Investment Banking puzzles, solved step by step

A always tells the truth and B always lies. Suppose A is a liar. Then A's statement is false, so neither of them is a liar, which means A tells the truth: a contradiction. So A is truthful, the statement is true, at least one of the two is a liar, and since A is not, B is. B never needed to speak.

Why does assuming A is a liar fall apart?

Picture someone who says, with a straight face, "I am lying right now." If that is a lie, they are telling the truth, and if it is the truth, they are lying; the sentence eats itself. A's statement is a milder cousin. A liar can never truthfully announce that there is a liar among us, because the announcement would then be true. So test the liar case first: if A lies, the statement is false, which means neither person is a liar, which means A is truthful. The assumption contradicts itself and is thrown out.

How does B get classified without saying a word?

Once A is truthful, A's sentence becomes a fact about the pair: at least one of them lies. A has been cleared, so the only person left who can be the liar is B, and B's silence changes nothing because the information came from A. The question is built this way to see whether you notice that evidence about a person can come from someone else. Say the two steps in order, A first and then B, and name the contradiction out loud rather than skipping to the answer.

Why do interviewers like this shape of puzzle?

It rewards the method you need for any logic test: assume a case, chase it to a contradiction, and move on. The same discipline is how you check a model: assume the circular reference resolves, trace it, and if it cannot, the assumption was wrong. Candidates who fail it usually try to argue from B's silence, which carries no information, instead of from A's sentence, which carries all of it.

24 trailing zeros. Each zero at the end is a factor of 10, which is a 2 paired with a 5. Among the numbers 1 to 100 there are 97 factors of 2 but far fewer 5s, so the 5s decide it. The 20 multiples of 5 contribute one 5 each, and the four multiples of 25 contribute a second, giving 20 + 4 = 24. There is no multiple of 125 below 100, so there is no third layer.

Why is the answer about fives and not about tens?

Think of making pairs of shoes from a pile of lefts and rights. If you have 97 left shoes and 24 right shoes, you can make 24 pairs; the extra lefts do not help. A trailing zero is a pair of one 2 and one 5 multiplied together, and in 1 to 100 the 2s vastly outnumber the 5s, so the number of 5s sets the number of zeros. Counting multiples of 10 misses the zeros built from a 5 in one number and a 2 in another, such as 5 x 2 = 10 or 15 x 4 = 60.

Where do the extra four come from?

A number like 50 is 2 x 5 x 5: it carries two fives, not one. Every multiple of 25 hides a second five, and 25, 50, 75 and 100 are the four of them below 100, so the count is 20 from the multiples of 5 plus 4 from the multiples of 25. The next layer would be multiples of 125, which carry a third five, but 125 is beyond 100, so the count stops at 24. The pattern continues for larger factorials: divide by 5, then 25, then 125, and add the whole-number parts.

How do you check it without writing out 100 factorial?

Test the method on a factorial you can write. 10 factorial is 3,628,800, with two trailing zeros, and the rule gives 10 over 5 = 2, plus 10 over 25 = 0, which matches. Showing the rule on a small case you can verify is the fastest way to earn trust for the big case you cannot. Then say the general version: the number of trailing zeros of n factorial is the whole-number sum of n over 5, n over 25, n over 125 and so on, and for 1,000 factorial that is 200 + 40 + 8 + 1 = 249.

The company spending 20% is cheaper, and EV over EBITDA less capex shows it. Give both Rs 100 crore of EBITDA, so each is worth Rs 800 crore. The first keeps Rs 80 crore after capex and trades at 10x that cash figure; the second keeps Rs 40 crore and trades at 20x. EV/EBITDA cannot see capital intensity, so when capex differs this much, compare value against cash earnings instead.

Why does the same 8x mean different things?

Two taxi drivers each earn Rs 1 lakh a month before costs. One drives a new car that needs a service a year; the other drives an old one that eats Rs 60,000 a month in repairs. You would not pay the same for both businesses. EBITDA is earnings before the spending needed to keep the assets working, so two companies with equal EBITDA can hand their owners very different amounts of cash. At 8x, both are valued at Rs 800 crore on Rs 100 crore of EBITDA. The first spends Rs 20 crore of that on capex, the second Rs 60 crore, and that difference never appears in the headline multiple.

Which multiple fixes it?

Take the capex off before dividing. EV over EBITDA less capex puts the two companies on cash they can actually keep, and the gap opens up to 10x against 20x. EV over EBIT does a rougher version of the same job, because depreciation is yesterday's capex spread over time, but it lags when a company is spending more than it depreciates. The cleanest version is EV over unlevered free cash flow, which also takes tax and working capital off. Name the limitation too: capex can be lumpy, so use a normal year or an average, and growth capex that adds capacity deserves a different reading from maintenance capex that merely keeps the lights on.

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

Say the case against your own answer. If company B's Rs 60 crore of capex is building capacity that will lift EBITDA sharply, the market may be paying for growth that company A will never deliver. The question has no growth figures, so the honest answer is that on today's cash the 20% company is cheaper, and that the interviewer would need to tell you about growth and the split between maintenance and growth capex before you could defend an equal multiple.

About 42% of the gross ending wealth. Rs 1 compounding at 10% for 30 years becomes Rs 17.45. With a 2% fee the money compounds at 8% and becomes Rs 10.06. The gap is Rs 7.39, and 7.39 over 17.45 is 42%. A fee that sounds like a fifth of the return takes more than two fifths of the final pot, because every rupee of fee also forfeits the growth it would have earned.

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

Imagine a farmer who hands over two in every hundred sacks of grain each harvest. The loss is not two sacks; it is two sacks plus everything those sacks would have grown into had they been sown. A fee is taken from a growing pot, and each slice removed stops compounding, so the cost builds year on year. After one year the fee has cost about 2% of the pot. After 10 years it has cost 17%, after 20 years 31%, and after 30 years 42%. The number is still climbing at year 30; it only stops when the money is withdrawn.

How do you get the number in your head?

Use the rule of 72 for the two growth rates. At 10% money doubles roughly every 7.2 years, so in 30 years it doubles a little over four times, about 17x. At 8% it doubles every 9 years, so just over three times, about 10x. The ratio of the two end points, 10 over 17, is roughly 0.58, so the fee has taken about 42% of the gross pot. The exact figures are 17.45x and 10.06x, and the exact share is 42.3%, but the rule of 72 gets you within a point in ten seconds.

What does an interviewer want after the number?

The link to the desk. Carried interest, management fees and transaction costs all work the same way: a small annual drag compounds into a large share of the terminal value, which is why net-of-fee returns, not gross, are what an investor should compare. Say the limitation too. The 10% gross return is an illustration, not a forecast, and a fee is worth paying if the manager's gross return beats the alternative by more than the fee. The question is about arithmetic, not about whether any fund is worth its fee.

It falls by about 14.3%, one seventh. Terminal value is next year's cash flow over the spread between WACC and growth. At 10% the spread is 6 points, so the value is Rs 104 crore over 0.06, about Rs 1,733 crore. At 11% the spread is 7 points, giving Rs 1,486 crore. The new value is 6 over 7 of the old, a fall of 14.3%. The spread, not the WACC, is what the value is sensitive to.

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

Think of a shop whose rent is Rs 10 a month and whose takings are Rs 16: the owner keeps Rs 6. If rent goes up by one rupee the owner's margin shrinks by a sixth, far more than the rent went up in percentage terms. The terminal value divides by the gap between WACC and growth, and when that gap is thin, a small move in either rate is a large move in the gap. Here the gap goes from 6 points to 7, so the value falls to six sevenths of itself.

What is the shortcut, and when does it break?

You do not need either terminal value to answer. The ratio of new to old value is the ratio of the old spread to the new spread, 6 over 7, so the percentage change is the spread change divided by the new spread. The same shortcut runs the other way: a fall in WACC to 9% narrows the spread to 5 points, and the value rises by 6 over 5 minus 1, 20%. Notice the asymmetry. A one point fall lifts value 20% while a one point rise cuts it 14.3%, which is why DCF sensitivity tables look lopsided.

Why does this matter beyond the arithmetic?

The terminal value is usually most of a DCF, often two thirds or more of the enterprise value. If a one point move in WACC moves the terminal value by a seventh, the whole valuation is a judgement about two inputs that nobody knows to within a point. That is why a banker shows a sensitivity table rather than a single number, and why a growth assumption of 4% against a WACC of 10% should be cross-checked against the implied exit multiple before anyone trusts it.

About 4%, below the unlevered 7%. Per Rs 100 of property, a 7% cap rate means Rs 7 of net operating income. A 60% loan is Rs 60, and at 9% interest-only it costs Rs 5.4 a year. That leaves Rs 1.6 of cash for the Rs 40 of equity, which is 4%. Because the loan costs more than the property yields, this is negative leverage: borrowing lowers the equity return rather than raising it.

Why is the cash-on-cash below the cap rate?

Suppose you borrow money at 9% to put in a deposit paying 7%. Every rupee borrowed loses you 2 paise a year, and the more you borrow the worse it gets. Leverage lifts the equity return only when the asset yields more than the debt costs; when the cap rate sits below the interest rate, every rupee of debt subtracts from the equity's return. Here the Rs 60 of debt earns 7% inside the property, Rs 4.2, but costs 9%, Rs 5.4, so it drags Rs 1.2 off the Rs 7 that the equity would otherwise keep on Rs 100 unlevered.

How do you say the answer in a way the interviewer can follow?

Fix the value at Rs 100 so every input becomes a rupee figure. Cap rate times value is the income; loan-to-value times value is the debt; debt times the rate is the interest; the rest is cash on the equity, and the equity is value minus debt. That is four multiplications and two subtractions, and saying each one aloud keeps you honest. Then say the comparison that matters: 4% levered against 7% unlevered, so the loan has cost the equity 3 points of yield.

Why would anyone buy with negative leverage?

Because cash-on-cash is a snapshot of year one. A buyer accepting 4% today is betting on rent growth lifting the income, on refinancing when rates fall, or on selling at a lower cap rate, none of which is in this calculation. Say that the figure ignores principal repayment, because the loan is interest-only, and ignores any appreciation. A total return would add both, and a lender would also look at the debt service coverage, which here is 7 over 5.4, about 1.3x, thin for most lenders.

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.

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.

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.

The equity falls by about 33%, a third. Equity value is enterprise value less net debt: 10 minus 7, so 3 turns of EBITDA. A 10% fall takes enterprise value to 9 turns, and the lenders are still owed 7, so equity is 2 turns. From 3 to 2 is a one third fall. At 7 turns of debt on a 10 turn valuation the equity is a thin slice, and every move in the business's value is magnified 3.33 times on it.

Why does the equity fall more than the business?

Picture a flat bought for Rs 1 crore with a Rs 70 lakh loan. If the flat's price falls 10% to Rs 90 lakh, the bank is still owed Rs 70 lakh, so the owner's stake falls from Rs 30 lakh to Rs 20 lakh, a third. Net debt is fixed in rupees, so the whole of any fall in enterprise value comes out of the equity, and the thinner the equity slice, the larger the percentage hit. Here the equity is 3 turns of EBITDA under 7 turns of debt, so a 1 turn fall in value is a third of it.

What is the general rule, and what does 7x tell you about sensitivity?

The percentage move in equity is the percentage move in EV times EV over equity. At 10x EV and 7x debt that multiplier is 10 over 3, 3.33, so each 1% in enterprise value is 3.33% on the equity, up or down. That is what an interviewer means by sensitivity: 7 turns of leverage on a 10 turn valuation makes the equity a geared bet on the business. If EBITDA or the multiple slips 30%, the equity is worth nothing and the lenders start taking losses, which is why debt at that level prices as if it carried some of the equity risk.

Why does a banker ask a tech company this?

Because 7 turns is a lot of debt for a business valued on growth rather than assets. If the valuation multiple compresses, as growth multiples do when rates rise or growth slows, the enterprise value can fall 10% with no change in EBITDA at all, and the equity takes a third of that on the chin. Say the assumption you leaned on: net debt stays fixed, which holds over a short window but not if the company is burning or generating cash. The question tests whether you can see leverage as a magnifier before you build a single model.

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.

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.

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.

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.

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.

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.