Investment Banking puzzles, solved step by step

At Rs 120, count it as 5 crore extra shares and not as debt; at Rs 80, count it as Rs 500 crore of debt and no new shares. Rs 500 crore at a Rs 100 conversion price is 5 crore shares. At Rs 120 those shares are worth Rs 600 crore, so the holder will convert. At Rs 80 they are worth Rs 400 crore, so the holder takes the Rs 500 crore back. Treat the bond as whichever it is most likely to become, never as both.

Why does the share price decide whether a bond is debt?

Suppose a friend lends you Rs 5 lakh and you agree they can take either the money back or a share of your shop. If the shop is booming, they will take the share; if it is struggling, they will take the cash. You plan around what they will choose. A convertible gives the holder a choice between repayment and shares, and the valuation should assume the choice a rational holder makes at today's price. The comparison is simple: conversion value, shares received times share price, against face value.

At Rs 120, converting gives 5 crore shares worth Rs 600 crore, more than the Rs 500 crore face value. The holder converts, so add 5 crore shares to the diluted count and remove the bond from debt. With, say, 50 crore existing shares, diluted equity value becomes 55 crore x 120 = Rs 6,600 crore. At Rs 80, converting gives shares worth only Rs 400 crore, so the bond stays as Rs 500 crore of debt and the share count stays at 50 crore.

What goes wrong if you count it the other way?

Counting it as debt at Rs 120 puts Rs 500 crore into the bridge when the holder will in fact take Rs 600 crore of value in shares, understating the claim by Rs 100 crore. The serious error is counting it twice, as debt and as diluted shares, which overstates enterprise value by the full Rs 500 crore. That happens when the share count comes from one source that already includes the conversion and the debt figure from a balance sheet that still shows the bond.

Say the limitation. Near the conversion price the choice is close and the bond carries option value either way, so some banks value it at fair value rather than flipping between debt and shares. Settlement terms also matter: some convertibles repay the face value in cash and deliver only the excess in shares, which needs a different share count.

Bet 20% each time; bet 40% and long-run growth falls to about zero, slightly negative. With even money, the growth-maximising fraction is the win probability minus the loss probability, 0.6 - 0.4 = 0.2. That earns about 2.0% compound growth a bet. At 40%, three wins and two losses leave you with 0.988 of what you started, a small loss, even though every bet still has a positive expected value.

If every bet has positive expected value, why not bet more?

A shopkeeper with a healthy margin who stakes the whole shop on each delivery will, sooner or later, lose one delivery and the shop with it. The margin was real; the sizing ruined it. When you reinvest the results, what compounds is the typical outcome, not the average one, and big stakes make one loss cost more than one win earns. At a 40% stake a win takes you to 1.4 and a loss to 0.6; one of each leaves 0.84, a 16% loss on a pair that broke even in count.

So test a typical stretch: in five bets you expect three wins and two losses. Betting 20%, 1.2 cubed times 0.8 squared is 1.106, about 11% richer. Betting 40%, 1.4 cubed times 0.6 squared is 0.988, slightly poorer. Same coin, same edge, and only the stake changed.

Where does 20% come from?

This is the Kelly criterion, named after John Kelly, who published it in 1956. Over 100 bets the typical bankroll at a 20% stake is about 7.5 times the start; at 40% it is about 0.78 times. Over-betting a real edge can destroy growth as surely as having no edge. In practice many people bet half the Kelly fraction: at 10% growth is 1.50% a bet, about three quarters of the maximum, with far smaller swings, which matters because real edges are estimated, not known.

About 84.58 rupees per dollar. Rs 83 invested at 7% grows to 88.81. Converted to 1 dollar and invested at 5% it grows to 1.05 dollars. Both routes are riskless once the forward is agreed, so they must end equal: 1.05 x F = 88.81, giving F of about 84.58. If the quoted forward sat far above that, you could borrow rupees, invest in dollars and sell them forward to lock in a gain.

Why is the forward set by interest rates and not by a forecast?

Imagine two banks next door, one paying 7% on rupee deposits and one paying 5% on dollar deposits, and a counter that promises today to swap your dollars back into rupees in a year at a fixed rate. If that promised rate were 83, everyone would ignore the dollar bank. The forward rate must exactly cancel the interest difference, otherwise one route earns more than the other with no risk, and the gap would be traded away. That rule is called covered interest parity, covered because the forward removes the currency risk.

What would you actually do if the forward were quoted at 86?

At 86, dollars are expensive in the forward market relative to parity, so sell dollars forward and build the dollars cheaply today. Borrow Rs 83 at 7%, owing 88.81 in a year. Convert to 1 dollar at spot and deposit it at 5%, which grows to 1.05 dollars. Sell those 1.05 dollars forward at 86 for Rs 90.30. After repaying the rupee loan you keep about Rs 1.49 per dollar, without taking any view on where the rupee goes. Traders doing exactly this pushes the quote back toward 84.58.

The forward rupee is about 1.9% weaker than spot, roughly the 2 point rate difference. A quick approximation is spot times one plus the difference, 83 x 1.02 = 84.66, close enough to sanity-check. The limitation: in practice, capital controls, transaction costs and limits on who can borrow in each currency let small gaps persist, so the arbitrage is cleaner on paper than on a dealing desk.

Yearly gives Rs 2.00, monthly about Rs 2.61, daily about Rs 2.71, and continuous compounding gives e, about Rs 2.718. Splitting 100% into n periods gives (1 + 1/n) to the power n. Each increase in frequency adds less than the last, and as n grows without limit the result levels off at the constant e rather than growing forever.

Why does compounding more often help at all?

Think of a savings jar where interest is added at the end of each month instead of once a year. From February onward, the interest from January is itself earning interest. More frequent compounding lets earlier interest start earning sooner, so the same headline rate produces more by year end. Monthly at 100% means adding 1/12 twelve times, (1 + 1/12) to the twelfth power, which is about 2.61 rather than 2.00.

Why does it stop at e instead of growing forever?

Because each extra slice adds interest on smaller and smaller amounts of fresh interest. Going from yearly to monthly adds 0.61; from monthly to daily adds only about 0.10; from daily to continuous about 0.0037. The gains shrink fast enough that their total converges, and the limit is the number e, about 2.71828. That is where e comes from: it is what 100% growth becomes when it compounds continuously.

Why it matters on a desk: the same quoted rate means different money depending on the compounding convention. A loan quoted at 12% compounded monthly costs about 12.68% a year in effective terms, so when comparing a monthly-pay loan with an annual-pay bond, convert both to an effective annual rate first. Continuous compounding is also why option and rates models use e: it is the clean limit of compounding very often.

MOIC is 2.72x: equity grows from Rs 320 crore to Rs 870 crore. Of the Rs 550 crore gain, EBITDA growth contributes 240 (30 x 8x), multiple expansion 130 (one turn on 130), and debt paydown 180 (480 - 300). The three add exactly to 550. Over five years that would be an IRR of about 22%, though the question gives no holding period.

How do you get entry and exit equity quickly?

Equity is what is left of the business's value after the debt. Entry: 8 x 100 is an enterprise value of 800, less 480 of debt, so the sponsor put in 320. Exit: 9 x 130 is 1,170, less 300 of debt, leaving 870. MOIC is exit equity over entry equity, 870 / 320 = 2.72x. Say both enterprise values out loud before subtracting debt, so the interviewer can follow each step.

Why split the gain into three sources?

Think of a house you bought, renovated and sold. Your profit came partly from adding a room, partly from the neighbourhood getting more popular, and partly from paying down the mortgage. Only the first is a skill you can repeat on the next house. Attribution shows which part of the return came from the sponsor's own work and which came from the market, which is what investors in the fund want to know. EBITDA growth and debt paydown come from running the business; multiple expansion mostly comes from market conditions at exit.

Say the convention. The growth and multiple pieces share a cross term: 30 of extra EBITDA times one extra turn, which is 30. Here it sits inside multiple expansion. Value growth at the exit multiple instead and growth becomes 270 while multiple expansion falls to 100. The total does not change, but the story does, so state which order you used.

Capitalising makes EBITDA 50 higher and net income 30 higher, but free cash flow is identical. Expensed, the 50 hits EBITDA and net income falls 37.5 after tax. Capitalised, only 10 of amortisation hits profit, so net income falls 7.5, and the 50 moves into investing cash flow as capex. Operating cash flow is 50 higher, capex 50 higher, and free cash flow is minus 37.5 under both.

If the same cash goes out, what actually changes?

Buying a laptop for work: you can think of it as this month's expense, or as a tool that lasts five years and costs a fifth each year. Your bank balance falls by the full price either way. Capitalising versus expensing changes which profit lines carry the cost and which cash flow section shows the outflow; it does not change the cash that left. That is why analysts compare companies on free cash flow when accounting choices differ.

Expensed: EBITDA falls 50, tax falls 12.5, net income falls 37.5. Capitalised: EBITDA is untouched, the year's amortisation is 50 over 5, which is 10, and net income falls 7.5. So capitalising shows EBITDA 50 higher and net income 30 higher in year one. In operating cash flow the capitalised company shows +12.5 against -37.5, a 50 swing, but it also shows 50 of capex in investing.

What assumption is doing the work on tax?

The question says the spend is deductible in year one either way, so the cash tax saving is 12.5 under both. In the capitalised books, tax expense follows the smaller 10 charge, and the 10.0 difference becomes a deferred tax liabilityTax that the accounts recognise as owed later because taxable profit was lower than book profit this year. that is added back in operating cash flow. If the tax authority instead required capitalisation too, the cash tax saving would fall to 2.5 and free cash flow under capitalisation would be -47.5, 10 lower: tax timing is the only route by which the choice reaches cash. State which rule you are assuming.

Why bankers care: two otherwise identical software companies, one capitalising development and one expensing it, will show different EBITDA margins. Valuing both on EV/EBITDA without adjusting flatters the one that capitalises. Look at capitalised development in the cash flow statement and either add it back to the expensing peer or deduct it from the capitalising one.

About 10%. The Gordon growth model says price equals next year's dividend over (cost of equity minus growth). Rearranged, cost of equity equals dividend yield plus growth. The dividend yield is payout over P/E, 0.6 / 20, which is 3%. Adding 7% growth gives 10%. If the 20x were on trailing earnings, the answer would be slightly higher, about 10.2%.

How does a price turn into a return?

If a flat costs Rs 1 crore and rents for Rs 3 lakh a year, rising 7% a year, a buyer is earning 3% in rent now and 7% a year from the rent growing, about 10% in total. A buyer of a growing income stream earns its current yield plus its growth, so a price plus a growth assumption tells you the return the buyer must be expecting. For a share, the income is the dividend.

Why is the dividend yield payout over P/E?

Dividend over price is (payout x earnings) over (P/E x earnings), and earnings cancel, leaving 0.6 / 20 = 3%. You never need the share price or the earnings per share, only the two ratios. That is the trick the interviewer is looking for: spotting that the units cancel.

How much should you trust the 10%?

Only as far as the growth input. The dividend yield is observable; the 7% perpetual growth is an opinion. Drop it to 6% and the same price implies 9%. Check consistency too: growth of 7% with 40% of earnings retained implies a return on retained equity of about 17.5%, since growth equals retention times return on equity. If the company cannot earn that, the 7% is too high.

7/16, about 43.8%. Put one arrival time on each axis of a square from 12:00 to 1:00. They meet when the two times are within 15 minutes, a band along the diagonal. They miss in two corner triangles with legs of 45 minutes, each covering (3/4)^2 / 2 = 9/32 of the square. So the chance of meeting is 1 - 9/16 = 7/16.

Why turn a timing problem into a square?

If you and a friend each pick a random seat in a cinema, every pair of choices is equally likely, so the chance of sitting together is the share of seat pairs that are neighbours. Arrival times work the same way, except they are continuous. When two independent times are uniformly likely, every point in the square of arrival pairs is equally likely, so a probability is simply an area. The question becomes: what fraction of the square has the two times within 15 minutes?

Why is the corner easier to measure than the band?

The band has an awkward shape, but its complement is two right triangles. In the lower right corner, associate 1 arrives more than 15 minutes after associate 2, who has already left. That triangle runs from 12:15 to 1:00 along each side, 45 minutes, or 3/4 of the hour. Each triangle covers half of (3/4) squared, which is 9/32, so the two together cover 9/16 and the band covers the remaining 7/16.

The formula gives quick answers to the follow-ups. If each waits 30 minutes, w is a half and they meet with probability 1 - 1/4 = 75%. The limitation is the assumption of uniform arrivals: real people arrive close to the agreed time, which raises the chance of meeting, so the answer is for the model, not for actual associates.

14.5%, from the project's own beta, not the company's 10% WACC. The project is all-equity, so its discount rate is its cost of equity at the peers' asset beta: 7% + 1.5 x 5% = 14.5%. Using 10% would accept projects that cannot earn their own risk. A project returning 12% looks like a winner at 10% but destroys value at 14.5%.

Why is the company's own WACC the wrong rate?

A bank with a low cost of funds still charges a risky borrower a high interest rate, because the rate should reflect the loan, not the bank. Projects work the same way. The discount rate is the return investors demand for the risk of these particular cash flows, so it belongs to the project, not to the company that happens to own it. The company's 10% WACC is the right rate for a project as risky as the company's average business, and this one is riskier.

Where does the risk come from? Pure-play peers, companies that do only this riskier business, have an asset betaThe beta of a business with its debt stripped out: how much its operating cash flows move with the market, regardless of how it is financed. of 1.5. Because the project is all-equity, its equity beta equals that asset beta. Plug it into CAPM: 7% + 1.5 x 5% = 14.5%. For comparison, the parent's own cost of equity at a beta of 1.0 is 12%, and its WACC is lower still because it blends in cheaper debt.

What does the wrong rate cost in practice?

Take a project costing 100 that pays 12 a year forever, an IRR of 12%. At 10% it is worth 120, an NPV of +20. At 14.5% it is worth 82.8, an NPV of -17.2. A company that discounts everything at its own WACC systematically accepts risky projects that destroy value and rejects safe ones that would create it. The limitation: peer betas are estimated with error, so use several peers and a median, and if the project were part-funded with debt you would relever the asset beta for that financing.

Division C grew fastest, at 25%, but Division A contributed most, adding Rs 42 crore of the Rs 111 crore increase. Growth rates are 42/420 = 10%, 39/260 = 15% and 30/120 = 25%. Rupee increases are 42, 39 and 30. A's slower rate on a much bigger base adds more. Total revenue grew 13.9%, and A supplied 37.8% of that growth.

Why are these two different questions?

A child's pocket money rising from 100 to 150 is 50% growth; a parent's salary rising from 1 lakh to 1.1 lakh is 10%. The household budget still cares far more about the salary. A growth rate measures change relative to each part's own size; contribution measures change in rupees, which is what adds up to the total. Online tests ask both in one question because candidates who rush answer the same division twice.

How do you do it fast under a timer?

Compute the increases first, because they are subtractions: 42, 39, 30. Then compute rates only where needed, using easy fractions: 42 on 420 is a tenth, 30 on 120 is a quarter, 39 on 260 sits between them at 15%. Write the two added columns next to the table rather than holding the numbers in your head, because the next question in the set often reuses them.

That weighting is the third trap a test can set. The simple average of 10%, 15% and 25% is 16.7%, which is wrong because it gives the small division the same say as the large one. Total revenue went from 800 to 911, a rise of 13.9%. If an option shows 16.7%, it is there to catch exactly that shortcut.