Investment Banking puzzles, solved step by step

Dilutive, by about 5.3%. At Rs 45 a share, Rs 1,200 crore needs 26.7 crore new shares, taking the count to 126.7 crore. Earnings rise to Rs 360 crore, so EPS falls from Rs 3.00 to about Rs 2.84. The shortcut: in an all stock deal, buying at a higher P/E than your own always dilutes, because each rupee of earnings bought costs more than each rupee handed over. It takes Rs 20 crore of after-tax synergies to break even.

What is the shortcut that answers it in ten seconds?

Picture swapping a shop for a share in a friend's shop. If your shop is priced at 15 times its yearly profit and your friend's at 20 times, every rupee of your profit you hand over buys only three quarters of a rupee of theirs. In an all stock deal the acquirer gives up its own earnings yield and receives the target's, so a target on a higher P/E means dilution. Here the acquirer's earnings yieldEarnings divided by price, the inverse of the P/E. A P/E of 15 is an earnings yield of 6.7%. is 1 over 15, 6.7%, and the target's is 1 over 20, 5.0%.

How do the full numbers work?

New shares: Rs 1,200 crore divided by Rs 45 is 26.7 crore, so the count rises from 100 to 126.7 crore. Earnings rise from Rs 300 crore to Rs 360 crore. When the share count grows faster than earnings, EPS falls: Rs 360 crore over 126.7 crore shares is Rs 2.84, down 5.3% from Rs 3.00.

What would make the deal break even?

Turn it round. The enlarged share count needs Rs 3.00 of earnings per share, so Rs 3.00 times 126.7 crore, which is Rs 380 crore. The deal needs Rs 20 crore a year of after-tax synergies, a third of the target's own earnings, before EPS is back where it started. Saying that number tells the interviewer how big the problem is, which matters more than the sign. Close with the limit: one year of EPS is not value, and a dilutive deal can still be worth doing if the target grows much faster than the acquirer.

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.

Your personal beta is 0.8. Salary does not move with the market, a beta of 0; the bonus moves twice as much, a beta of 2. Weighted by share of pay, 0.6 x 0 plus 0.4 x 2 gives 0.8. If the market falls 20%, the bonus falls 40% and total pay falls 16%. Your earnings already carry most of the market risk of an index fund before you invest a rupee of savings.

How do you get a beta for a person?

Think of a household with two earners, a schoolteacher on a fixed salary and a salesperson paid on commission. In a downturn the teacher's pay holds while the commission falls sharply, and the household's income falls by a blend of the two. The beta of a mix is the weighted average of the betas of its parts, so a pay packet's beta is each component's beta weighted by its share of pay. Salary has a beta of 0 and the bonus a beta of 2, so the blend is 0.8.

What does 0.8 look like in rupees?

Take pay of Rs 25 lakh: Rs 15 lakh of salary and Rs 10 lakh of bonus. If the market falls 20%, the bonus falls 40% to Rs 6 lakh, and total pay falls from Rs 25 lakh to Rs 21 lakh, a 16% drop, which is 0.8 times the market's fall. The salary cushions the swing, but the bonus is exactly the part that shrinks in the years the market is weakest.

Why does it matter for your savings?

Your future earnings are an asset, often the largest one you own early in a career, and finance calls it human capitalThe value today of the earnings a person expects to make over the rest of a working life.. If that asset already has a beta of 0.8, your total wealth carries market risk before your savings add any, and the worst years for markets are the years your bonus, and perhaps your job, are most at risk. This is a way of counting risk, not a recommendation for anyone's savings: the point the interviewer wants is that risk is measured across everything you own, including your pay.

About 13.6%. The peers' 0.8 is an unlevered beta, the risk of the business with no debt. The target carries debt equal to 50% of its equity, so relever it: 0.8 times (1 plus 75% times 0.5) is 1.1. CAPM then gives 7% plus 1.1 times 6%, which is 13.6%. Using the unlevered 0.8 directly would understate the cost at 11.8%.

Why can you not use the peer beta as it is?

Think of two people with the same job and salary, one with no loan and one with a large home loan. A pay cut hurts the borrower more, because the loan payment does not shrink. Debt magnifies the swings that reach shareholders, so the same business carries a higher equity beta the more it borrows. Peers' betas are first stripped of their own debt to give an unlevered betaThe beta a business would have with no debt: the risk of its operations alone., then relevered at the target's own mix of debt and equity.

How does the relevering work?

Use the standard formula, which assumes the debt itself carries no market risk. Levered beta equals unlevered beta times one plus (one minus the tax rate) times debt over equity: 0.8 times (1 + 0.75 x 0.5), which is 0.8 times 1.375, or 1.1. The tax term is there because interest is tax deductible, so the tax saving softens a quarter of the debt's burden on shareholders. Then CAPM: 7% plus 1.1 times 6% is 13.6%.

Which slips change the answer?

Three, each worth naming. Skipping the relevering gives 11.8%, using debt over total capital instead of debt over equity gives 13.0%, and ignoring tax gives 14.2%. A spread of 2.4 points in the cost of equity moves a DCF a long way, so the interviewer wants to hear which ratio you are plugging in. State the limit too: the rates here are illustrations, and a live model takes the current government bond yield and the premium the bank uses.

Enterprise value stays the same and equity value rises by Rs 10 crore. Shareholders own the extra cash, so their equity is worth Rs 10 crore more. Enterprise value is equity plus debt minus cash: equity is up 10 and cash is up 10, so EV does not move. EV measures the operating business, and picking money up off the street does not change what that business earns. Ignoring tax on the windfall keeps the numbers clean.

What is enterprise value actually measuring?

Think of buying a shop whose till holds Rs 10,000. You would pay for the business plus the cash in the till, and the cash is worth exactly its face value to you, no more and no less. Enterprise value prices the operating business on its own, so cash, which is worth its face value to whoever holds it, is taken out of the bridge. Equity value is what shareholders own, and they own both the business and the cash.

How does the bridge move?

Take a company with equity worth Rs 500 crore, debt of Rs 300 crore and cash of Rs 50 crore, so enterprise value is Rs 750 crore. Find Rs 10 crore: cash becomes Rs 60 crore and equity Rs 510 crore, and EV is 510 plus 300 minus 60, still Rs 750 crore. Nothing in the operating business changed, so multiples of EBITDAEarnings before interest, tax, depreciation and amortisation: a rough measure of the cash profit the operating business produces. or revenue do not change either.

What assumptions should you say out loud?

Two. First, tax: the windfall is probably taxable income, so at a 25% rate equity and cash each rise by Rs 7.5 crore rather than Rs 10 crore. Whatever amount sticks, cash and equity move together and enterprise value stays put. Second, the cash sits idle. If the company used it to repay debt, EV would still not change, but the split between lenders and shareholders would. Naming both shows the interviewer you know which line each event touches.

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.

About 2.67%. A P/E of 15 means each Rs 100 of share price buys Rs 6.67 of yearly earnings, an earnings yield of 1 over 15. The company pays out 40% of that, Rs 2.67, as dividend. Dividend yield is dividend over price, Rs 2.67 over Rs 100. In one line: the payout ratio divided by the P/E, 0.40 over 15.

How do you turn a P/E into something you can multiply?

Flip it. A P/E of 15 says you pay 15 rupees for every rupee of yearly profit, so every rupee of price buys one fifteenth of a rupee of profit. Inverting the P/E gives the earnings yieldEarnings per share divided by the share price: the inverse of the P/E., 1 over 15 or about 6.67%, and every other per-share ratio then falls out by multiplication. Picking a round share price of Rs 100 makes it concrete: earnings per share are Rs 6.67.

Why is the dividend only part of the shareholder's return?

Think of a shopkeeper who takes home 40% of each year's profit and leaves 60% in the shop to buy more stock. The retained Rs 4.00 is not lost to shareholders; it is reinvested, and if it earns a decent return it should lift future earnings and dividends. That is why a low dividend yield on its own says little about whether a share is cheap. A company paying out everything would show a 6.67% yield and little room to grow.

Say the general rule after the number, because the follow-up usually changes one input. If the P/E doubles to 30 with the same payout, the yield halves to 1.33%; if the payout doubles to 80% at the same P/E, the yield doubles to 5.33%. Having the one-line formula ready means you answer those in a breath.

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.