Investment Banking puzzles, solved step by step

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.