Investment Banking puzzles, solved step by step

Put one white ball alone in bowl A and the other 99 balls in bowl B. Bowl A gives white every time. Bowl B gives white 49 times in 99. Each bowl is picked half the time, so the chance of white is 0.5 x 1 + 0.5 x 49/99, about 74.7%, against 50% for an even split.

Why does the number of balls in a bowl not matter?

Think of two queues at a ticket counter, one with a single person and one with ninety nine. If a coin decides which queue gets served, the lone person gets served half the time. The bowls work the same way. The bowl is chosen with even odds, so a bowl holding one ball carries exactly as much weight as a bowl holding ninety nine. That is the lever: you can make one half of the draw certain at the cost of a single white ball.

How do you prove no other split beats it?

Bowl A can never beat 100%, and one ball is the cheapest way to get there. Every extra white ball you move into bowl A is a white ball taken out of bowl B, which lowers bowl B's odds without raising bowl A's. Adding black balls to bowl A only hurts it. So the best bowl A is one white ball, and bowl B's best case is whatever is left: 49 white out of 99.

Say the 74.7% figure out loud, then say the general rule: with n white and n black balls, the answer is one half plus one half of (n minus 1) over (2n minus 1), which creeps towards 75% as n grows. Interviewers like hearing that you see the limit.

The tax rate is 50%. P/E of 20 on equity of 200 gives net income of 10. Interest of 20 at 5% means debt of 400, so enterprise value is 600 and EBITDA is 60. Take off depreciation of 20 and interest of 20 to reach pre-tax profit of 20. Tax is 20 minus 10, which is 10, so the rate is 50%.

Where do you start when every number looks equally useful?

Start from what the answer is made of. A tax rate is tax divided by pre-tax profit, so the job is to find the two profit lines either side of the tax charge. Everything in the question is a road to one of those two numbers. Say that out loud first: it tells the interviewer you have a plan before you touch the arithmetic.

Net income is one step: equity value over the P/E, 200 over 20, is 10. Pre-tax profit takes longer. The enterprise valueThe value of the whole business to all its funders: equity value plus debt, less cash. multiple is on EBITDA, so you need EV, which needs debt, which the interest line hands you: 20 of interest at 5% means 400 of debt.

What assumption are you making, and should you say it?

You are assuming the company holds no cash, so enterprise value is simply equity plus debt. Say the assumption before you use it, because with any cash on the balance sheet EV falls, EBITDA falls, and the tax rate changes. A 50% rate is also high for most countries, which is worth one sentence: the interviewer built round numbers, not a real company.

About Rs 450 crore. At full run-rate the synergies add Rs 75 crore of after-tax earnings, which the market capitalises at 8x: Rs 600 crore. Two things the headline ignores come off. Year 1 delivers only half, a shortfall of Rs 37.5 crore after tax, and the one-off cost of Rs 150 crore is Rs 112.5 crore after its tax saving.

Why can you not just multiply Rs 100 crore by 8?

A shop owner expects a new supplier to cut her costs by Rs 1 lakh a year. Before celebrating, she has to pay tax on the extra profit, wait for the new contract to start, and pay the one-off cost of switching. A synergy is worth the after-tax earnings it adds, capitalised, less everything it costs to achieve and every rupee that arrives late. The 8x multiple applies to after-tax earnings, so the pre-tax Rs 100 crore first becomes Rs 75 crore at 25% tax, and 75 x 8 is Rs 600 crore.

What comes off the Rs 600 crore?

Two things. Phasing: the market value assumes the full run-rate, but year 1 delivers only half, so Rs 50 crore of pre-tax savings, Rs 37.5 crore after tax, never arrive. Cost to achieve: Rs 150 crore of severance, systems work and site closures, assumed tax deductible, so it costs Rs 112.5 crore after tax. Net of phasing and the cost to achieve, the synergies are worth about Rs 450 crore, a quarter less than the after-tax figure and 44% less than the pre-tax headline.

What does this rough answer leave out?

Three refinements, each worth naming. The year-1 items fall a year from now, so discounting them would trim the deductions slightly. The 8x assumes the market gives synergies the same multiple as the core earnings, while in practice cost synergiesSavings from combining two businesses, such as closing duplicate offices or merging purchasing. earn more credit than revenue synergies and both are discounted for execution risk. The deductibility of integration costs is an assumption to confirm for the jurisdiction. The answer to give is the method and Rs 450 crore, then one line on which assumption you would test first.

The number matters because it caps what a buyer can sensibly pay away. A premium above roughly Rs 450 crore hands the target's shareholders more than the whole value the deal creates, before any risk that the savings slip.

About Rs 8. With near-zero rates, an at-the-money call is worth roughly 0.4 times the price times the volatility times the square root of time: 0.4 x 100 x 0.20 x 1 = 8.0. The full Black-Scholes formula gives Rs 7.97. The intuition: the option collects the upside of a bell-shaped spread of outcomes and nothing on the downside, and the average of that upside half is about 0.4 of one standard deviation.

What is Black-Scholes actually doing?

Picture a ticket that pays you whatever a stock gains over the next year and nothing if it falls. You would pay something for it, even though an at-the-moneyAn option whose strike price equals the current stock price, so exercising it today would pay nothing. ticket is worth nothing if cashed in today. Black-Scholes prices that ticket as the average payoff across all the ways the stock could move, with the spread of those moves set by volatility and time. Its inputs are the stock price, strike, time, volatility and interest rate, and with zero rates the formula reduces to S times N(d1) minus K times N(d2), where the two N terms come from the bell curve.

Where does the 0.4 come from?

Over a year at 20% volatility, the stock's likely outcomes form a bell curve around Rs 100 with a standard deviation of about Rs 20. The call keeps the right half and treats the left half as zero. The average of the positive half of a bell curve, counting the negative half as zero, is the standard deviation divided by the square root of 2 pi, about 0.4 of it. So the call is worth about 0.4 x Rs 20, which is Rs 8. Halve the time to six months and the spread shrinks by the square root of 2, so the option is worth about Rs 5.6, not Rs 4.

What does the shortcut leave out?

It holds for at-the-money options with modest volatility and low rates. Away from the strike, the payoff is no longer the clean right half of the curve, and with meaningful rates the forward price drifts above the spot, which raises a call's value. The rule is a sanity check on a quoted price, not a replacement for the formula. Black-Scholes itself assumes constant volatility and smooth price moves, which is why traders quote options in implied volatility and adjust for the skew they see in the market. Doubling volatility to 40% roughly doubles this call's value, to about Rs 15.9.

Give B heavy depreciation and more debt, so much less of its EBITDA reaches shareholders. Both earn EBITDA of Rs 100 crore. A has D&A of 2 and interest of 18, keeps net income of 60, and at equity value 720 trades on 12x; its EV of 900 is 9x EBITDA. B has D&A of 25 and interest of 35, keeps only 30, and at equity value 450 trades on 15x, though its EV of 800 is 8x.

How can one company be cheaper on one multiple and dearer on another?

Two shops each take in Rs 1 lakh a month before equipment and loan costs. One works from a simple counter with a small loan; the other leases costly machines it must keep replacing and carries a large loan. Priced on takings before those costs, they look alike; priced on what the owner keeps, they do not. EV/EBITDA values the business before depreciation and before financing, while P/E values what is left for shareholders after both, so a company with heavy D&A or heavy interest can look cheap on one and dear on the other.

Which line explains how much of the gap?

Work down from EBITDA. B carries Rs 23 crore more D&A and Rs 17 crore more interest than A, which after 25% tax is Rs 17.25 crore and Rs 12.75 crore of net income: together exactly the Rs 30 crore gap between A's 60 and B's 30. Here depreciation does more of the work than debt, which you can show with EV/EBIT, a multiple that charges D&A but ignores financing: A is 9.2x and B 10.7x, so A is cheaper there too.

So which company is actually cheaper?

It depends on whether B's D&A is a real cost. If it reflects equipment that must be replaced, which capital expenditureCash spent on buying or replacing long-term assets such as machines and buildings. running close to depreciation would confirm, then EBITDA flatters B and A is better value on every measure that counts the cost of staying in business. If B's D&A is mostly amortisation of acquired intangibles with no cash replacement, EBITDA is the fairer lens and B may be the cheap one. The answer the interviewer wants is the diagnosis: check capex against D&A and check leverage before trusting either multiple.

The value falls, by about 4.8% here, because the cut to cash flow outweighs the cheaper debt. One: unlevered free cash flow falls, as NOPAT drops from Rs 150 crore to Rs 140 crore. Two: the after-tax cost of debt falls from 6.0% to 5.6%, taking the WACC from 10.20% to 10.08%. Three: the terminal value, which carries both effects, falls from about Rs 2,516 crore to Rs 2,395 crore at 4% growth.

Why does a tax rate touch both the top and the bottom of the fraction?

A landlord pays more tax on her rent, so the rent she keeps falls; but the interest on her mortgage is deductible, so the loan costs her a little less after tax. Two opposite effects from one change. A DCF is the same fraction: cash flow on top, discount rate below. A higher tax rate shrinks the cash flows directly and shrinks the WACC indirectly through the interest tax shield, and the first effect is almost always larger because it hits every rupee of operating profit while the second touches only the debt slice. Interviewers ask for three ways so that you show both sides and then say which wins.

How big is each effect on these numbers?

Cash flow. NOPAT is EBIT times one minus the tax rate: Rs 200 crore x 0.75 = Rs 150 crore before and Rs 200 crore x 0.70 = Rs 140 crore after, a fall of Rs 10 crore, or 6.7%, in every forecast year, treating depreciation, capex and working capital as cancelling out. Discount rate. The after-tax cost of debt is 8% x (1 - t): 6.0% before and 5.6% after. Weighted at 30% debt and 70% equity at 12%, the WACC moves from 10.20% to 10.08%, down 12 basis points. Terminal value. At 4% growth, a Gordon growthA terminal value that treats the final cash flow as growing at a constant rate forever: the next cash flow divided by the discount rate less the growth rate. terminal value is Rs 2,516 crore before and Rs 2,395 crore after. Separating the two, the cash flow cut alone takes Rs 168 crore off the terminal value and the lower WACC alone adds Rs 46 crore back, so the net is Rs 121 crore, 4.8% lower.

Is there a subtler channel the interviewer might be after?

Yes: the cost of equity. When you relever a beta, the formula carries a (1 - t) term, so a higher tax rate lowers the levered beta a little and the cost of equity with it. With debt to equity of 3 to 7 and a levered beta of 1.0 at 25%, the unlevered beta is 0.757, and relevering at 30% gives 0.984; with an assumed 6% risk-free rate and 6% equity risk premium, the cost of equity slips from 12% to 11.90% and the WACC to 10.01%. Even with that help the value is still about 3.7% lower, so the ranking of the effects does not change. Say the limitation too: the WACC relief assumes the company actually pays tax and keeps its interest deductible, and interest deductions are capped in many jurisdictions, a rule to confirm for the country in question.

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.

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.

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.

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.