Investment Banking puzzles, solved step by step

EBITDA rises to Rs 120 crore, enterprise value to Rs 950 crore, and EV/EBITDA eases from 8.0x to about 7.9x. Rent leaves operating costs and comes back as depreciation and interest, which sit below EBITDA, so EBITDA gains the full Rs 20 crore. The lease liability is debt-like, so it joins EV. The multiple barely moves because both sides change together, and it slips only because the lease enters at 7.5x.

Why does EBITDA rise when nothing about the business changed?

Think of a family that rents its flat. If you decide to treat the flat as bought with a loan, the monthly rent disappears from the household budget and reappears as loan interest plus wear and tear on the flat. The family pays the same cash; only the labels moved. Under the new standard, rent leaves operating costs and returns as depreciation and interest, both below EBITDA, so EBITDA rises by the whole Rs 20 crore.

Why must enterprise value move too?

Enterprise value has to match the earnings line it is divided into. Once EBITDA no longer bears the rent, the claim that rent used to pay for, the lease liabilityThe present value of the rent a company has committed to pay over the lease term, carried on the balance sheet like a loan., must be counted like debt and added to EV, taking it from Rs 800 crore to Rs 950 crore. Move one side without the other and the multiple jumps to 9.5x or collapses to 6.7x, though the business is exactly the same. Those two numbers are what candidates produce when they mix bases.

What does this mean for a comps table?

Some peers report under standards that capitalise leases and some under rules that leave rent in operating costs. A comps table means something only if every company sits on the same basis, either all before lease capitalisation or all after. For a retailer or an airline with large leases the gap is far bigger than here, so the adjustment can move the valuation more than any view on growth. Indian companies report under an equivalent standard; confirm which rule each peer follows before you build the table.

The equity is worth 30 and the debt 70. If assets reach 160, lenders get 100 and shareholders 60; if assets fall to 40, lenders take all 40 and shareholders get nothing, but never less than nothing. Half of 60 is 30 for equity; half of 100 plus half of 40 is 70 for debt. Together they equal the 100 the assets are worth. Limited liability makes equity a call option on the assets.

Why is the equity worth anything when assets only cover the debt?

Think of a student who borrows to start a food stall and can walk away from the loan if the stall fails. In a good year the student keeps everything above the loan; in a bad year the lender keeps the stall and the student loses nothing more. Because shareholders can lose at most what they put in but keep every rupee above the debt, equity is a call optionThe right, but not the obligation, to buy an asset at a fixed price. It pays the amount by which the asset ends above that price, or nothing. on the assets with a strike price equal to the debt. An option has value even when it sits exactly at the money, which is where this company stands today.

How do you price the two claims?

With zero interest rates and risk-neutral pricing, each claim is worth its average payoff. Equity pays 60 or 0 and is worth 30; debt pays 100 or 40 and is worth 70; the two add back to the asset value of 100. The debt trades at 70 for a promise of 100, a yield of about 42.9%, which is how a market prices distress.

What happens if the company takes more risk?

Widen the outcomes to 190 or 10, with the same average of 100. Equity now pays 90 or 0 and is worth 45; debt pays 100 or 10 and is worth 55: extra risk moves 15 of value from lenders to shareholders without the company being worth a rupee more. That is why lenders to weak companies write covenants against new risky projects, and why restructuring bankers ask who gains from each option the board is weighing.

Under the fixed ratio, target holders receive Rs 90 of value per share instead of Rs 100. They still get 0.5 acquirer shares, but each is now worth Rs 180. Under a fixed price they would still receive Rs 100, and the acquirer would issue 100 over 180, about 0.556 shares per target share, 11.1% more than planned. A fixed ratio shares the price risk with the target; a fixed price keeps it with the acquirer and its shareholders.

What does each structure actually fix?

Think of agreeing to swap your car for 50 grams of gold, rather than for Rs 5 lakh paid in gold. If the gold price falls before the swap, the first deal still hands you 50 grams, now worth less; the second hands you more grams so that you still get Rs 5 lakh. A fixed exchange ratioThe number of acquirer shares paid for each target share in a stock deal. fixes the number of shares the target receives; a fixed price fixes their rupee value, and the share count moves instead. Whichever number floats carries the risk of the acquirer's share price between signing and closing.

How do the two structures play out across the whole deal?

Assume the target has 10 crore shares and the acquirer 50 crore. Fixed ratio: target holders receive Rs 900 crore of value instead of Rs 1,000 crore, and own 5 crore of 55 crore shares, 9.1% of the combined company. Fixed price: they still receive Rs 1,000 crore, so the acquirer issues 5.56 crore shares, 11.1% more than planned, and the target's holders own 10.0%. The acquirer's existing shareholders give up more of the company to pay the same rupees.

Which side wants which, and how do bankers split the difference?

A target worried that the acquirer's shares will fall wants a fixed price; an acquirer worried about issuing too many shares wants a fixed ratio. A collar is the usual compromise: in one common form the ratio is fixed while the acquirer's price stays inside a band and the value is fixed outside it, so each side carries the risk only up to a point. Interviewers ask this to see whether you think about who carries risk between signing and closing, not just the headline value.

Number the bottles 1 to 1,000, write each number in 10-digit binary, and have tester k sip from every bottle whose kth digit is a 1. After 24 hours, write a 1 for each tester who is ill and a 0 for each who is not. That 10-digit binary number is the poisoned bottle. It works because 10 yes-or-no results can tell apart 2 to the 10th, 1,024 cases, which is more than 1,000.

Why does giving each tester a separate batch fail?

Picture 10 friends each tasting their own 100 bottles. One friend falls ill and you know the poison is among their 100, but not which one, and the day is over. When testers taste separate batches, each result answers one question about one batch, so 10 testers can single out at most 10 groups, not 1,000 bottles. The fix is to let testers overlap, so that every bottle is sipped by its own unique combination of testers.

How does the binary code name the bottle?

Give each tester a place value: tester 1 is worth 512, tester 2 is worth 256, and so on down to tester 10, worth 1. Bottle 357 equals 256 + 64 + 32 + 4 + 1, so testers 2, 4, 5, 8 and 10 sip from it. If exactly those five fall ill, adding their place values gives back 357, and no other bottle has the same set of tasters. Every bottle number is a different set of testers, so the pattern of illness points to one bottle only.

What is the general lesson?

Each tester is one yes-or-no question asked of every bottle at once. Ten yes-or-no answers can tell apart 1,024 possibilities, so the right design makes each answer carry one binary digit of the bottle's number. The same idea finds one faulty item among many with very few tests, and it is why 2,000 bottles need only one more tester. With two days, each tester has three outcomes, ill on day one, ill on day two, or never, and 7 testers would be enough.

About 8% a year, forever. In a dividend discount model, a bank that earns its ROE on book and grows at g must retain g over ROE of its earnings and pays out the rest, which makes its fair price to book (ROE minus g) over (cost of equity minus g). Setting (16% minus g) over (12% minus g) equal to 2.0 gives g of 8%, with 50% of earnings paid out. Whether 8% forever is believable is the real question.

Where does the price-to-book formula come from?

Think of a fruit tree. The fruit you eat is the dividend; the fruit you replant grows the orchard. A bank earning its ROE on book value can grow only by retaining earnings, so growth equals ROE times the share retained, and the rest is paid out as dividends. Put that dividend into the dividend discount modelA valuation that prices a share as the present value of its future dividends; with constant growth it becomes the next dividend divided by the cost of equity minus growth. and divide by book value, and price to book becomes (ROE minus g) over (r minus g).

How do you solve for the growth?

Set the formula equal to the price. (16% minus g) over (12% minus g) equals 2.0, so 16 minus g equals 24 minus 2g, and g is 8%. Check it with rupees: on book of Rs 100 the bank earns Rs 16, keeps Rs 8 to grow book by 8%, and pays Rs 8. Rs 8 over (12% minus 8%) is Rs 200, twice book.

Is 8% growth forever reasonable?

That is the judgement the interviewer is waiting for. Look at the curve: near 8% the price is very sensitive to growth, so a small shortfall, 7% instead of 8%, would justify only 1.8x book. Compare 8% with the nominal growth the economy could sustain over decades, a figure to look up rather than assume. And note what the formula says about returns: if the bank earned only its 12% cost of equity, it would be worth exactly book value, whatever its growth.

About 28%. Take 1,000 companies. Twenty commit fraud and the test flags 19 of them. The other 980 are clean, and the test wrongly flags 5% of them, which is 49. So 68 companies are flagged and only 19 are frauds: 19 over 68 is 27.9%. The test is accurate, but fraud is rare, so a small error rate on the large clean group produces more false flags than true ones.

Why is the answer not 95%?

Think of a smoke alarm that sounds for 95% of real fires and stays quiet 95% of the time there is no fire. In a building where fires are rare, most alarms you hear are burnt toast, because the quiet days vastly outnumber the fires and 5% of them still ring. A test's accuracy tells you how it behaves when the truth is known; the question asks the reverse, how likely the truth is given the test, and the base rateHow common the thing is before any test is run. Here, 2% of companies commit fraud. sits between the two. With 2% frauds, a 5% error on the 98% of clean companies creates 49 false flags for every 19 true ones.

How do you work it with whole numbers?

Start with 1,000 companies and walk them down the tree. Frauds: 2% of 1,000 is 20, and 95% of those, 19, are flagged. Clean: 980, and 5% of those, 49, are wrongly flagged. 19 true flags and 49 false flags make 68 flagged companies, so the chance that a flagged company is a fraud is 19 over 68, about 28%. Working in counts rather than percentages is faster in the room and harder to get wrong, and it shows the interviewer where every number came from.

What moves the number, and what does that mean for a screen?

Two things. The base rate: if 10% of the sample committed fraud, the same test would make a flag 68% reliable. The false-flag rate: cutting it from 5% to 1% lifts the answer to 66% even at a 2% base rate. A screen for a rare event is a filter that earns a closer look, not a verdict, and the second look does the real work: run an independent test of the same accuracy on the 68 flagged companies and a double flag is 88% likely to be a fraud. That is how diligence red flags, transaction monitoring and credit early-warning lists are meant to be read. The working assumes the two tests err independently; if they look at the same ratios, the second adds less than the arithmetic suggests.

Roughly 34,050 to 41,950, with about two-thirds confidence. Four months is a third of a year, and volatility scales with the square root of time, so 18% a year becomes 18% times 0.577, about 10.4%, over four months. One standard deviation is about 3,949 points either side of 38,000. If returns are roughly normal, about 68% of outcomes land inside that band and about 95% inside 30,100 to 45,900. Give the range and the confidence, never a single number.

Why does volatility shrink with the square root of time?

Picture someone leaving a party and taking steps at random, forward or back. After four steps they are not four steps from the door; some steps cancelled, and on average they are about two away. Random moves partly cancel, so the spread of where you end up grows with the square root of the number of moves, not with the number itself. Four months is a third of a year, the square root of a third is 0.577, and 18% times 0.577 is 10.4%. Dividing 18% by three to get 6% would be right only if every month moved the same way, which is the opposite of random.

How do you turn 10.4% into a range?

10.4% of 38,000 is about 3,949 points. One standard deviationA measure of how far outcomes typically land from the centre. Under a normal distribution about 68% of outcomes fall within one of the centre and 95% within two. either side is 34,050 to 41,950; two is 30,100 to 45,900. A one standard deviation band is a two-in-three bet, so the confidence attached to the width is the answer, and a range without one is just a guess with margins. Say the band, say the two in three, and offer the wider nineteen-in-twenty band if the interviewer wants more certainty.

What do you add to sound like a trader rather than a textbook?

Three refinements, each one sentence. Returns compound, so the band is slightly lopsided: 38,000 grown and shrunk by 10.4% continuously is about 34,250 to 42,150, a little more room on the upside. The expected drift over four months is a percent or two, small next to 10.4%, so ignoring it is fair. Real markets have fatter tails than the bell curve, so the two standard deviation band gets breached more often than one time in twenty. Say the range, say the confidence, then say what would make you wrong: the 18% itself is a historical figure, and the volatility the options market implies for the next four months may be higher or lower. This is a way of describing uncertainty, not a forecast, and the interviewer is testing whether you think in distributions.

204 squares. A square of side k can start in (9 minus k) columns and (9 minus k) rows, so there are 64 squares of size 1, 49 of size 2, 36 of size 3, and so on down to a single 8 x 8. The total is the sum of the first eight square numbers, 1 + 4 + 9 + ... + 64, which is 204.

How do you count one size without listing every square?

Picture sliding a 3 x 3 photo frame across the board. Its left edge can sit on column 1, 2, 3, 4, 5 or 6; on column 7 it would hang off the side. That is 6 positions across and, by the same logic, 6 down. Count where a square's top-left corner can sit, not the squares themselves: a k x k square has (9 minus k) choices in each direction, so (9 minus k) squared positions. For k = 3 that is 36, for k = 7 it is 4, and the whole board is the single 8 x 8.

How do you add 64 + 49 + ... + 1 quickly?

Add from the top and say the running total: 64 and 49 is 113, plus 36 is 149, plus 25 is 174, plus 16 is 190, then 9, 4 and 1 take it to 204. If you know the formula for a sum of squares, use it as a check rather than a starting point. The interviewer scores the moment you see that each size contributes a square number, not the formula you remember.

Why is 1,296 the wrong answer here?

1,296 is the number of rectangles of every shape. A rectangle is fixed by choosing two of the 9 vertical grid lines and two of the 9 horizontal ones, 36 ways each, and 36 x 36 is 1,296. Squares are the rectangles whose two sides are equal, which is why the count falls from 1,296 to 204. Offering that link unprompted shows you understand the counting rather than a memorised trick.

Option two, by about 10 points: roughly 34.8% against 24.6%. Option one triples the money in five years, so its IRR is 3 to the power of one fifth, minus 1, about 24.6%. Option two returns a third of the proceeds in year 2. Test 35%: the year-2 cheque is worth 54.9 today and the year-5 cheque 44.6, together 99.5, just under 100, so the IRR is a shade under 35%.

Why does the same multiple give two different IRRs?

Imagine lending a friend Rs 100 and getting Rs 300 back. If the Rs 300 arrives in one go after five years, fine. If Rs 100 comes back after two years, you have your stake back early and can lend it again. The money multiple counts rupees; the IRR counts rupees and how long each one spends away from you. Option two returns a third of the proceeds three years sooner, so the money is tied up for less time on average and the yearly rate is higher.

How do you get the second IRR without a calculator?

Option one is clean. You know 2x in five years is about 15% a year and 3x is about 25%, and the exact figure is 24.6%. Option two has no neat formula, so pick a round guess and test it. At 35%, the year-2 cheque is worth 100 over 1.35 squared, about 54.9, and the year-5 cheque is worth 200 over 1.35 to the fifth, about 44.6: 99.5 in total, just short of the 100 invested, so the IRR sits just under 35%. One trial, said out loud, is all the interviewer needs.

Why do sponsors like recaps, and what is the catch?

Funds are often judged on IRR, so pulling cash out early through a dividend recapThe company borrows new debt and pays the money out to its owners as a dividend. lifts the number without changing the multiple. The catch is that the recap is paid for with new borrowing, so the company carries more debt for its last three years and the exit equity is smaller and riskier. A strong answer names all three effects: a higher IRR, the same multiple, more leverage. IRR rewards speed and the multiple rewards size, and investors look at both.

107 crore diluted shares. Only options with a strike below the Rs 50 share price are exercised. The Rs 20 tranche brings in Rs 200 crore, which buys back 4 crore shares, so it adds 6 crore net. The Rs 40 tranche also brings in Rs 200 crore, buying back 4 crore, so it adds 1 crore net. The Rs 60 tranche is out of the money and adds nothing.

Why do only some options count?

An option to buy a share at Rs 60 when it trades at Rs 50 is like a voucher to buy a shirt for more than its shelf price: nobody uses it. Only in-the-money options, those with a strike below the current share price, would be exercised, so only they add shares. Here the Rs 20 and Rs 40 tranches are in the money and the Rs 60 tranche is not. The 8 crore Rs 60 options are ignored today, but they come back into play if the price rises past Rs 60 or a bidder offers more than that.

Why does each tranche add fewer shares than its option count?

When holders exercise, they pay the strike price to the company. The treasury stock methodA way to count dilution that assumes the company uses the cash from option exercises to buy back its own shares at the current price. assumes the company uses that cash to buy back shares at today's price. Each in-the-money tranche adds its option count minus the shares its exercise cash can buy back. The Rs 20 tranche pays 10 crore x Rs 20 = Rs 200 crore, which buys 4 crore shares at Rs 50, so 6 crore net. The Rs 40 tranche pays 5 crore x Rs 40 = Rs 200 crore, again 4 crore bought back, so 1 crore net.

Notice what the formula says. The deeper in the money an option is, the closer it comes to a full new share: the Rs 20 tranche dilutes at 60% of its count, the Rs 40 tranche at only 20%. Diluted equity value at Rs 50 is 107 crore x Rs 50 = Rs 5,350 crore, Rs 350 crore above the basic Rs 5,000 crore, and that diluted figure is the one that goes into an enterprise value bridge.