Financial Analysis puzzles, solved step by step

Puzzles: 100
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001Two store chains run identical stores and earn the same Rs 100 crore a year before property costs. One owns its stores; the other leases them at Rs 12 crore a year for 10 years. Once the lease is capitalised at 9%, the leaser's EBITDA rises by Rs 12 crore and a lease liability of about Rs 77 crore appears. How do you compare the two on EV/EBITDA fairly?Accounting flow riddlesHardMizuhoNew York · 2026
Try it first
The owner trades at 8.0x. The leaser's shares and debt are worth Rs 723 crore and its EBITDA, rent added back, is Rs 100 crore. Which multiple is the fair one to set beside 8.0x?
Show the worked solution
Put the lease liability inside EV whenever the rent is outside EBITDA. After capitalisation the leaser's EBITDA is Rs 100 crore, like the owner's. Its shares and debt are worth Rs 723 crore, so dividing those alone gives 7.2x and makes it look cheaper. Add the Rs 77 crore lease and it is 8.0x, the owner's multiple. The numerator and denominator must describe the same claims.
Why does a lease behave like debt?
Think of two families in identical flats. One bought with a home loan; the other rents on a ten year agreement it cannot walk away from. Both owe fixed payments for years. A long lease is a loan from the landlord, repaid in rent, so the rent contains both the use of the asset and the financing of it. Under IFRS 16, and Ind AS 116 in India, the leaser now shows that promise as a lease liabilityThe present value of the rent the company is contractually committed to pay over the lease term, carried on the balance sheet like a borrowing.: Rs 12 crore a year for ten years, discounted at 9%, is about Rs 77 crore.
The relationship
$L = 12 \times \frac{1 - 1.09^{-10}}{0.09} = 12 \times 6.418 \approx 77.0$
12 yearly rent, Rs crore
0.09 the discount rate applied to the lease
10 years left on the lease
What it says in wordsThe lease liability is the rent stream discounted back to today, exactly as you would value a loan's repayments.
What changes in the numbers, and what does not?
Capitalisation moves the rent out of operating costs. It comes back as depreciation on a right-of-use asset and interest on the lease, both below EBITDA. So the leaser's EBITDA jumps by the full Rs 12 crore while nothing about its stores, customers or cash has changed. The same move puts about Rs 77 crore of lease on the balance sheet. The two changes are a pair, and a fair multiple has to use both halves or neither.
After capitalisation the leaser's EBITDA rises from Rs 88 crore to Rs 100 crore and a Rs 77 crore lease appears. Dividing only its Rs 723 crore of shares and debt by Rs 100 crore gives 7.2x, while adding the lease to EV gives 8.0x, the same as the owner.
Where does this bite in real comparables work?
Data providers and peer tables do not always treat leases the same way, and a peer set can mix companies reporting under different standards. Before trusting a multiple, check whether its EV includes lease liabilities and whether its EBITDA is before or after rent, then make every company in the table match. A retailer, airline or restaurant chain that leases most of its sites can look 11% cheaper than an owner purely from this mismatch, which is the whole of the gap in this example.
Say the limitation too. The capitalised figure depends on the discount rate and the lease term the company chose, so two leasers with the same rent can carry different liabilities. Lease-adjusted multiples are better, not exact.
Where candidates lose it
The common loss is quoting the leaser at 7.2x and calling it cheap. The candidate has taken the EBITDA uplift from the new standard and forgotten the liability that came with it, so the comparison rewards a company for renting instead of owning.
The second miss is going the other way and deducting rent from one company's EBITDA while leaving the other's untouched. Whichever basis you choose, say it once and apply it to every company in the set.
What the interviewer asks next
Before lease capitalisation, how would you have compared the two chains, and what is EBITDAR?
What happens to the leaser's net income in year 1 compared with the old rent expense?
Does lease capitalisation change the leaser's free cash flow?
How should a DCF treat lease payments if EBITDA already excludes rent?
Asked at Mizuho, Generalist, New York, 2026 (Wall Street Oasis): How does a $10 increase for depreciation Finance lease vs operating lease (which effect valuation)
002Company X trades at 12x earnings but 9x EV/EBITDA. Its peers trade at 15x earnings and 7x EV/EBITDA. Give two reasons, with numbers, that make both facts true at once.Valuation and multiples riddlesHardBarclaysNew York · 2026
Try it first
Which single fact could, on its own, push X's P/E down and its EV/EBITDA up at the same time?
Show the worked solution
X carries more debt and owns a stake in an associate. Debt of Rs 750 crore at 8% costs 5.6% after tax, less than the 6.7% its operations earn on their value, so levering lowers the P/E. The associate adds Rs 22 crore to net income but nothing to EBITDA, while its Rs 300 crore value sits inside EV. Strip it out and X is 7.0x, like its peers.
Why can two multiples on the same company disagree?
Picture a house with a mortgage. Its price compared with the rent it earns is one ratio; your equity in it compared with the rent left after the mortgage payment is another. The two only agree if there is no loan. EV/EBITDA looks at the whole business before financing, while P/E looks at the shareholders' slice after interest, tax and anything non-operating. Every gap between them is explained by something that sits between EBITDA and net income, or between EV and equity value.
What numbers make both facts true?
Give both companies the same operations: EBITDA Rs 150 crore, depreciation Rs 50 crore, operating profit Rs 100 crore, tax 30%. The peer has no debt, earns Rs 70 crore and is worth Rs 1,050 crore: 15.0x earnings and 7.0x EBITDA. X borrows Rs 750 crore at 8%, so interest of Rs 60 crore leaves Rs 28 crore from operations, and it books Rs 22 crore as its share of an associateA company in which the group holds a significant minority stake, usually 20% to 50%. The group books its share of that company profit below operating profit, never in revenue or EBITDA.'s profit. Net income of Rs 50 crore at 12x is equity of Rs 600 crore, and with the debt that is an EV of Rs 1,350 crore, 9.0x EBITDA.
Company X's EV of Rs 1,350 crore is funded by Rs 600 crore of equity and Rs 750 crore of debt, and pays for the same Rs 1,050 crore of core operations as the peer plus a Rs 300 crore associate stake. Taking the stake out brings X back to 7.0x EBITDA.
Which reason moves which multiple?
Separate them, because the follow-up always asks. Debt lowers the P/E when its after-tax cost is below the earnings yield of the operations. Here debt costs 8% times 0.7, which is 5.6%, while the operations earn 70 on 1,050, or 6.7%. Swapping expensive equity for cheaper debt leaves the operating slice at about 10.7x earnings. The associate does the other job: its Rs 300 crore of value sits inside EV while its profit sits below EBITDA, which lifts EV/EBITDA from 7.0x to 9.0x.
Name a third candidate if you have time: a lower tax rate than peers raises net income without touching EBITDA, so it also lowers P/E alone. Say what you would check to choose between them: the notes on debt, associates and the effective tax rate.
Where candidates lose it
Most candidates say X must have more debt and stop. Debt alone explains the lower P/E, but it does not raise EV/EBITDA if the business is worth the same, because EV is the same whoever funds it. The interviewer is waiting for something that sits inside EV but outside EBITDA.
The other loss is giving reasons with no numbers. Build one small example in which both multiples land where the question says; it proves the reasons work together.
What the interviewer asks next
Would you subtract the associate stake from X's EV in a comps table, and at what value?
If X's debt cost 11% instead of 8%, would its P/E still be below its peers'?
Which of the two multiples would you use to value X, and why?
Asked at Barclays, Investment Banking, New York, 2026 (Wall Street Oasis): A company is trading at a lower P/E but a higher EV/EBITDA than peers
003A company earns a 20% after-tax return on the capital it invests and wants operating profit to grow 15% a year. What share of operating profit must it reinvest each year? What changes if the return on capital is 10%?Ratio and margin riddlesHardEquity researchCorporate finance
Try it first
At a 20% return on capital, how much of each year's operating profit has to go back into the business to grow 15%?
Show the worked solution
It must reinvest 75% of operating profit at a 20% return, and 150% at a 10% return. Growth equals return on capital times the share of profit reinvested, so the reinvestment rate is 15% divided by the return. At 20% that leaves Rs 25 of every Rs 100 free. At 10% the company must invest Rs 150 for every Rs 100 it earns, raising Rs 50 from lenders or shareholders to keep growing.
Where does growth in operating profit come from?
Picture a tailor who earns 20% a year on every rupee of sewing machines she owns. If she wants next year's profit 15% higher, she needs 15% more machines' worth of profit, and each rupee of machines earns only 20 paise. Growth equals the return on new capital multiplied by the share of profit put back in. That share is the reinvestment rate, and it is the price the company pays today for tomorrow's growth.
The relationship
$g = \text{ROC} \times b \quad\Rightarrow\quad b = \frac{g}{\text{ROC}} = \frac{15\%}{20\%} = 75\%$
g growth in operating profit, 15% a year
ROC after-tax return on the new capital invested
b the share of after-tax operating profit reinvested
What it says in wordsDivide the growth you want by the return you earn, and that is the share of profit you must plough back.
To grow operating profit 15% a year, a company earning 30% on capital reinvests 50% of its profit, one earning 20% reinvests 75%, one earning 15% reinvests all of it, and one earning 10% must invest 150%, raising the gap from outside.
What does the 10% case tell an analyst?
Below a 15% return, 15% growth cannot be paid for out of profit at all. At 10%, for every Rs 100 earned the company must invest Rs 150, so Rs 50 comes from new debt or new shares every year and free cash flow is negative. The table shows how the free cash left for owners changes with the return on capital.
Return on capital Reinvestment rate Free cash per Rs 100 of profit
30% 50% 50
20% 75% 25
15% 100% 0
10% 150% (50)
Same 15% growth in every row. Free cash is operating profit less reinvestment; a bracket means cash must be raised from outside.
Growth at a low return can still be worth having if the return beats the cost of capital, and it destroys value if it does not. Say the limitation: the formula assumes new capital earns what old capital earns. Price increases and efficiency gains are growth that needs no reinvestment, so real companies sometimes beat it.
Where candidates lose it
The quick wrong answer is 15%, treating the growth rate as the share you reinvest. That ignores the return: the same growth costs very different amounts of capital at 10% and at 30%.
The second loss is stopping at 150% without saying what it means. A reinvestment rate above 100% is a funding need, and the interviewer wants to hear that growth at a low return on capital burns cash.
What the interviewer asks next
At a 12% cost of capital, is 15% growth at a 10% return good or bad for shareholders?
How would you estimate return on capital for a company from its annual report?
What happens to the valuation if growth falls to 5% with the return held at 20%?
007You have a biased coin that lands heads one third of the time. How can you use it to produce a fair 50:50 result, and how many flips of the biased coin does each fair result take on average?Probability and expected valueHardD.E. ShawNew York · 2026
Try it first
Flip in pairs, keep heads-tails and tails-heads, discard the rest. On average, how many single flips does one fair result take?
Show the worked solution
Flip twice: heads then tails counts as heads, tails then heads counts as tails, and anything else is thrown away and flipped again. The two mixed orders each have probability 2/9, so they are equally likely whatever the bias. A pair succeeds 4/9 of the time, so each fair result takes 9/4 pairs, or 4.5 flips. Knowing the bias is exactly 1/3 lets you cut that to 2.25.
Why are heads-tails and tails-heads always equally likely?
Picture two friends flipping the same lopsided coin, one after the other. The chance the first gets heads and the second tails is the heads chance times the tails chance. The chance of the reverse is the tails chance times the heads chance. Multiplication does not care about order, so the two mixed outcomes are exactly equally likely, whatever the bias. That symmetry is the whole trick, known as the von Neumann method. Here each mixed pair has probability 1/3 times 2/3, which is 2/9.
Flipping the biased coin twice gives heads-tails and tails-heads with probability 2/9 each, so calling one heads and the other tails is fair. Discarding the matching pairs means a pair works 4/9 of the time, which costs 4.5 flips per fair result on average.
How do you get the average of 4.5 flips?
Each pair either works or does not, independently of the last. Waiting for a success that happens with probability q takes 1/q tries on average, the same reason a die takes six rolls on average to show a six. A pair works with probability 4/9, so you need 9/4 pairs, and two flips a pair makes 4.5 flips. The method pays for its fairness with waste: 5 pairs in 9 are thrown away.
The relationship
$E[\text{flips}] = \frac{2}{P(HT)+P(TH)} = \frac{2}{2p(1-p)} = \frac{2}{4/9} = 4.5$
p the chance of heads on one flip, 1/3
2p(1-p) the chance a pair is mixed, 4/9
2 flips used by each pair
What it says in wordsDivide the flips per attempt by the chance an attempt succeeds.
Can you do better if you know the bias exactly?
Yes, and this is usually the follow-up. With p exactly 1/3, tails-tails has probability 4/9, the same as the two mixed pairs together. Call tails-tails one side and either mixed pair the other, and only heads-heads, 1/9 of pairs, is wasted, so each fair result costs 2 times 9/8, or 2.25 flips. The von Neumann method is still the better answer when nobody tells you the bias, because it works for any p. The limit for any scheme is set by how much randomness one flip carries: about 0.92 of a fair bit here, so no method can beat roughly 1.09 flips per fair result on average.
Where candidates lose it
The common loss is trying to build fairness from single flips, for example calling heads on one flip and tails on two in a row. Those schemes depend on the exact bias and usually fail the moment you write out the probabilities.
The second loss is giving the method and not the cost. The interviewer reported here went straight on to efficiency, so have 4.5 flips ready, then say why the known-bias grouping halves it and why the order trick is still the safe answer.
What the interviewer asks next
Your fair-result method uses 4.5 flips. How could you reuse the discarded heads-heads and tails-tails pairs to get more fair results from the same flips?
How would you simulate a fair six-sided die with this coin?
If the coin's bias is unknown and drifts slowly over time, does the pair method still work?
Asked at D.E. Shaw, Research, New York, 2026 (Wall Street Oasis): How can I make an effective fair coin given a biased coin with p_heads = 1/3?
008A company's D&A is Rs 80 crore and its capex Rs 200 crore. Revenue is Rs 1,600 crore and growing 10% a year, and fixed asset turnover is 2.0x. Estimate how much of the capex is growth capex and how much is maintenance capex.Working capital and cash riddlesHardEquity researchCorporate FP&A
Try it first
How much of the Rs 200 crore is maintenance capex?
Show the worked solution
Growth capex is about Rs 80 crore and maintenance capex about Rs 120 crore. Revenue grows by Rs 160 crore, and at a fixed asset turnover of 2.0x each rupee of new revenue needs 50 paise of new assets, so growth capex is Rs 80 crore. The rest, Rs 120 crore, replaces worn assets. That is Rs 40 crore more than D&A, because D&A records assets at the older prices paid for them.
Why not just say maintenance capex equals D&A?
Think of a taxi owner who bought a car eight years ago for Rs 6 lakh and has been setting aside Rs 75,000 a year as depreciation. When the car dies, the same model costs Rs 9 lakh, not Rs 6 lakh. D&A spreads the price paid for old assets, while maintenance capex pays today's price to replace them, so with any inflation the two drift apart. D&A is a reasonable floor for maintenance spending, not an estimate of it.
How does fixed asset turnover split the capex?
Fixed asset turnoverRevenue divided by net fixed assets. At 2.0x, each rupee of plant and equipment supports two rupees of yearly revenue. tells you how much plant each rupee of revenue needs. At 2.0x, the company's Rs 1,600 crore of revenue sits on about Rs 800 crore of net fixed assets. If the new revenue needs assets at the same ratio, the Rs 160 crore of growth needs Rs 80 crore of new capacity, and that is growth capex. Whatever is left of the Rs 200 crore went on keeping the existing Rs 1,600 crore of revenue alive.
The relationship
$\text{growth capex} = \frac{\Delta \text{revenue}}{\text{FAT}} = \frac{160}{2.0} = 80 \qquad \text{maintenance} = 200 - 80 = 120$
Δ revenue next year's extra revenue, 10% of 1,600
FAT fixed asset turnover, revenue over net fixed assets
What it says in wordsNew capacity is the new revenue divided by how much revenue each rupee of assets supports; the rest of capex is upkeep.
Treating D&A as maintenance splits the Rs 200 crore into 80 of maintenance and 120 of growth. The turnover method gives the opposite split, 120 of maintenance and 80 of growth, so maintenance runs Rs 40 crore above D&A.
Is Rs 120 crore believable, and what is the catch?
Check it against inflation. If the average asset was bought about eight years ago and equipment prices rose 5% a year, replacing it costs 1.05 to the power 8, about 1.48 times its original price. Rs 80 crore of D&A at today's prices is about Rs 118 crore, close to the Rs 120 crore estimate, so the split hangs together. The catch is that turnover is measured on net book value, which is itself at old prices. New capacity bought at today's prices may need more than 50 paise per rupee of revenue, which would make growth capex larger and maintenance smaller. Give the estimate as a range, and say why it matters: free cash flow before growth spending is Rs 40 crore lower than the D&A shortcut suggests.
Where candidates lose it
The common loss is the shortcut: maintenance equals D&A, so growth capex is 200 less 80, which is 120. It gives exactly the reverse of the turnover answer and overstates how much cash the business could release if it stopped growing.
The second loss is giving 120 and stopping. Say why D&A understates replacement cost, then name the weakness in your own method, the book-value turnover, before the interviewer does.
What the interviewer asks next
If growth stopped tomorrow, how much free cash flow would the business release each year?
How would you estimate maintenance capex from five years of the company's own history?
Why might a company with ageing assets report rising margins while its true earnings power falls?
012A supplier lists a part at Rs 100. It offers two schemes: 5% off every unit if you buy 1,000 or more, or 10% off only the units above 1,000. At what volume do the two schemes cost the same? And why might a buyer who needs 990 units order 1,000?Pricing, costing and unit economicsHardCorporate FP&ACost accounting
Try it first
At what order size do the two schemes cost exactly the same?
Show the worked solution
The schemes cost the same at 2,000 units, Rs 1,90,000 each. The all-units deal costs 95 times the quantity; the incremental deal costs Rs 1,00,000 for the first 1,000 and Rs 90 after that, which equals 95Q only at 2,000. Below that, all-units is cheaper. It also creates a cliff: 990 units cost Rs 99,000 but 1,000 cost Rs 95,000, so anyone ordering more than 950 should round up to 1,000.
How do you set the two schemes side by side?
Think of two mobile data plans, one with a flat lower price on everything once you cross a usage level and one that only discounts the extra data. They feel similar, but they reward different amounts of use. Write each scheme as total cost against quantity, then find where the two formulas are equal. All-units: 95Q once Q reaches 1,000. Incremental: 1,00,000 for the first 1,000 units, plus 90 for each unit beyond.
The relationship
$95Q = 100{,}000 + 90(Q - 1{,}000) \;\Rightarrow\; 5Q = 10{,}000 \;\Rightarrow\; Q = 2{,}000$
Q units ordered, at least 1,000
95 all-units price after 5% off
90 incremental price on units above 1,000
What it says in wordsThe incremental deal saves Rs 5 a unit more on every unit above 1,000, and needs 2,000 units in total to make up the Rs 5,000 head start the all-units deal gives at 1,000.
A quicker way to see it: at 1,000 units the all-units scheme is Rs 5,000 cheaper. Each extra unit costs Rs 95 under all-units and Rs 90 under incremental, so the incremental deal claws back Rs 5 a unit. It needs another 1,000 units to recover Rs 5,000, so the schemes meet at 2,000. Above that the incremental scheme wins: at 3,000 units it costs Rs 2,80,000 against Rs 2,85,000.
The all-units scheme drops the average price from Rs 100 to Rs 95 the moment an order reaches 1,000 units, while the incremental scheme lowers it gradually. The incremental average only reaches Rs 95 at 2,000 units, which is where the two schemes cost the same.
Why would anyone order more than they need?
Because the all-units scheme makes buying more cost less. 990 units at Rs 100 cost Rs 99,000, while 1,000 units at Rs 95 cost Rs 95,000, so ten extra parts arrive with Rs 4,000 back in your pocket. The same logic holds for any order above 950 units, because 1,000 units cost Rs 95,000 and anything from 951 to 999 costs more. A buyer with somewhere to store the spares should always round up.
Under the all-units scheme, 990 units cost Rs 99,000 but 1,000 units cost only Rs 95,000. Any order between 951 and 999 units costs more than rounding up to 1,000, which is the cliff an all-units discount creates.
Say the limit from both sides of the deal. Extra units carry storage, handling and the risk they are never used, so the saving is only real if the spares have a use. For the supplier, the cliff means giving away Rs 4,000 on a sale that would have happened anyway, which is why incremental schemes are common where buyers order close to a threshold.
Where candidates lose it
The common loss is answering 1,000, the point where the discounts start, instead of finding where the costs meet. Writing both costs as formulas takes ten seconds and makes 2,000 obvious.
The second loss is missing the cliff. Interviewers add the 990 question to see whether you notice that an all-units discount makes total cost fall as quantity rises, which is the opposite of what a cost curve normally does.
What the interviewer asks next
What is the smallest order at which rounding up to 1,000 saves money?
If storing each spare part costs Rs 3 a year, does the 990 buyer still round up?
Which scheme would you offer as the supplier, and why?
018What is the beta of a slot machine that pays back Rs 92 on average for every Rs 100 staked? And why does its expected return not match what the capital asset pricing model would give an asset with that beta?Cost of capital, leverage and ratesHardRothschild & CoNew York · 2021
Try it first
What is the slot machine's beta?
Show the worked solution
Its beta is zero, yet its expected return is minus 8%. Beta measures movement with the market, and a slot machine's payouts are random and unrelated to the market. CAPM would give a zero-beta asset the risk-free rate, say 7% a year, so the machine falls short by at least 15 points. There is no contradiction: CAPM prices assets bought as investments, and a slot machine is bought as entertainment.
How can something so risky have a beta of zero?
Think of an umbrella seller and an ice cream seller in the same town. Each has a volatile income, but whether it rains has nothing to do with the stock market. BetaHow much an asset tends to move when the market moves, measured as its covariance with the market divided by the variance of the market. measures how an asset moves with the market, not how much it moves, so a gamble driven by a random number generator has a beta of zero. The slot machine is about as volatile as anything in a town, but all of that risk is the kind a diversified owner can spread away, and in this case the casino does exactly that across thousands of players.
Slot machine sessions plotted against the market's return show no slope, so the fitted beta is zero and the average session loses 8%. CAPM gives a zero-beta asset the risk-free rate, so the slot machine falls well short of what its beta alone would predict.
Why does CAPM not give it the risk-free rate?
CAPM says expected return equals the risk-free rate plus beta times the equity risk premium. With beta of zero that is just the risk-free rate, 7% a year in this example. The machine instead returns minus 8% on every stake, and a stake lasts seconds, so over a year of play the gap is far wider than the 15 points the two headline numbers suggest. In CAPM language that is a large negative alphaThe return an asset earns above or below what its beta implies under CAPM..
The relationship
$E[r] = r_f + \beta\,(E[r_m] - r_f) = 7\% + 0 \times 6\% = 7\% \qquad \text{slot}: \frac{92 - 100}{100} = -8\%$
r_f risk-free rate, 7% a year in this example
β the slot machine's beta, zero
E[r_m] - r_f equity risk premium, 6%
What it says in wordsCAPM gives a zero-beta asset the risk-free rate; the slot machine returns 92 for every 100 staked.
So is CAPM wrong?
No, it is answering a different question. CAPM describes the prices of assets that diversified investors hold to earn a return, where anyone could sell an overpriced asset short. Nobody plays a slot machine for return; players pay 8% of each stake for entertainment, the way a cinema ticket has a negative return. And you cannot short a single slot machine to collect the edge. The only way to take the other side is to own the casino, which needs licences, buildings and capital, and the casino's return on that capital is what an investor would compare with CAPM. Say that and the interviewer hears that you know where a model applies, not just its formula.
Where candidates lose it
The common loss is saying the beta is high because a slot machine is risky. That confuses total risk with market risk, which is the exact distinction CAPM is built on.
The second loss is answering zero and then claiming the machine should earn the risk-free rate, or that CAPM is broken. Close with why the model does not apply: a consumption good with no way to short it is outside the model's world.
What the interviewer asks next
What is the beta of the casino company's shares, and why is it not zero?
Can you think of an investment asset with a beta below zero? What return would CAPM give it?
What is your own personal beta, if your salary depends on the stock market?
Asked at Rothschild & Co, Mergers and Acquisitions, New York, 2021 (Wall Street Oasis): what is the beta of a slot machine?
026A company trades at 15x earnings and 6x EBITDA. Market cap is Rs 300 crore, net debt is Rs 180 crore, interest expense is Rs 18 crore and depreciation and amortisation is Rs 30 crore. What is the implied effective tax rate, and what does it suggest about the company?Valuation and multiples riddlesHardEquity researchTransaction advisory
Try it first
Before you calculate: which two numbers must you find first?
Show the worked solution
The implied tax rate is 37.5%. Market cap of 300 at 15x earnings gives net income of 20. Adding net debt of 180 gives EV of 480, so EBITDA at 6x is 80. Less D&A of 30 is EBIT of 50; less interest of 18 is pre-tax profit of 32. Tax is 32 minus 20, which is 12, and 12 over 32 is 37.5%, high enough to ask why.
Where do you start when six numbers arrive at once?
Think of working out a friend's take-home pay from what they spend and what they save. You do not start with their job title; you start with the two numbers that pin the answer. A tax rate is tax divided by pre-tax profit, so every other number in the question is a road to net income or to pre-tax profit. Say that first. Net income is one step: 300 over 15 is 20. Pre-tax profit needs EBITDA, which needs EV, which needs the net debt you were handed.
Market cap of 300 at 15x gives net income of 20, and adding net debt of 180 gives EV of 480 and EBITDA of 80 at 6x. EBITDA less D&A of 30 and interest of 18 leaves pre-tax profit of 32, so a tax charge of 12 is a 37.5% rate.
The relationship
$t = \frac{\text{PBT} - \text{NI}}{\text{PBT}} = \frac{(480/6 - 30 - 18) - 300/15}{480/6 - 30 - 18} = \frac{12}{32} = 37.5\%$
PBT pre-tax profit: EBITDA less D&A less interest
NI net income: market cap over the P/E
480 enterprise value: market cap 300 plus net debt 180
What it says in wordsBuild pre-tax profit from the EV multiple, build net income from the P/E, and the tax rate is the share of pre-tax profit that did not survive.
What does a 37.5% rate tell you, and what have you assumed?
You assumed there are no minority interests, no associates and no interest income on cash, so EV is just market cap plus net debt and every rupee of pre-tax profit belongs to shareholders. Say that. Then read the number. An effective rate well above the statutory rate usually means some costs are not tax deductible, some losses sit in units that cannot use them, or there are one-off tax charges. Compare it with the statutory rate the company actually pays, which you should confirm for the year in question. A second check: interest of 18 on net debt of 180 is 10%, which is plausible, so the inputs hang together.
Where candidates lose it
Candidates reach EBITDA of 80, subtract tax from somewhere and forget one of the two lines between EBITDA and pre-tax profit, usually D&A. Walk the income statement in order, one line per step, and the missing line has nowhere to hide.
The second loss is stopping at 37.5%. The interviewer asked what it suggests. One sentence on non-deductible costs or loss-making units turns arithmetic into analysis.
What the interviewer asks next
If the company also held Rs 40 crore of cash earning 5%, how does the implied rate change?
Which items typically push an effective tax rate above the statutory rate?
What happens to the implied rate if the P/E rises to 20x with everything else fixed?
028The correlation between X and Y is 0.2 and the correlation between Y and Z is 0.5. What range of values can the correlation between X and Z take?Data and statistics intuitionHardTower Research CapitalNew York · 2019
Try it first
Pick the range before any algebra.
Show the worked solution
Anywhere from about -0.75 to 0.95. Read each correlation as the cosine of an angle between two arrows. X is 78.5 degrees from Y, and Z is 60 degrees from Y. Z can swing to the same side as X, leaving them 18.5 degrees apart, or to the other side, 138.5 degrees apart. The cosines of those angles are 0.9485 and -0.7485.
Why can a correlation be drawn as an angle?
Think of three people walking away from the same lamp post. Knowing how far apart the first two point, and how far apart the second and third point, limits how far apart the first and third can point, but only loosely. If you standardise each variable, its correlation with another is the cosine of the angle between them, so correlations must behave like angles in space. A correlation of 0.2 is an angle of 78.5 degrees; 0.5 is 60 degrees.
X sits 78.5 degrees from Y and Z sits 60 degrees from Y, so the angle between X and Z runs from 18.5 to 138.5 degrees, which puts their correlation anywhere from -0.75 to 0.95; two correlations of 0.9 would pin it much tighter, from 0.62 to 1.
The relationship
$\rho_{XZ} \in \rho_{XY}\rho_{YZ} \pm \sqrt{(1-\rho_{XY}^2)(1-\rho_{YZ}^2)} = 0.10 \pm 0.849$
rho XY, rho YZ the two known correlations, 0.2 and 0.5
square root how much room the weak links leave, the product of the two sines
What it says in wordsThe range is centred on the product of the known correlations and is as wide as the product of how far each is from perfect.
Where does the formula come from, and when does it bite?
The three-by-three correlation matrix must not imply a negative variance for any mix of X, Y and Z, which means its determinant cannot go below zero. Solving that condition gives the formula above, the cosine rule for the difference and sum of two angles. The constraint is loose when the known correlations are weak and tight when they are strong. With 0.2 and 0.5, almost the whole scale stays open. With 0.9 and 0.9, X and Z must correlate at least 0.62: two things each closely tied to Y cannot drift far from each other. Say the practical use: a risk model that fills in missing correlations by hand can break this rule and produce a matrix that is not valid.
Where candidates lose it
The fast wrong answer is 0.10, the product, as if correlation passed along a chain. The product is the centre of the range, not the answer. The other wrong answer is that nothing can be said, which ignores that angles must fit together.
Candidates who know the formula often cannot say why it holds. Draw the arrows, say cosine of an angle, and the formula follows in one line.
What the interviewer asks next
If X and Y have correlation 0.9 and Y and Z 0.9, what is the minimum correlation of X and Z?
Can three variables all have pairwise correlation of -0.6?
Why might a hand-edited correlation matrix in a risk model fail to invert?
Asked at Tower Research Capital, Prop Trading, New York, 2019 (Wall Street Oasis): What if the correlation between X and Y is 0.2 and the correlation between Y and Z is 0.5.
029A bowl holds 100 noodles. You pick two free ends at random and tie them together, and keep doing this until no free ends are left. What is the expected number of loops in the bowl at the end?Probability and expected valueHardConsulting-style caseKPO research support
Try it first
Before you compute: roughly how many loops do you expect from 100 noodles?
Show the worked solution
About 3.28 loops. Every tie cuts the number of strands by one, so there are exactly 100 ties. With k strands left, the second end you grab is one of 2k - 1 free ends, and only one of those closes a loop, so that tie adds 1 over (2k - 1) to the expected count. Summing 1 + 1/3 + 1/5 + ... + 1/199 gives 3.2843.
How do you avoid tracking the whole bowl?
Think of shaking hands at a party where each person has two hands. You do not need the seating plan to know that each handshake joins two hands. Whatever the bowl looks like, every tie reduces the count of strands by exactly one, and a strand is either an open piece or a closed loop that has left the game. So you can ignore the tangled history and ask one question per tie: does this tie close a loop, yes or no?
Pick any free end first; which one does not matter. With k strands there are 2k free ends, so 2k - 1 remain for the second pick, and exactly one of them is the other end of the same strand. That tie closes a loop with chance 1 over (2k - 1). Otherwise two strands fuse into one longer strand.
Early ties almost never close a loop: with 100 strands the chance is 1 in 199, and the expected count is only 1.15 after 90 ties. The last three ties add 0.2, 0.33 and 1, bringing the total to 3.28 loops.
Why can you add the step expectations when the steps depend on each other?
Whether an early tie closes a loop changes nothing about how many strands remain, because every tie removes one strand either way. Expected values add even when the events are linked, so the total is simply the sum of the per-tie chances. This is linearity of expectation, and it is what makes the puzzle a two-minute answer instead of a simulation. For large n the sum is close to half the natural log of n plus about 0.98; for 100 noodles that gives 3.28, a useful sanity check.
The relationship
$E[\text{loops}] = \sum_{k=1}^{n} \frac{1}{2k-1} = 1 + \tfrac13 + \tfrac15 + \cdots + \tfrac{1}{199} \approx 3.28$
n the number of noodles, 100
k strands left before a tie
1/(2k-1) the chance the second end chosen belongs to the same strand
What it says in wordsAdd, tie by tie, the chance that the tie closes a loop.
Where candidates lose it
Candidates try to picture how the strands grow and get lost in cases. The problem is only tractable when you notice that each tie does exactly one of two things and that the strand count falls by one either way.
The second loss is getting the chance wrong as 1 over 2k. You have already picked one end, so the pool for the second pick is 2k - 1, not 2k.
What the interviewer asks next
With just two noodles, what is the chance of ending with two loops?
Roughly how many loops would you expect from 10,000 noodles?
Where else does linearity of expectation save you from tracking dependence?

12	yearly rent, Rs crore
0.09	the discount rate applied to the lease
10	years left on the lease

g	growth in operating profit, 15% a year
ROC	after-tax return on the new capital invested
b	the share of after-tax operating profit reinvested

Return on capital	Reinvestment rate	Free cash per Rs 100 of profit
30%	50%	50
20%	75%	25
15%	100%	0
10%	150%	(50)

p	the chance of heads on one flip, 1/3
2p(1-p)	the chance a pair is mixed, 4/9
2	flips used by each pair

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