Investment Banking puzzles, solved step by step

The lottery ticket has a beta of zero, and the half-and-half portfolio has a beta of 0.5. Beta is the covariance of an asset's return with the market's return divided by the market's variance. A lottery draw does not move with the index at all, so its covariance is zero, however wild its payoff. Betas combine in proportion to weights, so half an index fund at beta 1 and half tickets at beta 0 give 0.5 x 1 + 0.5 x 0 = 0.5.

Why is a wildly risky asset given a beta of zero?

Think of two shops on one street. One sells umbrellas, whose takings rise and fall with the weather; the other sells birthday cards, whose takings have nothing to do with it. The card shop's sales swing plenty, but not with the rain. Beta measures only the part of an asset's movement that follows the market, so an asset whose payoff is decided by a draw of numbered balls has a beta of zero no matter how large its swings. On a Rs 10 ticket with an assumed one in a crore chance of a Rs 5 crore prize, the standard deviation of the return is about 1,600 times the stake and the beta is still zero. Volatility is total riskThe full spread of an asset's returns, usually measured by standard deviation, including both the part linked to the market and the part unique to the asset.; beta is only its market-linked part.

How does the portfolio beta follow?

Covariance is linear in the weights, so the beta of a portfolio is the weighted average of the betas inside it. An index fund tracking the market has a beta of one by construction. Half at beta 1 and half at beta 0 gives 0.5, the same beta as a portfolio that is half the index and half cash. That comparison is the point: in the capital asset pricing model the lottery half and a cash half demand the same return, the risk-free rate, because neither adds market risk; the lottery's enormous extra risk is diversifiable in theory and simply unrewarded.

What does the security market line say about the blend?

Plot required return against beta. With an assumed 6% risk-free rate and 6% equity premium, the line runs from 6% at beta 0 to 12% at beta 1, and the blend at beta 0.5 would be required to earn 9%. Its expected return is nothing like that: if a ticket pays out an assumed half of its price in prizes, its expected return is -50%, and the blend's is 0.5 x 12% plus 0.5 x -50%, which is -19%. Beta tells you where an asset sits along the line, and the lottery ticket sits about 56 points below it: a zero-beta asset with an appalling expected return. That is the limitation to state: beta says nothing about expected return outside the model's assumptions, and nothing about risk the market does not share.

Why a banker asks it: the same logic decides what goes into a cost of equity. A project whose cash flows do not move with the economy, a regulated pipeline or a contracted toll road, carries a low beta and a low cost of equity even if its engineering risk is large, because that risk is specific and diversifiable. Candidates who confuse volatility with beta tend to over-discount exactly those assets, and to under-discount a cyclical business that happens to look calm.

About Rs 27.3 lakh, the value of selling it. Keeping the elephant forever costs Rs 20 lakh a year, minus Rs 2 crore in today's money at 10%. Selling means one more year of upkeep and Rs 50 lakh at the end, a net Rs 30 lakh worth Rs 27.3 lakh today. Donating is worth zero. A gift is worth its best use, and the free exit means it can never be worth less than zero.

Why does the elephant not have one value?

Think of being handed an old car that needs Rs 30,000 of repairs before anyone will buy it. The car is not worth its sticker price; it is worth what you can do with it, net of what doing that costs. An asset with a carrying cost has no value of its own: each option for it has a value, and the asset is worth the best of them. So the first move is to list the options out loud, keep it, sell it or give it away, and then price each one at the same 10% rate.

How do you price each option?

Keeping is a perpetuityA payment that repeats every year forever. Its value today is the yearly amount divided by the discount rate. of costs: Rs 20 lakh a year divided by 10% is minus Rs 200 lakh, or minus Rs 2 crore. Selling means paying one more year of upkeep while the paperwork clears, then collecting Rs 50 lakh. Net of upkeep, the sale delivers Rs 30 lakh at the end of year one, and Rs 30 lakh a year away is worth Rs 27.3 lakh today. Donating costs nothing and brings nothing. The working assumes upkeep is paid at the year end; if it is paid upfront, the sale is worth Rs 25.5 lakh instead, and the choice does not change.

When would the gift be worth nothing?

Change one number. If the best sale price were Rs 15 lakh, the sale would net minus Rs 5 lakh at the year end, minus Rs 4.5 lakh today, and donating would win at zero. The free exit puts a floor under the gift: it can be worth nothing, but never less than nothing, as long as someone will take it away for free. Remove the sanctuary and the floor goes too. That is why bankers ask about exit options before valuing a loss-making plant, a lease that cannot be broken or a subsidiary with heavy fixed costs.

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.

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.

Take the 10% revenue rise: it adds Rs 0.4 crore of EBITDA. Revenue rises by Rs 1 crore, and with variable COGS at 60%, Rs 0.4 crore of it reaches EBITDA, a 20% jump. A 5% cut in COGS saves 5% of Rs 6 crore, Rs 0.3 crore. A 5% rise in EBITDA is 5% of the smallest number, Rs 0.1 crore. Translate every option into rupees of EBITDA before choosing.

Why can't you compare the three percentages directly?

Ten per cent off a car and ten per cent off a cup of coffee are not the same saving. Each percentage is taken on a different base, so convert all three into one unit, rupees of EBITDA, before you choose. Revenue is the biggest base at Rs 10 crore, COGS is Rs 6 crore and EBITDA is the smallest at Rs 2 crore. A small percentage of a big number can beat a big percentage of a small one, and here it does.

What should you ask before you answer?

Two readings change the numbers, and asking about them is part of the answer. First, is the revenue rise from volume or from price? Selling 10% more units drags 60% of the new revenue away as COGS, but a 10% price rise costs nothing extra, so all Rs 1 crore reaches EBITDA. Second, does a 5% fall in COGS mean 5% of COGS, or COGS falling from 60% to 55% of revenue? The first saves Rs 0.3 crore; the second saves Rs 0.5 crore and would beat a volume-led revenue rise.

Close with what a banker adds beyond the arithmetic. Operating leverageHow much profit swings for a given change in sales, because fixed costs do not move with volume. is why the revenue rise is so strong: with Rs 2 crore of fixed costs, a 10% rise in sales lifts EBITDA by 20%. The same leverage works in reverse, so a 10% fall in volume would cut EBITDA by 20% as well.

About 4%, below the unlevered 7%. Per Rs 100 of property, a 7% cap rate means Rs 7 of net operating income. A 60% loan is Rs 60, and at 9% interest-only it costs Rs 5.4 a year. That leaves Rs 1.6 of cash for the Rs 40 of equity, which is 4%. Because the loan costs more than the property yields, this is negative leverage: borrowing lowers the equity return rather than raising it.

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

Suppose you borrow money at 9% to put in a deposit paying 7%. Every rupee borrowed loses you 2 paise a year, and the more you borrow the worse it gets. Leverage lifts the equity return only when the asset yields more than the debt costs; when the cap rate sits below the interest rate, every rupee of debt subtracts from the equity's return. Here the Rs 60 of debt earns 7% inside the property, Rs 4.2, but costs 9%, Rs 5.4, so it drags Rs 1.2 off the Rs 7 that the equity would otherwise keep on Rs 100 unlevered.

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

Fix the value at Rs 100 so every input becomes a rupee figure. Cap rate times value is the income; loan-to-value times value is the debt; debt times the rate is the interest; the rest is cash on the equity, and the equity is value minus debt. That is four multiplications and two subtractions, and saying each one aloud keeps you honest. Then say the comparison that matters: 4% levered against 7% unlevered, so the loan has cost the equity 3 points of yield.

Why would anyone buy with negative leverage?

Because cash-on-cash is a snapshot of year one. A buyer accepting 4% today is betting on rent growth lifting the income, on refinancing when rates fall, or on selling at a lower cap rate, none of which is in this calculation. Say that the figure ignores principal repayment, because the loan is interest-only, and ignores any appreciation. A total return would add both, and a lender would also look at the debt service coverage, which here is 7 over 5.4, about 1.3x, thin for most lenders.

The equity falls by about 33%, a third. Equity value is enterprise value less net debt: 10 minus 7, so 3 turns of EBITDA. A 10% fall takes enterprise value to 9 turns, and the lenders are still owed 7, so equity is 2 turns. From 3 to 2 is a one third fall. At 7 turns of debt on a 10 turn valuation the equity is a thin slice, and every move in the business's value is magnified 3.33 times on it.

Why does the equity fall more than the business?

Picture a flat bought for Rs 1 crore with a Rs 70 lakh loan. If the flat's price falls 10% to Rs 90 lakh, the bank is still owed Rs 70 lakh, so the owner's stake falls from Rs 30 lakh to Rs 20 lakh, a third. Net debt is fixed in rupees, so the whole of any fall in enterprise value comes out of the equity, and the thinner the equity slice, the larger the percentage hit. Here the equity is 3 turns of EBITDA under 7 turns of debt, so a 1 turn fall in value is a third of it.

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

The percentage move in equity is the percentage move in EV times EV over equity. At 10x EV and 7x debt that multiplier is 10 over 3, 3.33, so each 1% in enterprise value is 3.33% on the equity, up or down. That is what an interviewer means by sensitivity: 7 turns of leverage on a 10 turn valuation makes the equity a geared bet on the business. If EBITDA or the multiple slips 30%, the equity is worth nothing and the lenders start taking losses, which is why debt at that level prices as if it carried some of the equity risk.

Why does a banker ask a tech company this?

Because 7 turns is a lot of debt for a business valued on growth rather than assets. If the valuation multiple compresses, as growth multiples do when rates rise or growth slows, the enterprise value can fall 10% with no change in EBITDA at all, and the equity takes a third of that on the chin. Say the assumption you leaned on: net debt stays fixed, which holds over a short window but not if the company is burning or generating cash. The question tests whether you can see leverage as a magnifier before you build a single model.

Rs 400 crore, and the coverage test binds. Leverage of 5.0x allows Rs 500 crore. Coverage of 2.5x caps interest at 100 divided by 2.5, which is Rs 40 crore, and at 10% that interest supports Rs 400 crore of debt. The borrower must pass both tests, so the lower number is the ceiling. Below an 8% rate, leverage would bind instead.

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

A bank deciding your home loan asks two things: is the loan too large against your salary, and can you afford the monthly payment? You can pass one and fail the other. Lenders to a company ask the same pair. Leverage limits the size of the debt; coverage limits the cost of carrying it, and the borrower must clear both, so the binding test is whichever gives less. Compute both every time, then name the one that bites.

Leverage is one step: 5.0 times EBITDA of 100 is Rs 500 crore. Coverage works backwards from the payment. Interest coverEBITDA divided by interest expense: how many times over the operating profit pays the interest bill. of at least 2.5 means interest can be at most 100 divided by 2.5, which is Rs 40 crore. At a 10% rate, Rs 40 crore of interest corresponds to Rs 400 crore of debt.

When would the answer flip to the leverage test?

The coverage ceiling moves with the rate; the leverage ceiling does not. Set the two equal: interest of 40 on 500 of debt is an 8% rate. Below 8%, leverage binds; above it, coverage binds, which is why debt capacity shrinks when rates rise even if earnings do not move. At 7%, coverage would allow about Rs 571 crore, so the 500 leverage cap becomes the limit.

Say the limitation too. Real covenants often test cover on EBITDA less capex, or on a fixed charge basis that includes leases, and lenders size a deal with headroom below both ceilings rather than at them. The two-test logic stays the same.