Investment Banking puzzles, solved step by step

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

Showing 71–80 of 100

071Final-year free cash flow is Rs 100 crore, long-run growth is 4% and WACC is 10%. By what percentage does the terminal value fall if WACC rises to 11%?DCF and cost of capitalCoreBulge bracket IBMiddle market IB
Try it first
Before you compute: how much does the terminal value fall?
Show the worked solution
It falls by about 14.3%, one seventh. Terminal value is next year's cash flow over the spread between WACC and growth. At 10% the spread is 6 points, so the value is Rs 104 crore over 0.06, about Rs 1,733 crore. At 11% the spread is 7 points, giving Rs 1,486 crore. The new value is 6 over 7 of the old, a fall of 14.3%. The spread, not the WACC, is what the value is sensitive to.
Why does a one point move in WACC cost a seventh of the value?
Think of a shop whose rent is Rs 10 a month and whose takings are Rs 16: the owner keeps Rs 6. If rent goes up by one rupee the owner's margin shrinks by a sixth, far more than the rent went up in percentage terms. The terminal value divides by the gap between WACC and growth, and when that gap is thin, a small move in either rate is a large move in the gap. Here the gap goes from 6 points to 7, so the value falls to six sevenths of itself.
With growth fixed at 4%, the terminal value of Rs 100 crore of final-year cash flow falls from Rs 1,733 crore at a 10% WACC to Rs 1,486 crore at 11%, a drop of 14.3%, because the spread in the denominator grows from 6 points to 7.
What is the shortcut, and when does it break?
You do not need either terminal value to answer. The ratio of new to old value is the ratio of the old spread to the new spread, 6 over 7, so the percentage change is the spread change divided by the new spread. The same shortcut runs the other way: a fall in WACC to 9% narrows the spread to 5 points, and the value rises by 6 over 5 minus 1, 20%. Notice the asymmetry. A one point fall lifts value 20% while a one point rise cuts it 14.3%, which is why DCF sensitivity tables look lopsided.
The relationship
$TV = \frac{FCF \times (1+g)}{WACC - g} \qquad \frac{TV_{11\%}}{TV_{10\%}} = \frac{0.10 - 0.04}{0.11 - 0.04} = \frac{6}{7} \Rightarrow -14.3\%$
FCF final-year free cash flow, Rs 100 crore
g long-run growth, 4%
WACC the discount rate, 10% rising to 11%
What it says in wordsThe cash flow cancels; the terminal value changes by the ratio of the old spread to the new spread.
WACC Spread over 4% growth Terminal value, Rs crore Change from the 10% case
8% 4 points 2,600 +50.0%
9% 5 points 2,080 +20.0%
10% 6 points 1,733 +0.0%
11% 7 points 1,486 -14.3%
12% 8 points 1,300 -25.0%
Each one point step in WACC moves the terminal value by a different percentage, larger on the way down in WACC than on the way up, because the value is a cash flow divided by a thin and changing spread.
Why does this matter beyond the arithmetic?
The terminal value is usually most of a DCF, often two thirds or more of the enterprise value. If a one point move in WACC moves the terminal value by a seventh, the whole valuation is a judgement about two inputs that nobody knows to within a point. That is why a banker shows a sensitivity table rather than a single number, and why a growth assumption of 4% against a WACC of 10% should be cross-checked against the implied exit multiple before anyone trusts it.
Where candidates lose it
The common loss is answering 10%, from the WACC moving from 10 to 11, or 1%, from the one point move itself. Both forget that the value divides by the spread, not by the WACC.
The other loss is computing both terminal values with the Rs 104 crore numerator and losing a minute. Say the spreads, 6 and 7, and the answer is one division.
What the interviewer asks next
What happens to the terminal value if growth rises from 4% to 5% with WACC fixed at 10%?
What exit EV/EBITDA multiple does a Rs 1,733 crore terminal value imply if final-year EBITDA is Rs 150 crore?
Why would a WACC of 10% with 4% growth be hard to defend for a mature company?
072A property is bought at a 7% cap rate with a 60% loan-to-value loan at 9% interest-only. What is the cash-on-cash return on the equity?Rates, risk and optionsCoreWells Fargo SecuritiesNew York · 2026
Try it first
Before you work it: does borrowing raise or lower the equity return here?
Show the worked solution
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.
Per Rs 100 of property, Rs 7 of net operating income less Rs 5.4 of interest on Rs 60 of debt at 9% leaves Rs 1.6 for Rs 40 of equity, a 4% cash-on-cash return, and the same loan at 5% would give 10% while at 7% it would leave the return unchanged at 7%.
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.
The relationship
$\text{CoC} = \frac{\text{NOI} - \text{interest}}{\text{equity}} = \frac{7 - 60 \times 9\%}{100 - 60} = \frac{1.6}{40} = 4\%$
NOI net operating income, the cap rate times the value
interest the loan times its rate, interest-only so no principal
equity value less the loan, here Rs 40 per Rs 100
What it says in wordsCash-on-cash is the income left after interest, divided by the cash the equity put in.
Loan rate Interest on Rs 60 Cash to Rs 40 of equity Cash-on-cash
5% 3.0 4.0 10%
7% 4.2 2.8 7%
9% 5.4 1.6 4%
The cash-on-cash return crosses the unlevered 7% exactly when the loan rate equals the cap rate, rising above it at cheaper rates and falling below it, to 4%, at the 9% in the question.
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.
Where candidates lose it
The common loss is assuming leverage must raise the return and answering something above 7%. The interviewer has set the loan rate above the cap rate on purpose to see whether you check the sign before you multiply.
The second loss is dividing the cash by the whole Rs 100 of value instead of the Rs 40 of equity, which gives 1.6% and a confused look. Cash-on-cash is measured on the cash the equity actually put in.
What the interviewer asks next
At what loan rate does the cash-on-cash equal the cap rate, and why?
Now the loan amortises over 25 years. Does the cash-on-cash rise or fall, and what does it leave out?
If rents grow 3% a year, what is the cash-on-cash in year 3?
Asked at Wells Fargo Securities, Real Estate, New York, 2026 (Wall Street Oasis): What is the CoC when you have _ Interest Rate, _ LTV, and _ Cap rate?
073A company buys a Rs 100 crore asset. It depreciates it over 10 years in its books and over 5 years for tax, at a 25% tax rate. What deferred tax balance exists at the end of year 1 and year 5, and what happens to it afterwards?Accounting riddlesHardBulge bracket IBMiddle market IB
Try it first
Before you work it: what sits on the balance sheet at the end of year 5?
Show the worked solution
A deferred tax liability of Rs 2.5 crore at year 1, Rs 12.5 crore at year 5, and it unwinds to zero by year 10. Book depreciation is Rs 10 crore a year; tax depreciation is Rs 20 crore for five years then nothing. In years 1 to 5 taxable profit is Rs 10 crore below book profit, so the company pays Rs 2.5 crore less tax than it charges, and the difference accrues as a liability. From year 6 the position reverses and the liability drains at Rs 2.5 crore a year.
Why is there a liability when the company has paid less tax?
Think of a shopkeeper allowed to pay this year's electricity bill next year. Cash looks better today, but the bill has not vanished; it sits as something owed. Faster tax depreciation lets the company deduct the asset's cost sooner, so it pays less tax now and more later, and the accounts record the later tax as a liability the moment the saving is taken. The income statement charges tax on book profit, Rs 2.5 crore a year more than the cash actually paid in years 1 to 5, and that extra charge is what builds the balance.
The asset's book value falls by Rs 10 crore a year while its tax value falls by Rs 20 crore, so the gap between them reaches Rs 50 crore at year 5 and 25% of that, Rs 12.5 crore, is the deferred tax liability, which then unwinds by Rs 2.5 crore a year as book depreciation continues with no tax deduction left.
How do you get the balance without a schedule?
Compare the two values of the asset. The deferred tax liability is the tax rate times the gap between the asset's book value and its tax value, because that gap is profit the tax authority has not yet taxed. At year 1 the book value is Rs 90 crore and the tax value Rs 80 crore, a gap of 10, and 25% of 10 is Rs 2.5 crore. At year 5 the book value is Rs 50 crore and the tax value is zero, a gap of 50, so the liability is Rs 12.5 crore. At year 10 both are zero and so is the liability.
Year end Book value Tax value Gap Deferred tax liability at 25%
1 90 80 10 2.5
3 70 40 30 7.5
5 50 0 50 12.5
7 30 0 30 7.5
10 0 0 0 0.0
Rs crore. The liability is always a quarter of the gap between the book value and the tax value of the asset, which is why it peaks at Rs 12.5 crore when the tax value hits zero at year 5 and disappears when the book value catches up at year 10.
The relationship
$DTL = t \times (\text{book value} - \text{tax value}) \qquad 0.25 \times (50 - 0) = 12.5$
DTL the deferred tax liability, Rs crore
t the tax rate, 25%
book value cost less book depreciation, Rs 50 crore at year 5
tax value cost less tax depreciation, zero at year 5
What it says in wordsThe liability is the tax rate applied to the profit the tax authority has not yet taxed.
Why does a banker care about a timing difference?
Because it is cash. In years 1 to 5 the company keeps Rs 2.5 crore a year more cash than its income statement suggests, which shows up as an increase in deferred tax liabilities in operating cash flow, and in years 6 to 10 the same amount drains out. A model that uses book tax on EBIT misses both legs, and a buyer of a company with a large and growing deferred tax liability should ask whether it is still growing because the company keeps buying assets, or about to reverse. Say the limitation: the balance reverses only if the company stops adding new assets, and tax rules on depreciation differ by country and change, so confirm the current rates before building this into a model.
Where candidates lose it
The common loss is calling the balance a deferred tax asset, because the company has paid less tax and that feels like a benefit. Paying less now means paying more later, which is a liability.
The second loss is saying the two methods cancel out and so nothing appears. They cancel only over the full ten years; at every year end in between, the gap is real and sits on the balance sheet.
What the interviewer asks next
Now the asset is sold at the end of year 5 for Rs 60 crore. What happens to the deferred tax liability?
What would create a deferred tax asset instead of a liability?
The tax rate rises to 30% at the start of year 3. What happens to the balance, and where does the change hit?
074A corridor has 100 closed lockers. Student 1 opens every locker. Student 2 toggles every second locker, student 3 every third, and so on up to student 100, who toggles only locker 100. Which lockers end up open?Logic and brainteasersCoreConsulting style brainteasersSales and trading
Try it first
Before you reason it out: how many lockers end up open?
Show the worked solution
The 10 perfect squares: lockers 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Locker n is toggled once by every student whose number divides n, so it ends open if n has an odd number of divisors. Divisors come in pairs, d and n over d, so the count is even unless one divisor pairs with itself, which happens only when n is a perfect square. Those ten lockers are open; the other 90 are shut.
What decides whether a locker ends open or closed?
Think of a light switch flicked by a series of people walking past. Only the number of flicks matters: an odd count leaves the light on, an even count leaves it off. Locker n is flicked by student d exactly when d divides n, so its final state depends only on how many divisors n has. Locker 12 is touched by students 1, 2, 3, 4, 6 and 12, six flicks, and ends closed. Locker 16 is touched by 1, 2, 4, 8 and 16, five flicks, and ends open.
Of the 100 lockers only the ten perfect squares stay open, because locker 12 is toggled by its six divisors and ends closed while locker 16 is toggled by its five divisors, 4 pairing with itself, and ends open.
Why do only perfect squares have an odd number of divisors?
Divisors arrive in pairs. If d divides n then so does n over d: for 12 the pairs are 1 and 12, 2 and 6, 3 and 4. Pairs give an even count, and the only way to break a pair is for a divisor to be its own partner, d equal to n over d, which means n is d squared. So 16 has the pairs 1 and 16, 2 and 8, and then 4 standing alone. Every perfect square has exactly one such unpaired divisor, and no other number has any, which is why the open lockers are exactly the squares.
The relationship
$\text{toggles}(n) = \#\{d : d \mid n\} \qquad \text{odd} \iff n = m^2$
toggles(n) the number of times locker n changes state
d | n d divides n, so student d touches locker n
m^2 a perfect square, the only case where a divisor pairs with itself
What it says in wordsA locker is toggled once per divisor, and the count is odd only for perfect squares.
How should you present it in the room?
Do not simulate 100 students. Work one small locker aloud, say the pairing rule, name the exception, and list the ten squares: that is a ninety second answer that shows structure rather than stamina. Then offer the check: the number of open lockers is the whole-number part of the square root of 100, which is 10, and for 1,000 lockers it would be 31. Interviewers use this puzzle to see whether you look for the rule behind a process instead of running the process, which is the same instinct a model builder needs when a spreadsheet has a thousand rows and one formula.
Where candidates lose it
The usual loss is trying to simulate the first twenty lockers by hand under time pressure and never reaching the rule. The puzzle rewards stepping back and asking what one locker's fate depends on.
The other loss is spotting that the squares stay open and being unable to say why. Have the pairing argument ready: divisors pair up, and only a square root pairs with itself.
What the interviewer asks next
How many lockers end open if there are 1,000 lockers and 1,000 students?
Which lockers are toggled exactly twice, and what kind of numbers are they?
Student 1 now skips the corridor entirely. Which lockers end open?
075A tech company is worth 10x EBITDA and carries net debt of 7x EBITDA. If its enterprise value falls 10%, what happens to its equity value?Enterprise value and dilutionCoreMoelis & CompanyNew York · 2026
Try it first
Before you work it: how far does the equity fall?
Show the worked solution
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.
With net debt fixed at 7 turns of EBITDA, a fall in enterprise value from 10 turns to 9 takes the equity from 3 turns to 2, a 33% fall, and the multiplier of 3.33 works the same way upwards, where a 10% rise in EV would lift the equity 33%.
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.
The relationship
$\frac{\Delta E}{E} = \frac{\Delta EV}{EV} \times \frac{EV}{E} = -10\% \times \frac{10}{3} = -33.3\%$
E equity value, EV less net debt, 3 turns of EBITDA
EV enterprise value, 10 turns of EBITDA
Delta EV / EV the 10% fall in enterprise value
What it says in wordsThe equity moves by the EV move scaled up by how many times EV covers the equity.
Move in EV EV, x EBITDA Net debt, x EBITDA Equity, x EBITDA Move in equity
+10% 11.0 7.0 4.0 +33%
+0% 10.0 7.0 3.0 +0%
-10% 9.0 7.0 2.0 -33%
-20% 8.0 7.0 1.0 -67%
-30% 7.0 7.0 0.0 -100%
With net debt fixed at 7 turns, each 10% step in enterprise value is a one third step in the equity, and a 30% fall in enterprise value leaves the equity worth nothing.
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.
Where candidates lose it
The common loss is answering 10%, as if equity and enterprise value moved together. The debt sits between them and does not move, so the equity absorbs the whole fall.
The second loss is giving 33% and stopping. The interviewer asked what 7x tells you about sensitivity, so name the multiplier, 10 over 3, and say that it works both ways.
What the interviewer asks next
EBITDA falls 10% and the multiple stays at 10x. What happens to the equity, and is it the same answer?
At what fall in enterprise value is the equity worth nothing?
How does this magnifier relate to the beta of a levered company?
Asked at Moelis & Company, Generalist, New York, 2026 (Wall Street Oasis): A tech company has leverage rate of 7X. What does it tell you about the about the impact on sensitivity?
076An office building earns net operating income of Rs 10 crore a year and trades at an 8% cap rate. If cap rates fall to 7%, what happens to its value?Valuation riddlesCoreElite boutique IBBulge bracket IB
Try it first
Before you compute: how much does the value move?
Show the worked solution
The value rises about 14%, from Rs 125 crore to Rs 142.9 crore. A cap rate is the income yield a buyer demands, so value is net operating income divided by the cap rate. Rs 10 crore over 8% is Rs 125 crore; the same Rs 10 crore over 7% is Rs 142.9 crore. The ratio of the two values is 8 over 7, a rise of 14.3%. Nothing happened to the building; the market now pays more for each rupee of its rent.
Why does a lower cap rate mean a higher value?
A fixed deposit paying Rs 10,000 a year is worth Rs 1.25 lakh if savers demand 8%, and Rs 1.43 lakh if they will accept 7%. The cheque has not changed; the price people pay for it has. A cap rate is the yield a buyer requires on a building's income, so value is income divided by the cap rate, and a fall in the required yield raises the price of the same income. Rs 10 crore over 0.08 is Rs 125 crore; over 0.07 it is Rs 142.9 crore. The cap rateCapitalisation rate: net operating income divided by the property value, the income yield a buyer requires. works exactly like a bond yield, and value moves the other way.
With net operating income fixed at Rs 10 crore, a fall in the cap rate from 8% to 7% lifts the building's value from Rs 125 crore to Rs 142.9 crore, a rise of 14.3%, because value is income divided by the cap rate and 8 over 7 is 1.143.
How do you get the percentage without computing either value?
The income cancels. The ratio of new to old value is the old cap rate over the new cap rate, 8 over 7, so the rise is one seventh, 14.3%. The move is not symmetric: the same one point rise, from 8% to 9%, would cut value by 8 over 9 minus 1, about 11.1%. Small cap rates make the swings bigger: from 5% to 4% is a 25% jump in value, from 10% to 9% only 11.1%. Say that pattern after the number, because the follow-up is usually about a different starting rate.
The relationship
$V = \frac{NOI}{c} \qquad \frac{V_{7\%}}{V_{8\%}} = \frac{0.08}{0.07} = 1.143 \Rightarrow +14.3\%$
V the building's value
NOI net operating income, Rs 10 crore a year
c the cap rate, 8% falling to 7%
What it says in wordsValue is income over the cap rate, so the value changes by the ratio of the old cap rate to the new one.
Cap rate Value of Rs 10 crore of NOI, Rs crore Change from the 8% case
6% 166.7 +33.3%
7% 142.9 +14.3%
8% 125.0 +0.0%
9% 111.1 -11.1%
10% 100.0 -20.0%
Each one point step in the cap rate moves the value by a different percentage, larger as the cap rate gets lower, because the value is a fixed income divided by a shrinking rate.
What would a banker add about why cap rates move?
Cap rates track the cost of money and the perceived risk of the rent. A fall from 8% to 7% usually means lower interest rates, more buyers chasing the asset, or more confidence in the tenant, and none of those is something the owner did. That is why a real estate team separates value created by raising income from value handed over by the market, and why an owner who bought at 8% and sells at 7% has earned a 14% gain on cap rate compression alone. Mention the limit: the formula assumes a stable income, so a building with leases about to expire does not deserve the same cap rate as one with ten years of rent locked in.
Where candidates lose it
The common loss is reading a falling cap rate as bad news for the value, because falling sounds like a fall. A cap rate is a yield, and a lower required yield means a higher price for the same income.
The second loss is saying 12.5%, from 1 over 8, or 1%, from the one point move. The value scales by the ratio of the rates, 8 over 7, and saying that ratio aloud is the answer.
What the interviewer asks next
NOI also rises 5% as the cap rate falls to 7%. What is the value now?
Why is a one point move in the cap rate a bigger deal at 5% than at 10%?
If the building was bought with 60% debt at the 8% cap rate, what did the cap rate move do to the equity?
077In your head, and out loud: what is 48 x 52, and what is 97 x 103?Mental maths and countingWarm upBulge bracket IBMiddle market IB
Try it first
Pick the pair.
Show the worked solution
2,496 and 9,991. Both pairs sit the same distance either side of a round number, so use the difference of squares: (a minus b)(a plus b) is a squared minus b squared. 48 x 52 is 50 squared less 2 squared, 2,500 minus 4, 2,496. 97 x 103 is 100 squared less 3 squared, 10,000 minus 9, 9,991. Say the middle number and the gap out loud, then subtract.
Why is the product always a little less than the middle number squared?
Take a square courtyard 50 paces on each side, 2,500 square paces. Cut a strip 2 paces wide off one side and lay it along the bottom. The courtyard is now 48 by 52, but the strip you moved was only 48 long, so a 2 by 2 corner is left bare. Two numbers either side of a round one multiply to the round number squared minus the gap squared, because the strip you move never quite fills the corner. 48 x 52 is 2,500 less 4, which is 2,496; the bare corner is the 4.
A 50 by 50 square with a strip moved from its side to its base becomes 48 by 52 with a 2 by 2 corner missing, so 48 x 52 is 2,500 less 4, 2,496, and the same picture at 100 by 100 with a 3 by 3 corner missing gives 97 x 103 as 10,000 less 9, 9,991.
How do you spot when the trick applies?
Add the two numbers and halve: if the result is round, the trick is on. 48 plus 52 is 100, half is 50, gap 2. 97 plus 103 is 200, half is 100, gap 3. Any pair that straddles a round number is a difference of squares: 73 x 67 is 70 squared less 3 squared, 4,900 minus 9, 4,891. If the pair does not straddle a round number, fall back on splitting one factor: 48 x 53 is 48 x 52 plus 48, 2,496 plus 48, 2,544. The reflex to look for the structure first is the skill the question tests.
The relationship
$(a-b)(a+b) = a^2 - b^2 \qquad 48 \times 52 = 50^2 - 2^2 = 2,496 \qquad 97 \times 103 = 100^2 - 3^2 = 9,991$
a the round number in the middle, 50 or 100
b the gap on each side, 2 or 3
What it says in wordsThe product of two numbers equally spaced around a middle is the middle squared minus the spacing squared.
Why does a banking interview bother with this?
Because the same reflex speeds up every number you say in a meeting. A 4% discount on a Rs 1,040 crore valuation, or a share price of Rs 97 times 103 crore shares, is a difference of squares in disguise, and getting it in two seconds without a calculator buys you credibility. Say your working out loud as you go: middle, gap, square, subtract. The interviewer is listening for the method as much as the answer, and a candidate who mumbles the right number silently earns less than one who narrates the route.
Where candidates lose it
The common loss is adding the small square instead of subtracting it, giving 2,504 and 10,009. The missing corner picture settles the sign: the product is always a little less than the middle squared.
The second loss is reaching for long multiplication and taking twenty seconds for 48 x 52. Check the sum of the two numbers first; if its half is round, the whole thing is one subtraction.
What the interviewer asks next
What is 995 x 1,005?
What is 48 x 53, and how does the trick help even though the pair is not symmetric?
Use the same idea to find 49 squared in your head.
078Estimate how many ATMs India needs. Walk me through the logic, then tell me whether your number is sensible.Estimation and market sizingCoreBulge bracket IBConsulting style brainteasers
Try it first
Which input turns a count of cash users into a count of machines?
Show the worked solution
About 2.7 lakh ATMs, on these assumptions. Take 100 crore adults, 80% with a bank account and half of those using ATMs: 40 crore users. Two withdrawals a month each is 80 crore a month, or 2.67 crore a day. If one machine handles about 100 withdrawals a day, you need roughly 2.67 lakh machines. Every input is an assumption to say out loud, not a fact.
Where does population stop being the useful number?
Think of a petrol pump. Nobody decides how many pumps a town needs by counting residents; they count cars, how often each one fills up, and how many fills one nozzle can manage in a day. ATMs work the same way. Population only tells you demand; capacity per machine is what turns demand into a number of machines. So the estimate has two halves: build daily withdrawals from people, then divide by what one machine does.
The demand half, with round numbers. Start with about 100 crore adults, rounded so the arithmetic stays easy. Say 80% hold a bank account with a debit card, and half of those actually take cash from a machine in a typical month: 40 crore users. Two withdrawals a month each gives 80 crore withdrawals, which over 30 days is about 2.67 crore a day.
Starting from 100 crore adults, the banked and ATM-using shares and two withdrawals a month give 2.67 crore withdrawals a day, and dividing by 100 uses per machine gives about 2.67 lakh ATMs, because one machine is busy only about 3.3 hours of its day.
How many withdrawals can one machine really handle?
A withdrawal takes about two minutes from card in to cash out, so a machine busy all 24 hours could do 720 a day. It never is. Demand bunches around mornings, lunch, evenings and salary days, and sits near zero at three in the morning. Size the machine for its realistic daily use, not its theoretical maximum, because idle night hours are capacity nobody can use. About 100 a day, a little over three hours of real use, is a fair working figure, and the one you should defend if pushed.
The relationship
$N = \frac{A \times b \times u \times w}{30 \times c} = \frac{100\text{ cr} \times 0.8 \times 0.5 \times 2}{30 \times 100} \approx 2.67\text{ lakh}$
A adults, 100 crore
b share with a bank account, 80%
u share of those who use ATMs, 50%
w withdrawals a month per user, 2
c withdrawals one machine handles a day, 100
What it says in wordsMachines needed equal daily withdrawals divided by what one machine serves in a day.
How do you sense-check 2.7 lakh?
Turn the answer into a ratio a person can judge. 2.67 lakh machines for 100 crore adults is about 27 ATMs per lakh adults, or one machine for every 3,750 adults. In a city that feels like a machine every few streets; in a village it feels sparse, which matches deployment leaning toward cities. RBI publishes the actual installed count, so say you would check your figure against the latest data. The second check is direction: digital payments cut cash withdrawals, so the honest answer drifts down over time, not up. If withdrawals fall from two a month to 1.5, the requirement falls to about 2.0 lakh.
Where candidates lose it
The common failure is starting from population and multiplying by an ATMs-per-person figure pulled from the air. That is a guess with arithmetic around it, and the interviewer has no step to follow or challenge.
The second loss is using a machine's theoretical capacity. Seven hundred and twenty uses a day assumes a queue at three in the morning, and it cuts the answer roughly seven times. Say why you picked 100 before anyone asks.
What the interviewer asks next
How would wider UPI use change your estimate over five years?
How would you size the cash a bank must load into these machines each week?
Which single assumption would you research first, and why?
079A company has EBITDA of Rs 100 crore and can borrow at 10%. Lenders require interest cover of at least 2.5x and leverage of no more than 5.0x EBITDA. How much can it borrow, and which test binds?Deal mathsCoreWells Fargo SecuritiesNew York · 2025
Try it first
Before you calculate: how much debt can the company raise?
Show the worked solution
Rs 400 crore, and the coverage test binds. Leverage of 5.0x allows Rs 500 crore. Coverage of 2.5x caps interest at 100 divided by 2.5, which is Rs 40 crore, and at 10% that interest supports Rs 400 crore of debt. The borrower must pass both tests, so the lower number is the ceiling. Below an 8% rate, leverage would bind instead.
Why are there two answers, and why take the smaller?
A bank deciding your home loan asks two things: is the loan too large against your salary, and can you afford the monthly payment? You can pass one and fail the other. Lenders to a company ask the same pair. Leverage limits the size of the debt; coverage limits the cost of carrying it, and the borrower must clear both, so the binding test is whichever gives less. Compute both every time, then name the one that bites.
Leverage is one step: 5.0 times EBITDA of 100 is Rs 500 crore. Coverage works backwards from the payment. Interest coverEBITDA divided by interest expense: how many times over the operating profit pays the interest bill. of at least 2.5 means interest can be at most 100 divided by 2.5, which is Rs 40 crore. At a 10% rate, Rs 40 crore of interest corresponds to Rs 400 crore of debt.
The leverage test allows Rs 500 crore but the coverage test, with interest capped at Rs 40 crore at a 10% rate, allows only Rs 400 crore, so the company can borrow Rs 400 crore and the extra 100 the leverage test permits is unusable.
When would the answer flip to the leverage test?
The coverage ceiling moves with the rate; the leverage ceiling does not. Set the two equal: interest of 40 on 500 of debt is an 8% rate. Below 8%, leverage binds; above it, coverage binds, which is why debt capacity shrinks when rates rise even if earnings do not move. At 7%, coverage would allow about Rs 571 crore, so the 500 leverage cap becomes the limit.
The relationship
$D_{\max} = \min\Big(L \times E,\; \frac{E / C}{r}\Big) = \min(500,\; 400) = 400$
E EBITDA, Rs 100 crore
L maximum leverage, 5.0x
C minimum interest cover, 2.5x
r interest rate, 10%
What it says in wordsDebt capacity is the smaller of the size limit and the debt the maximum affordable interest bill can carry.
Say the limitation too. Real covenants often test cover on EBITDA less capex, or on a fixed charge basis that includes leases, and lenders size a deal with headroom below both ceilings rather than at them. The two-test logic stays the same.
Where candidates lose it
Most candidates multiply 5.0 by 100, say Rs 500 crore and stop, because leverage is the covenant they have read about. The interviewer has deliberately set the rate high enough that coverage bites first.
The other loss is treating the two tests as alternatives to pick from. They are both conditions. Say both numbers, then say which binds and the rate at which it would switch.
What the interviewer asks next
What interest rate makes the two tests give the same answer?
If EBITDA falls 20%, which ceiling falls further?
How would a cash sweep or amortisation change the coverage picture in later years?
Asked at Wells Fargo Securities, Investment Banking, New York, 2025 (Wall Street Oasis): Simple 400 guide questions, walk through DCF, how much leverage is too much leverage, walk through 3 statemets.
080You roll a fair six-sided die repeatedly. On average, how many rolls will it take before every face from 1 to 6 has appeared at least once?Expected value and gamesHardBulge bracket IBConsulting style brainteasers
Try it first
Gut call before you work it: how many rolls on average?
Show the worked solution
About 14.7 rolls. Break the hunt into stages. The first roll always gives a new face. After that a new face comes up with chance 5/6, then 4/6, and so on down to 1/6. The average wait for an event of chance p is 1/p, so the stages take 1, 1.2, 1.5, 2, 3 and 6 rolls, which add to 6 x (1 + 1/2 + ... + 1/6) = 14.7.
Why does breaking it into stages make it easy?
Collecting cricket cards from cereal packets feels quick at first: almost every packet has a new card. The last card is the painful one, because nearly every packet repeats something you already own. The die is the same. Each stage is a simple waiting game with a fixed chance of success, and the average wait for a chance p is 1/p rolls. With k faces already found, a new one turns up with chance (6 minus k)/6, so that stage takes 6/(6 minus k) rolls on average.
Why is the wait 1/p? If a new face comes up one roll in three, you expect to wait three rolls, the same way a bus that comes one minute in ten keeps you waiting about ten. That one fact, applied six times, is the whole puzzle.
The six stages take 1, 1.2, 1.5, 2, 3 and 6 rolls on average, adding to 14.7, and the final face alone accounts for 6 rolls because only one roll in six finds it.
How do you add it up quickly in the room?
The relationship
$E[N] = \frac{6}{6} + \frac{6}{5} + \frac{6}{4} + \frac{6}{3} + \frac{6}{2} + \frac{6}{1} = 6 \sum_{k=1}^{6} \frac{1}{k} \approx 14.7$
E[N] the expected number of rolls to see all six faces
6/(6 minus k) the average wait for a new face once k faces are found
What it says in wordsAdd the average wait for each new face; the waits grow because new faces get rarer.
Pair the fractions to keep the arithmetic out loud: 1 plus 6 is 7, 1.2 plus 3 is 4.2, 1.5 plus 2 is 3.5, total 14.7. The last face costs six rolls on its own, more than the first four stages put together. That shape, cheap early progress and an expensive tail, is worth one sentence because it shows you see the structure, not just the sum.
One limitation: 14.7 is an average. The spread is wide, and plenty of runs take more than 20 rolls, so if the interviewer asks for a number you would bet on for a single run, say the average is not a likely outcome.
Where candidates lose it
The instinct is six, or a vague 10, because candidates picture one roll per face. Saying a number without the stage structure gives the interviewer nothing to follow, and an unexplained guess counts as a miss even if it is close.
The second loss is getting the stages right and fumbling the sum. Pair the terms that make round numbers and say 14.7, not fourteen point something.
What the interviewer asks next
How many rolls on average to see all faces of a 20-sided die?
What is the expected number of rolls until you see a 6?
Roughly how many rolls would you need to be 90% sure of seeing every face?

FCF	final-year free cash flow, Rs 100 crore
g	long-run growth, 4%
WACC	the discount rate, 10% rising to 11%

WACC	Spread over 4% growth	Terminal value, Rs crore	Change from the 10% case
8%	4 points	2,600	+50.0%
9%	5 points	2,080	+20.0%
10%	6 points	1,733	+0.0%
11%	7 points	1,486	-14.3%
12%	8 points	1,300	-25.0%

NOI	net operating income, the cap rate times the value
interest	the loan times its rate, interest-only so no principal
equity	value less the loan, here Rs 40 per Rs 100

Loan rate	Interest on Rs 60	Cash to Rs 40 of equity	Cash-on-cash
5%	3.0	4.0	10%
7%	4.2	2.8	7%
9%	5.4	1.6	4%

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

What is the shortcut, and when does it break?

Why does this matter beyond the arithmetic?

Where candidates lose it

What the interviewer asks next

Try it first

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

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

Why would anyone buy with negative leverage?

Where candidates lose it

What the interviewer asks next

Try it first

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

How do you get the balance without a schedule?

Why does a banker care about a timing difference?

Where candidates lose it

What the interviewer asks next

Try it first

What decides whether a locker ends open or closed?

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

How should you present it in the room?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the equity fall more than the business?

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

Why does a banker ask a tech company this?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a lower cap rate mean a higher value?

How do you get the percentage without computing either value?

What would a banker add about why cap rates move?

Where candidates lose it

What the interviewer asks next

Try it first

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

How do you spot when the trick applies?

Why does a banking interview bother with this?

Where candidates lose it

What the interviewer asks next

Try it first

Where does population stop being the useful number?

How many withdrawals can one machine really handle?

How do you sense-check 2.7 lakh?

Where candidates lose it

What the interviewer asks next

Try it first

Why are there two answers, and why take the smaller?

When would the answer flip to the leverage test?

Where candidates lose it

What the interviewer asks next

Try it first

Why does breaking it into stages make it easy?

How do you add it up quickly in the room?

Where candidates lose it

What the interviewer asks next