Investment Banking puzzles, solved step by step
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039A company earns Rs 10 a share, has a 10% cost of equity, does not grow and pays out everything. It decides to retain half its earnings and reinvest them at a 10% return. Does the share price change?Elite boutique IBBulge bracket IB
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Does the price move?
Show the worked solution
No, the price stays at Rs 100. Paying out Rs 10 forever at a 10% cost of equity is worth 10 over 10%, Rs 100. Retaining half and reinvesting at 10% gives growth of 50% times 10%, which is 5%, on a Rs 5 dividend: 5 over (10% minus 5%) is also Rs 100. Reinvesting at exactly the cost of capital swaps cash today for cash later at a fair rate, so value does not change.
Why does growth not add value here?
Imagine lending your bonus to a friend at exactly the rate your bank pays. You will have more money later, but you are no richer today, because the bank would have paid you the same. Retained earnings create value only if the company reinvests them at more than shareholders could earn elsewhere at the same risk, which is the cost of equity. At exactly 10% the company is just the bank: the growth is real, but it is paid for rupee for rupee by the dividend given up.
Rs 10 a year forever and Rs 5 a year growing at 5% are both worth Rs 100 at a 10% cost of equity, even though the growing stream only overtakes after about 14 years; reinvesting at 15% would lift the price to Rs 200, at 6% it would cut it to Rs 71. How do the two dividend streams compare?
The flat stream pays Rs 10 every year. The growing one starts at Rs 5 and rises 5% a year, so it overtakes Rs 10 only after about 14 years. Discounted at 10%, the early shortfall and the later surplus cancel exactly, and both streams are worth Rs 100. The tool is the Gordon growth modelA valuation of a stream that grows at a constant rate forever: the next payment divided by the discount rate minus the growth rate.: price equals next year's dividend over the cost of equity minus growth, and growth equals the share retained times the return on the money reinvested.
The relationshipD1 next year's dividend, half of Rs 10 r the cost of equity, 10% g growth: half retained, times a 10% return on it What it says in wordsA smaller dividend that grows is worth exactly the same as the full dividend when the growth is bought at the cost of capital.When would the decision change the price?
Change the return on the reinvested money. At 15%, growth is 7.5% and the price doubles to Rs 200; at 6%, growth is 3% and the price falls to about Rs 71. The same retention policy creates or destroys value depending only on whether the return beats 10%. That is the sentence the interviewer is waiting for: growth is not good in itself, profitable growth is.
Where candidates lose it
The usual answer is that the price rises because the company now grows. Growth sounds good, but it is bought with the dividend given up, and at a 10% return it costs exactly what it is worth.
The opposite slip is saying the price halves because the dividend halves. That ignores the growth the retained cash buys. Run both legs through the same formula and they cancel.
What the interviewer asks next
- What if the retained earnings are reinvested at 15%?
- Why might the market still cheer a company that announces growth at its cost of capital?
- How does this connect to return on invested capital in a DCF?
069Two companies both trade at 8x EV/EBITDA. One spends 20% of its EBITDA on capex each year and the other spends 60%. Which is cheaper, and which multiple shows it?Elite boutique IBBulge bracket IB
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Before you work it: which company is cheaper?
Show the worked solution
The company spending 20% is cheaper, and EV over EBITDA less capex shows it. Give both Rs 100 crore of EBITDA, so each is worth Rs 800 crore. The first keeps Rs 80 crore after capex and trades at 10x that cash figure; the second keeps Rs 40 crore and trades at 20x. EV/EBITDA cannot see capital intensity, so when capex differs this much, compare value against cash earnings instead.
Why does the same 8x mean different things?
Two taxi drivers each earn Rs 1 lakh a month before costs. One drives a new car that needs a service a year; the other drives an old one that eats Rs 60,000 a month in repairs. You would not pay the same for both businesses. EBITDA is earnings before the spending needed to keep the assets working, so two companies with equal EBITDA can hand their owners very different amounts of cash. At 8x, both are valued at Rs 800 crore on Rs 100 crore of EBITDA. The first spends Rs 20 crore of that on capex, the second Rs 60 crore, and that difference never appears in the headline multiple.
Both companies carry Rs 100 crore of EBITDA and an Rs 800 crore enterprise value, but after capex the first keeps Rs 80 crore and the second Rs 40 crore, so on EV over EBITDA less capex they trade at 10x and 20x and the first is the cheaper company. Which multiple fixes it?
Take the capex off before dividing. EV over EBITDA less capex puts the two companies on cash they can actually keep, and the gap opens up to 10x against 20x. EV over EBIT does a rougher version of the same job, because depreciation is yesterday's capex spread over time, but it lags when a company is spending more than it depreciates. The cleanest version is EV over unlevered free cash flow, which also takes tax and working capital off. Name the limitation too: capex can be lumpy, so use a normal year or an average, and growth capex that adds capacity deserves a different reading from maintenance capex that merely keeps the lights on.
Rs crore a year Company A Company B EBITDA 100 100 Capex 20 60 EBITDA less capex 80 40 Enterprise value 800 800 EV / EBITDA 8.0x 8.0x EV / (EBITDA less capex) 10.0x 20.0x On identical EBITDA and value, company A keeps Rs 80 crore after capex and company B Rs 40 crore, so company B costs twice as much per rupee of cash earnings. The relationshipEV enterprise value, Rs 800 crore for both EBITDA Rs 100 crore for both capex capital spending, Rs 20 crore and Rs 60 crore What it says in wordsDivide value by the cash left after capex, and the capital-hungry company looks twice as expensive.When would the market be right to pay 8x for both?
Say the case against your own answer. If company B's Rs 60 crore of capex is building capacity that will lift EBITDA sharply, the market may be paying for growth that company A will never deliver. The question has no growth figures, so the honest answer is that on today's cash the 20% company is cheaper, and that the interviewer would need to tell you about growth and the split between maintenance and growth capex before you could defend an equal multiple.
Where candidates lose it
The common loss is answering that neither is cheaper because the multiples match. The interviewer has deliberately picked a multiple that is blind to capex to see whether you know what EBITDA leaves out.
The second loss is naming the fix without the limitation. Capex is lumpy and some of it is growth, so say which capex you would use and why.
What the interviewer asks next
- Both companies also have the same EV/EBIT. What would that tell you about their depreciation?
- How would you separate maintenance capex from growth capex if the company does not disclose the split?
- Which industries tend to show the biggest gap between EV/EBITDA and EV/(EBITDA less capex)?
076An office building earns net operating income of Rs 10 crore a year and trades at an 8% cap rate. If cap rates fall to 7%, what happens to its value?Elite boutique IBBulge bracket IB
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Before you compute: how much does the value move?
Show the worked solution
The value rises about 14%, from Rs 125 crore to Rs 142.9 crore. A cap rate is the income yield a buyer demands, so value is net operating income divided by the cap rate. Rs 10 crore over 8% is Rs 125 crore; the same Rs 10 crore over 7% is Rs 142.9 crore. The ratio of the two values is 8 over 7, a rise of 14.3%. Nothing happened to the building; the market now pays more for each rupee of its rent.
Why does a lower cap rate mean a higher value?
A fixed deposit paying Rs 10,000 a year is worth Rs 1.25 lakh if savers demand 8%, and Rs 1.43 lakh if they will accept 7%. The cheque has not changed; the price people pay for it has. A cap rate is the yield a buyer requires on a building's income, so value is income divided by the cap rate, and a fall in the required yield raises the price of the same income. Rs 10 crore over 0.08 is Rs 125 crore; over 0.07 it is Rs 142.9 crore. The cap rateCapitalisation rate: net operating income divided by the property value, the income yield a buyer requires. works exactly like a bond yield, and value moves the other way.
With net operating income fixed at Rs 10 crore, a fall in the cap rate from 8% to 7% lifts the building's value from Rs 125 crore to Rs 142.9 crore, a rise of 14.3%, because value is income divided by the cap rate and 8 over 7 is 1.143. How do you get the percentage without computing either value?
The income cancels. The ratio of new to old value is the old cap rate over the new cap rate, 8 over 7, so the rise is one seventh, 14.3%. The move is not symmetric: the same one point rise, from 8% to 9%, would cut value by 8 over 9 minus 1, about 11.1%. Small cap rates make the swings bigger: from 5% to 4% is a 25% jump in value, from 10% to 9% only 11.1%. Say that pattern after the number, because the follow-up is usually about a different starting rate.
The relationshipV the building's value NOI net operating income, Rs 10 crore a year c the cap rate, 8% falling to 7% What it says in wordsValue is income over the cap rate, so the value changes by the ratio of the old cap rate to the new one.Cap rate Value of Rs 10 crore of NOI, Rs crore Change from the 8% case 6% 166.7 +33.3% 7% 142.9 +14.3% 8% 125.0 +0.0% 9% 111.1 -11.1% 10% 100.0 -20.0% Each one point step in the cap rate moves the value by a different percentage, larger as the cap rate gets lower, because the value is a fixed income divided by a shrinking rate. What would a banker add about why cap rates move?
Cap rates track the cost of money and the perceived risk of the rent. A fall from 8% to 7% usually means lower interest rates, more buyers chasing the asset, or more confidence in the tenant, and none of those is something the owner did. That is why a real estate team separates value created by raising income from value handed over by the market, and why an owner who bought at 8% and sells at 7% has earned a 14% gain on cap rate compression alone. Mention the limit: the formula assumes a stable income, so a building with leases about to expire does not deserve the same cap rate as one with ten years of rent locked in.
Where candidates lose it
The common loss is reading a falling cap rate as bad news for the value, because falling sounds like a fall. A cap rate is a yield, and a lower required yield means a higher price for the same income.
The second loss is saying 12.5%, from 1 over 8, or 1%, from the one point move. The value scales by the ratio of the rates, 8 over 7, and saying that ratio aloud is the answer.
What the interviewer asks next
- NOI also rises 5% as the cap rate falls to 7%. What is the value now?
- Why is a one point move in the cap rate a bigger deal at 5% than at 10%?
- If the building was bought with 60% debt at the 8% cap rate, what did the cap rate move do to the equity?
097A stock trades at 20x next year's earnings, pays out 60% of earnings as dividends, and its dividends are expected to grow at 7% a year forever. What cost of equity is the market implying?Elite boutique IBBulge bracket IB
Try it first
What return is a buyer at this price implicitly expecting?
Show the worked solution
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.
A P/E of 20 and a 60% payout give a dividend yield of 3%, and adding 7% expected dividend growth implies that buyers at this price expect a 10% return on equity. Why is the dividend yield payout over P/E?
The relationshipP share price, 20 times next year's earnings E D_1 next year's dividend, 60% of E k_e cost of equity implied by the price g perpetual dividend growth, 7% What it says in wordsThe return implied by a price is next year's dividend yield plus the dividend's growth rate.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.
Where candidates lose it
The fast wrong answer is 5%, the earnings yield, which treats all earnings as paid out and ignores growth. A close second is answering 7%, quoting the growth rate as if it were the return.
The quieter slip is mixing trailing and forward. Say which earnings the 20x is on; if trailing, grow the dividend one year before dividing.
What the interviewer asks next
- What P/E would the stock trade on if the cost of equity were 12% with the same payout and growth?
- Check the 7% growth against retention and return on equity: is it consistent?
- Why might the implied cost of equity differ from a CAPM estimate for the same company?