seats seats on sale each day, 5 x 200 = 1,000
days x occupancy summed over weekdays and weekends
avg price Rs 250 across all shows
What it says in wordsTicket revenue is daily seat capacity times how full it runs across the year times the average fare.
State the boundary of what you sized. This is gross ticket revenue before taxes, and the operator shares part of it with the film's distributor. Food, drinks and on-screen advertising sit outside it and can add a large share on top, so say which number you were asked for.
Where candidates lose it
The common slip is a single occupancy figure for the whole year, often too high because candidates picture the weekend shows they attend. That one number can put the estimate out by half.
The second loss is precision in the wrong place: debating whether the screen has 180 or 220 seats while guessing occupancy in a word. Spend the time where the uncertainty is.
What the interviewer asks next
How would you estimate food and drink revenue per screen?
What share of ticket revenue would you expect the cinema to keep?
How would you value a chain of 300 such screens?
090Two trains start 100 km apart on the same track and head toward each other at 50 km/h each. A bird starts at the front of one train and flies back and forth between them at 75 km/h until the trains meet. How far does the bird fly?Logic and brainteasersCoreConsulting style brainteasersSales and trading
Try it first
Answer before you start drawing zig-zags.
Show the worked solution
75 km. The trains close the 100 km gap at a combined 100 km/h, so they meet after exactly one hour. The bird flies without stopping for that hour at 75 km/h, so it covers 75 km. You never need to track the individual legs, though they do add up: 60 km, then 12, then 2.4, each a fifth of the one before, summing to 75.
What question should you ask first?
If a friend walks a dog in the park for an hour and the dog runs at 10 km/h the whole time, the dog covers 10 km however it zig-zags between trees. Nobody measures each dash. Distance is speed times time, so find how long the bird flies and the shape of its path stops mattering. The trains settle the time: they approach each other at 50 plus 50, which is 100 km/h, and 100 km apart means one hour.
The trains meet after one hour, so the bird flying at 75 km/h covers 75 km, and summing its legs of 60, 12 and 2.4 km and onward, each a fifth of the last, reaches the same 75 km by the long road.
If the interviewer insists on the legs, how do you sum them?
On the first leg the bird and the oncoming train close at 75 plus 50, which is 125 km/h, so they meet after 100 / 125 = 0.8 hours; the bird has flown 60 km. In that time the trains have closed 80 km, leaving 20 km between them. The next leg is the same problem a fifth the size: 12 km, then 2.4, and so on. Each leg is a fixed fraction of the last, which makes the total a geometric series that sums to 60 / (1 - 1/5) = 75 km.
The relationship
$60 + 12 + 2.4 + \dots = \frac{60}{1 - \tfrac{1}{5}} = 75 \text{ km}$
60 the first leg in km
1/5 the ratio of each leg to the one before
What it says in wordsA series where each term is a fixed fraction of the last sums to the first term over one minus the fraction.
Why ask this in a banking interview? It is a test of whether you look for the simpler quantity before diving into the mechanics, which is what a good analyst does with a messy model: find the number that settles the question and compute that. Answer the short way, then offer the series as a check if there is time.
Where candidates lose it
Candidates who start computing the first leg often keep going, and by the third leg they are lost in fractions with the clock running. The interviewer is watching whether you pause to ask what actually determines the answer.
The opposite loss is saying 75 instantly with no reason. Give the one line, combined speed 100 and so one hour, so the answer is audibly reasoned.
What the interviewer asks next
How many times does the bird turn around?
If train A runs at 60 km/h and train B at 40, does the answer change?
Where in a model have you found a shortcut quantity that made a messy calculation unnecessary?
091A company has a Rs 500 crore convertible bond with a conversion price of Rs 100, and its shares trade at Rs 120. In the enterprise value bridge, do you count the bond as debt or as shares? What changes if the share price falls to Rs 80?Enterprise value and dilutionCoreBulge bracket IBElite boutique IB
Try it first
At a share price of Rs 120, how should the bond enter the bridge?
Show the worked solution
At Rs 120, count it as 5 crore extra shares and not as debt; at Rs 80, count it as Rs 500 crore of debt and no new shares. Rs 500 crore at a Rs 100 conversion price is 5 crore shares. At Rs 120 those shares are worth Rs 600 crore, so the holder will convert. At Rs 80 they are worth Rs 400 crore, so the holder takes the Rs 500 crore back. Treat the bond as whichever it is most likely to become, never as both.
Why does the share price decide whether a bond is debt?
Suppose a friend lends you Rs 5 lakh and you agree they can take either the money back or a share of your shop. If the shop is booming, they will take the share; if it is struggling, they will take the cash. You plan around what they will choose. A convertible gives the holder a choice between repayment and shares, and the valuation should assume the choice a rational holder makes at today's price. The comparison is simple: conversion value, shares received times share price, against face value.
At Rs 120, converting gives 5 crore shares worth Rs 600 crore, more than the Rs 500 crore face value. The holder converts, so add 5 crore shares to the diluted count and remove the bond from debt. With, say, 50 crore existing shares, diluted equity value becomes 55 crore x 120 = Rs 6,600 crore. At Rs 80, converting gives shares worth only Rs 400 crore, so the bond stays as Rs 500 crore of debt and the share count stays at 50 crore.
At Rs 120 the 5 crore conversion shares are worth Rs 600 crore against Rs 500 crore of repayment, so the bond counts as shares, while at Rs 80 they are worth Rs 400 crore and it counts as Rs 500 crore of debt; counting it both ways double counts it.
What goes wrong if you count it the other way?
Counting it as debt at Rs 120 puts Rs 500 crore into the bridge when the holder will in fact take Rs 600 crore of value in shares, understating the claim by Rs 100 crore. The serious error is counting it twice, as debt and as diluted shares, which overstates enterprise value by the full Rs 500 crore. That happens when the share count comes from one source that already includes the conversion and the debt figure from a balance sheet that still shows the bond.
The relationship
$\text{conversion value} = \frac{\text{face}}{\text{conversion price}} \times \text{share price} = \frac{500}{100} \times 120 = 600 > 500$
face the bond's face value, Rs 500 crore
conversion price Rs 100 per share
share price today's price, Rs 120
What it says in wordsIf the shares a holder would receive are worth more than the repayment, assume conversion.
Say the limitation. Near the conversion price the choice is close and the bond carries option value either way, so some banks value it at fair value rather than flipping between debt and shares. Settlement terms also matter: some convertibles repay the face value in cash and deliver only the excess in shares, which needs a different share count.
Where candidates lose it
The common mistake is treating the convertible as debt because it is called a bond, whatever the share price. At Rs 120 the holder is not waiting to be repaid; they are holding shares in all but name.
The quieter loss is double counting when pulling numbers from different sources. Say out loud that the bond is in one place only, and which.
What the interviewer asks next
At what share price does the treatment switch?
How would you treat a convertible that settles the face value in cash and only the excess in shares?
How does a convertible change the accretion or dilution of a deal paid for in stock?
093The rupee trades at 83 per dollar. One-year interest rates are 7% in India and 5% in the US. What should the one-year forward rate be, and what could you do if the quoted forward were far from it?Rates, risk and optionsCoreDebt capital marketsSales and trading
Try it first
Where should the one-year forward sit?
Show the worked solution
About 84.58 rupees per dollar. Rs 83 invested at 7% grows to 88.81. Converted to 1 dollar and invested at 5% it grows to 1.05 dollars. Both routes are riskless once the forward is agreed, so they must end equal: 1.05 x F = 88.81, giving F of about 84.58. If the quoted forward sat far above that, you could borrow rupees, invest in dollars and sell them forward to lock in a gain.
Why is the forward set by interest rates and not by a forecast?
Imagine two banks next door, one paying 7% on rupee deposits and one paying 5% on dollar deposits, and a counter that promises today to swap your dollars back into rupees in a year at a fixed rate. If that promised rate were 83, everyone would ignore the dollar bank. The forward rate must exactly cancel the interest difference, otherwise one route earns more than the other with no risk, and the gap would be traded away. That rule is called covered interest parity, covered because the forward removes the currency risk.
Investing Rs 83 at 7% gives Rs 88.81 in a year, and converting to 1 dollar at 5% gives 1.05 dollars, so the forward must be about 84.58 for the two routes to match, and a quote of 86 would let a trader lock in Rs 1.49 per dollar.
What would you actually do if the forward were quoted at 86?
At 86, dollars are expensive in the forward market relative to parity, so sell dollars forward and build the dollars cheaply today. Borrow Rs 83 at 7%, owing 88.81 in a year. Convert to 1 dollar at spot and deposit it at 5%, which grows to 1.05 dollars. Sell those 1.05 dollars forward at 86 for Rs 90.30. After repaying the rupee loan you keep about Rs 1.49 per dollar, without taking any view on where the rupee goes. Traders doing exactly this pushes the quote back toward 84.58.
The relationship
$F = S \times \frac{1 + r_{\text{INR}}}{1 + r_{\text{USD}}} = 83 \times \frac{1.07}{1.05} \approx 84.58$
F the one-year forward, rupees per dollar
S spot, 83
r_INR, r_USD one-year rates in India and the US, 7% and 5%
What it says in wordsThe forward equals spot scaled by the ratio of the two currencies' interest growth.
The forward rupee is about 1.9% weaker than spot, roughly the 2 point rate difference. A quick approximation is spot times one plus the difference, 83 x 1.02 = 84.66, close enough to sanity-check. The limitation: in practice, capital controls, transaction costs and limits on who can borrow in each currency let small gaps persist, so the arbitrage is cleaner on paper than on a dealing desk.
Where candidates lose it
The common answer is that the forward is the market's forecast of the rupee, so it should be 83 or whatever you expect. Interviewers on rates and FX desks ask this precisely to hear you say it is an arbitrage relationship, not a forecast.
The second slip is flipping the ratio and getting a stronger forward rupee. Check the direction: the higher-yielding currency trades weaker forward.
What the interviewer asks next
What is the six-month forward on the same numbers?
An importer must pay 1 million dollars in a year. How does the forward help, and what does it cost?
Why might the quoted forward differ from parity for an extended period?
094A bank pays 100% interest a year. What does Rs 1 grow to in one year if interest is compounded yearly, monthly, daily, and continuously?Growth and compoundingCoreMiddle market IBPrivate equity
Try it first
If you compounded every second instead of every day, roughly what would Rs 1 grow to?
Show the worked solution
Yearly gives Rs 2.00, monthly about Rs 2.61, daily about Rs 2.71, and continuous compounding gives e, about Rs 2.718. Splitting 100% into n periods gives (1 + 1/n) to the power n. Each increase in frequency adds less than the last, and as n grows without limit the result levels off at the constant e rather than growing forever.
Why does compounding more often help at all?
Think of a savings jar where interest is added at the end of each month instead of once a year. From February onward, the interest from January is itself earning interest. More frequent compounding lets earlier interest start earning sooner, so the same headline rate produces more by year end. Monthly at 100% means adding 1/12 twelve times, (1 + 1/12) to the twelfth power, which is about 2.61 rather than 2.00.
Rs 1 at 100% grows to 2.00 compounded yearly, 2.61 monthly and 2.71 daily, and the curve flattens toward e, about 2.718, which is the most any compounding frequency can produce.
Why does it stop at e instead of growing forever?
Because each extra slice adds interest on smaller and smaller amounts of fresh interest. Going from yearly to monthly adds 0.61; from monthly to daily adds only about 0.10; from daily to continuous about 0.0037. The gains shrink fast enough that their total converges, and the limit is the number e, about 2.71828. That is where e comes from: it is what 100% growth becomes when it compounds continuously.
The relationship
$\Big(1 + \frac{1}{n}\Big)^n: \; n=1 \to 2.00,\; n=12 \to 2.613,\; n=365 \to 2.715,\; n \to \infty \to e \approx 2.718$
n the number of compounding periods in the year
1/n the interest rate applied each period
e the limit as compounding becomes continuous, about 2.718
What it says in wordsSplitting a 100% rate into ever more periods raises the result, but only up to e.
Why it matters on a desk: the same quoted rate means different money depending on the compounding convention. A loan quoted at 12% compounded monthly costs about 12.68% a year in effective terms, so when comparing a monthly-pay loan with an annual-pay bond, convert both to an effective annual rate first. Continuous compounding is also why option and rates models use e: it is the clean limit of compounding very often.
Where candidates lose it
The first trap is thinking the answer grows without limit as compounding gets more frequent. Candidates sense that more is better and miss that each step adds less.
The second is freezing on the daily figure. You do not need it to three decimals. Say it sits just under e, about 2.71, and explain why the gap is tiny.
What the interviewer asks next
What effective annual rate does 12% compounded monthly give?
At 10% compounded continuously, what does Rs 1 grow to in a year?
Why do option pricing models use continuous compounding?
096A company spends Rs 50 crore on product development. It can expense the spend, or capitalise it and amortise it over five years. The tax rate is 25%, and the spend is deductible for tax in year one either way. In year one, what happens to EBITDA, net income, operating cash flow and free cash flow under each choice?Accounting riddlesCoreBulge bracket IBMiddle market IB
Try it first
Compared with expensing, what does capitalising do to year-one free cash flow?
Show the worked solution
Capitalising makes EBITDA 50 higher and net income 30 higher, but free cash flow is identical. Expensed, the 50 hits EBITDA and net income falls 37.5 after tax. Capitalised, only 10 of amortisation hits profit, so net income falls 7.5, and the 50 moves into investing cash flow as capex. Operating cash flow is 50 higher, capex 50 higher, and free cash flow is minus 37.5 under both.
If the same cash goes out, what actually changes?
Buying a laptop for work: you can think of it as this month's expense, or as a tool that lasts five years and costs a fifth each year. Your bank balance falls by the full price either way. Capitalising versus expensing changes which profit lines carry the cost and which cash flow section shows the outflow; it does not change the cash that left. That is why analysts compare companies on free cash flow when accounting choices differ.
Expensed: EBITDA falls 50, tax falls 12.5, net income falls 37.5. Capitalised: EBITDA is untouched, the year's amortisation is 50 over 5, which is 10, and net income falls 7.5. So capitalising shows EBITDA 50 higher and net income 30 higher in year one. In operating cash flow the capitalised company shows +12.5 against -37.5, a 50 swing, but it also shows 50 of capex in investing.
Capitalising the Rs 50 crore development spend leaves EBITDA 50 higher, net income 30 higher and operating cash flow 50 higher than expensing it, but the 50 reappears as capex, so year-one free cash flow is minus 37.5 under both treatments.
What assumption is doing the work on tax?
The question says the spend is deductible in year one either way, so the cash tax saving is 12.5 under both. In the capitalised books, tax expense follows the smaller 10 charge, and the 10.0 difference becomes a deferred tax liabilityTax that the accounts recognise as owed later because taxable profit was lower than book profit this year. that is added back in operating cash flow. If the tax authority instead required capitalisation too, the cash tax saving would fall to 2.5 and free cash flow under capitalisation would be -47.5, 10 lower: tax timing is the only route by which the choice reaches cash. State which rule you are assuming.
Why bankers care: two otherwise identical software companies, one capitalising development and one expensing it, will show different EBITDA margins. Valuing both on EV/EBITDA without adjusting flatters the one that capitalises. Look at capitalised development in the cash flow statement and either add it back to the expensing peer or deduct it from the capitalising one.
Where candidates lose it
The common slip is saying capitalising improves cash flow because operating cash flow rises. It does, but the outflow is simply relabelled as capex, and free cash flow is unchanged.
The second loss is getting the net income difference wrong by forgetting tax: 50 minus 10 is 40 before tax, and 30 after. Say the after-tax number.
What the interviewer asks next
What happens in years two to five under each treatment?
How would you adjust EV/EBITDA when comparing a company that capitalises development with one that does not?
Which treatment would a company under covenant pressure prefer, and why should a lender care?

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

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%

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

365/365	the first person can have any birthday
364/365	the second avoids the first
336/365	the thirtieth avoids the 29 taken

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