Investment Banking puzzles, solved step by step

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.

EV/Revenue is 2.0x and price to sales is 1.8x. Take Rs 100 of revenue. A 20% margin gives EBITDA of Rs 20, and at 10x the enterprise value is Rs 200, two times revenue. Net debt equals one year of EBITDA, Rs 20, so equity value is Rs 200 minus Rs 20, which is Rs 180, or 1.8 times revenue.

How do multiples convert into one another?

Think of a corner shop priced at 10 years of profit, where profit is a fifth of takings. Priced against takings, it costs two years' worth: 10 times a fifth. An enterprise value multiple converts into another by the ratio of their denominators, so EV/Revenue equals EV/EBITDA times the EBITDA margin. Working per Rs 100 of revenue keeps every step visible: EBITDA Rs 20, enterprise value Rs 200.

Why is price to sales lower than EV to revenue here?

Price to sales is an equity multiple: the market value of the shares over revenue. To reach it from enterprise value you cross the equity bridgeThe steps from enterprise value to equity value: subtract debt and similar claims, add cash.. Equity value is enterprise value less net debt, so an equity multiple sits below the enterprise multiple when a company has net debt, and above it when it holds net cash. Net debt of one year's EBITDA is Rs 20 per Rs 100 of revenue, which takes Rs 200 down to Rs 180, or 1.8x.

The limitation to state is why bankers prefer one family over the other. Revenue belongs to every funder of the business, so comparing it with enterprise value is consistent; comparing it with equity value alone mixes a whole-business line with one funder's slice. Two companies with identical operations but different debt will show the same EV/Revenue and different price to sales ratios.

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.

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.

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.

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.