Investment Banking puzzles, solved step by step

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

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086A company writes down Rs 50 crore of goodwill, and the impairment is not tax deductible. Walk me through what happens to EBITDA, net income, cash and the balance sheet.Accounting riddlesWarm upBulge bracket IBMiddle market IB
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What happens to the company's cash?
Show the worked solution
EBITDA is unchanged, net income falls by 50, cash is unchanged, and goodwill and equity both fall by 50. The impairment sits below EBITDA, so operating profit and pre-tax profit drop 50. Because it is not deductible, tax does not change and net income falls the full 50. The cash flow statement adds the non-cash charge back. On the balance sheet goodwill drops 50 and retained earnings drop 50.
What is an impairment actually admitting?
Suppose you paid Rs 10 lakh for a used car three years ago and a dealer now offers 6 lakh. Writing the car down in your notebook does not take money out of your wallet; the money left when you bought it. A goodwill impairment admits that an acquisition was overpaid for in the past; it does not spend any new cash. GoodwillThe part of an acquisition price above the fair value of the identifiable net assets bought, carried as an asset on the buyer balance sheet. is the premium paid above the target's net assets, and the write-down says part of that premium is no longer supported by the business's expected cash flows.
The Rs 50 crore write-down cuts operating profit and net income by 50 with no tax change, is added back in the cash flow statement so cash is unchanged, and reduces goodwill and retained earnings by 50 each, so the balance sheet still balances.
Why does net income fall the full 50 rather than 37.5?
Tax is what makes most non-cash charges reach cash. Depreciation, for example, usually cuts the tax bill, so a 50 charge at 25% lowers tax by 12.5. Goodwill impairment is generally not deductible, so the tax line does not move and the whole 50 hits net income. Confirm the rule in the relevant tax regime before relying on it. If the charge were deductible at 25%, net income would fall 37.5, cash would rise 12.5 from the lower tax bill, and the balance sheet would show cash up 12.5, goodwill down 50 and equity down 37.5.
Why does an analyst care if nothing happened to cash?
Because it tells you something about management's past decisions and future cash flows. An impairment says an acquisition is earning less than was paid for it, which usually reflects weaker expected cash flows. It also lowers book equity, which can push up leverage ratios measured on book values and occasionally trip a covenant. Most analysts strip it out of adjusted earnings as a one-off, but a string of them is a pattern worth asking about.
Where candidates lose it
The common slip is applying a tax shield by habit: net income down 37.5 and cash up 12.5. That is the depreciation answer, and the question told you the charge is not deductible precisely to see if you listen.
The second loss is forgetting the balance sheet. Say both sides: goodwill down 50, retained earnings down 50, and it balances.
What the interviewer asks next
What changes if the impairment were tax deductible?
Can a company later reverse a goodwill impairment?
How would the impairment affect a leverage covenant measured on book equity?
087A DCF sets its terminal value at 10x final-year EBITDA. Free cash flow runs at half of EBITDA and the WACC is 10%. What perpetual growth rate is that exit multiple quietly assuming?Valuation riddlesHardElite boutique IBBulge bracket IB
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Roughly what growth rate does 10x EBITDA imply here?
Show the worked solution
About 4.8% a year, forever. Write the terminal value both ways. The multiple says 10 x EBITDA. The Gordon formula says 0.5 x EBITDA x (1 + g) / (0.10 - g). Setting them equal, EBITDA cancels: 1.0 - 10g = 0.5 + 0.5g, so g = 0.5 / 10.5 = 4.76%. Every exit multiple implies a growth rate, and the check is whether that rate is believable for this business.
Why does a multiple contain a growth rate at all?
If a buyer wants a 5% return and pays 25 times this year's rent for a flat, they are counting on the rent rising: with flat rent, a 5% return supports a price of only 20 times. A multiple is a shorthand price for a stream of future cash flows, so every multiple implies some combination of growth and discount rate. In a DCF the exit multiple sits beside an explicit WACC, which means the growth assumption can be solved for exactly.
Set the two terminal values equal. The multiple gives 10 x EBITDA. The perpetuity growth method gives next year's free cash flow over (r minus g), and next year's cash flow is 0.5 x EBITDA x (1 + g). EBITDA cancels from both sides, leaving one equation in g, which solves to 4.76%.
With free cash flow at half of EBITDA and a 10% WACC, a 10x exit multiple implies perpetual growth of 4.8%, and implied growth rises with the multiple while bending toward the 10% WACC, which it can never reach.
The relationship
$m = \frac{f(1+g)}{r-g} \;\Rightarrow\; g = \frac{mr - f}{m + f} = \frac{1.0 - 0.5}{10 + 0.5} \approx 4.76\%$
m exit multiple of EBITDA, 10x
f free cash flow as a share of EBITDA, 0.5
r WACC, 10%
g implied perpetual growth
What it says in wordsRearranging the perpetuity formula turns any exit multiple into the growth rate it assumes.
How do you judge whether 4.8% is believable?
Perpetual growth means growth forever, so it should not exceed the long-run nominal growth of the economy whose currency the cash flows are in, or the business eventually becomes larger than that economy. Whether 4.8% passes depends on that currency's inflation and real growth, so state the benchmark you are using rather than a number from memory. The useful habit is running the check both ways: an exit multiple that implies an unbelievable growth rate is a sign the multiple came from a peer set that does not match this business.
Say the convention too. If you apply the multiple to final-year EBITDA but take next year's cash flow without the (1 + g) uplift, the answer becomes 10% minus 0.5/10, which is 5.0%. A small difference, but naming the convention shows you know where it comes from.
Where candidates lose it
The common answer is that an exit multiple has no growth assumption, which is exactly backwards. Candidates who have only used the multiple method never think to translate it, and the interviewer is testing that translation.
The second loss is solving the algebra but not commenting on the result. Say the number, then say what you would compare it against and why.
What the interviewer asks next
What exit multiple implies zero growth here?
If free cash flow conversion rises to 70% of EBITDA, what growth does 10x imply?
Which method would you lead with in a fairness opinion, and why show both?
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
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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
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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
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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?
092You may bet any fraction of your bankroll, again and again, on a coin that lands in your favour 60% of the time and pays even money. What fraction maximises the long-run growth of your bankroll, and what happens if you bet twice that fraction?Expected value and gamesHardBulge bracket IBConsulting style brainteasers
Try it first
Which stake grows your bankroll fastest over many bets?
Show the worked solution
Bet 20% each time; bet 40% and long-run growth falls to about zero, slightly negative. With even money, the growth-maximising fraction is the win probability minus the loss probability, 0.6 - 0.4 = 0.2. That earns about 2.0% compound growth a bet. At 40%, three wins and two losses leave you with 0.988 of what you started, a small loss, even though every bet still has a positive expected value.
If every bet has positive expected value, why not bet more?
A shopkeeper with a healthy margin who stakes the whole shop on each delivery will, sooner or later, lose one delivery and the shop with it. The margin was real; the sizing ruined it. When you reinvest the results, what compounds is the typical outcome, not the average one, and big stakes make one loss cost more than one win earns. At a 40% stake a win takes you to 1.4 and a loss to 0.6; one of each leaves 0.84, a 16% loss on a pair that broke even in count.
So test a typical stretch: in five bets you expect three wins and two losses. Betting 20%, 1.2 cubed times 0.8 squared is 1.106, about 11% richer. Betting 40%, 1.4 cubed times 0.6 squared is 0.988, slightly poorer. Same coin, same edge, and only the stake changed.
Long-run growth per bet peaks at about 2.0% when you stake 20% of the bankroll and falls to slightly below zero at 40%, so doubling the right stake throws away all of the growth that a genuine 60% edge provides.
Where does 20% come from?
The relationship
$g(f) = p \ln(1+f) + q \ln(1-f), \qquad g'(f) = 0 \Rightarrow f^* = p - q = 0.6 - 0.4 = 0.2$
f the fraction of bankroll bet each time
p, q the chances of winning and losing, 0.6 and 0.4
g(f) the average log growth per bet, what compounds over time
What it says in wordsMaximise the average of the log of wealth, and with even-money odds the best stake equals your edge.
This is the Kelly criterion, named after John Kelly, who published it in 1956. Over 100 bets the typical bankroll at a 20% stake is about 7.5 times the start; at 40% it is about 0.78 times. Over-betting a real edge can destroy growth as surely as having no edge. In practice many people bet half the Kelly fraction: at 10% growth is 1.50% a bet, about three quarters of the maximum, with far smaller swings, which matters because real edges are estimated, not known.
Where candidates lose it
The first trap is answering 100% because expected value rises with the stake. It is a single-bet answer to a repeated-bet question, and it ends in ruin the first time you lose.
The second trap is answering 60%, confusing the win probability with the stake. Say the rule out loud, edge over odds, and show the three-wins-two-losses check so the interviewer sees why 40% fails.
What the interviewer asks next
What is the Kelly stake if the coin pays 2 to 1 and wins 40% of the time?
Why do practitioners bet half Kelly or less?
How does this sizing logic carry over to a portfolio manager's position limits?
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
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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?
095A sponsor bought a company at 8x EBITDA of Rs 100 crore, funded with Rs 480 crore of debt. It exits at 9x EBITDA of Rs 130 crore with Rs 300 crore of debt still outstanding. What is the MOIC, and how much of the equity gain came from EBITDA growth, multiple expansion and debt paydown?Deal mathsHardElite boutique IBPrivate equity
Try it first
Which source contributed least to the equity gain?
Show the worked solution
MOIC is 2.72x: equity grows from Rs 320 crore to Rs 870 crore. Of the Rs 550 crore gain, EBITDA growth contributes 240 (30 x 8x), multiple expansion 130 (one turn on 130), and debt paydown 180 (480 - 300). The three add exactly to 550. Over five years that would be an IRR of about 22%, though the question gives no holding period.
How do you get entry and exit equity quickly?
Equity is what is left of the business's value after the debt. Entry: 8 x 100 is an enterprise value of 800, less 480 of debt, so the sponsor put in 320. Exit: 9 x 130 is 1,170, less 300 of debt, leaving 870. MOIC is exit equity over entry equity, 870 / 320 = 2.72x. Say both enterprise values out loud before subtracting debt, so the interviewer can follow each step.
Why split the gain into three sources?
Think of a house you bought, renovated and sold. Your profit came partly from adding a room, partly from the neighbourhood getting more popular, and partly from paying down the mortgage. Only the first is a skill you can repeat on the next house. Attribution shows which part of the return came from the sponsor's own work and which came from the market, which is what investors in the fund want to know. EBITDA growth and debt paydown come from running the business; multiple expansion mostly comes from market conditions at exit.
Entry equity of Rs 320 crore grows by 240 from EBITDA growth, 130 from the multiple rising one turn and 180 from debt repaid, reaching Rs 870 crore, so more than three quarters of the gain came from operations and deleveraging rather than the multiple.
The relationship
$\Delta E = \underbrace{(130-100)\times 8}_{240} + \underbrace{(9-8)\times 130}_{130} + \underbrace{(480-300)}_{180} = 550$
(130 - 100) x 8 EBITDA growth valued at the entry multiple
(9 - 8) x 130 multiple expansion applied to exit EBITDA
480 - 300 debt repaid from the business's cash flow
What it says in wordsThe equity gain is growth at the old multiple, plus the new multiple's lift on the new EBITDA, plus debt repaid.
Say the convention. The growth and multiple pieces share a cross term: 30 of extra EBITDA times one extra turn, which is 30. Here it sits inside multiple expansion. Value growth at the exit multiple instead and growth becomes 270 while multiple expansion falls to 100. The total does not change, but the story does, so state which order you used.
Where candidates lose it
The common error is forgetting debt paydown, because it is not a change in value of the business. Candidates find 240 and 130, get 370, and cannot reconcile to the 550 gain.
The second loss is quoting an IRR without a holding period. MOIC needs no time; IRR does. Ask, or state the assumption, before giving a percentage.
What the interviewer asks next
What is the IRR if the holding period is three years instead of five?
What happens to the MOIC if the exit multiple falls back to 8x?
Which of the three sources would a fund's investors value most, and why?

m	exit multiple of EBITDA, 10x
f	free cash flow as a share of EBITDA, 0.5
r	WACC, 10%
g	implied perpetual growth

5,00,000	the rupee amount handed over
0.015	the fee, 1.5% of the rupee amount
83.50	rupees per dollar

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

60	the first leg in km
1/5	the ratio of each leg to the one before

Try it first

What is an impairment actually admitting?

Why does net income fall the full 50 rather than 37.5?

Why does an analyst care if nothing happened to cash?

Where candidates lose it

What the interviewer asks next

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Why does a multiple contain a growth rate at all?

How do you judge whether 4.8% is believable?

Where candidates lose it

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What is the question really testing?

Does it matter whether you take the fee before or after converting?

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What the interviewer asks next

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Why is occupancy the swing input and not seats?

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What the interviewer asks next

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Why does the share price decide whether a bond is debt?

What goes wrong if you count it the other way?

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What the interviewer asks next

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If every bet has positive expected value, why not bet more?

Where does 20% come from?

Where candidates lose it

What the interviewer asks next

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Why is the forward set by interest rates and not by a forecast?

What would you actually do if the forward were quoted at 86?

Where candidates lose it

What the interviewer asks next

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Why does compounding more often help at all?

Why does it stop at e instead of growing forever?

Where candidates lose it

What the interviewer asks next

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How do you get entry and exit equity quickly?

Why split the gain into three sources?

Where candidates lose it

What the interviewer asks next