Portfolio Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 31
Topics: 13
Hard: 30

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007A company trades at 20 times earnings. It uses cash that was earning 3% after tax to buy back 10% of its shares at the market price. Is the buyback accretive to earnings per share, and by how much?Valuation riddlesCoreFundamental asset managementAsset management
Try it first
Which comparison decides whether the buyback lifts earnings per share?
Show the worked solution
Yes, it is accretive, by about 4.4%. Take Rs 50 crore of earnings on 10 crore shares at Rs 100, so EPS is Rs 5.00. The buyback spends Rs 100 crore, which was earning Rs 3 crore, so earnings fall to Rs 47 crore while the share count falls to 9 crore. EPS becomes Rs 5.22. It is accretive because the 5% earnings yield beats the 3% return on cash.
What is the company actually swapping?
Suppose you hold a fixed deposit paying 3% after tax and use it to buy out a partner's share of a shop that earns 5% on its price. Your income goes up, because you replaced a 3% asset with a 5% one. A buyback is the same trade. The company gives up the after-tax return on its cash and gets back a slice of its own earnings, priced at the earnings yield, which is one over the P/E. At 20 times earnings that yield is 5%, so the swap raises earnings per share. At 33.3 times earnings the yield would be 3% and the buyback would leave EPS unchanged.
The buyback gives up cash earning 3% after tax and buys shares carrying 5% of earnings, so earnings fall from Rs 50 crore to Rs 47 crore while shares fall from 10 crore to 9 crore, and EPS rises from Rs 5.00 to Rs 5.22.
How do you get the exact number quickly?
Pick round figures and the answer falls out: Rs 1,000 crore of market value, Rs 50 crore of earnings and 10 crore shares. Earnings drop by 10% of the market value times 3%, which is Rs 3 crore, and shares drop by 10%, so EPS moves by 0.94 divided by 0.90. That is 1.0444, an accretion of 4.44%. The shortcut works for any size of company because only the ratios matter.
The relationship
$\frac{EPS_1}{EPS_0}=\frac{1-f\cdot PE\cdot y_c}{1-f}=\frac{1-0.1\times 20\times 0.03}{0.9}\approx 1.044$
f the fraction of shares bought back, 10%
PE the price to earnings multiple, 20
y_c the after-tax return on the cash spent, 3%
What it says in wordsEPS rises when the lost interest, as a share of earnings, is smaller than the share of the company bought back.
Does accretive mean the buyback was a good idea?
No, and a portfolio manager is expected to say so. Accretion is arithmetic about earnings per share; value is created only if the company paid less for its shares than they are worth. A company on a low P/E almost always gets an accretive buyback, even if the shares are overpriced, and one on a high P/E can create value with a dilutive buyback if the shares are cheap relative to future growth. Accretion also ignores risk: swapping cash for shares makes the remaining equity more levered.
Where candidates lose it
Candidates answer that a buyback always raises EPS because there are fewer shares. That forgets the lost income on the cash, and at a high enough P/E the buyback dilutes.
The second trap is stopping at accretive. On a buy-side desk the interviewer usually follows with whether it creates value, and the answer that separates candidates is that the two are different questions.
What the interviewer asks next
At what P/E does the same buyback become dilutive?
What if the buyback is funded with new debt at 8% before a 25% tax rate?
Why might a buyback that is accretive still destroy value for the remaining shareholders?
059Why should I buy your college, and how much would you sell it for? Suppose it earns an operating surplus of Rs 40 crore this year, the surplus grows 5% a year for the foreseeable future, and a buyer wants a 12% return.Valuation riddlesCoreWellington ManagementBoston · 2024
Try it first
What price does the growing surplus support, before land?
Show the worked solution
About Rs 600 crore for the operating business, before any value in the land. Next year's surplus is Rs 40 crore grown 5%, or Rs 42 crore. A surplus that grows forever at 5% and is valued at 12% is worth next year's amount divided by the 7-point gap: 42 over 0.07 is Rs 600 crore. The reason to buy is the durability of that surplus: steady demand for seats and fees that can rise with costs.
What is the question really asking?
It is a stock pitch in disguise. The interviewer wants two things in order: why this asset produces reliable cash, and what that cash is worth. Answer the why with the quality of the surplus, and the how much with a valuation you can do out loud. For a college the why is simple to say: students keep applying every year, fees are paid in advance, and a college with a good name can raise fees roughly in line with its costs. Those are the reasons the surplus can be treated as growing and durable.
Why divide by 12% less 5%?
Think of a rented flat whose rent rises every year. A buyer asking for a 12% return on a rent that grows 5% needs only 7% from the current rent; the other 5% arrives through growth. A cash flow growing at g forever, valued at a required return r, is worth next year's cash flow divided by r minus g. Here that is Rs 42 crore over 0.07, which is Rs 600 crore. Dividing this year's Rs 40 crore instead gives Rs 571 crore, a common small slip: the buyer receives next year's surplus, not this year's.
The relationship
$V = \frac{S_1}{r - g} = \frac{40 \times 1.05}{0.12 - 0.05} = \frac{42}{0.07} = 600$
S_1 next year's surplus, Rs crore
r the buyer's required return
g the permanent growth rate of the surplus
What it says in wordsA growing perpetuity is worth next year's payment divided by the gap between the required return and the growth rate.
With growth fixed at 5%, the college is worth Rs 600 crore at a 12% required return, but Rs 700 crore at 11% and Rs 525 crore at 13%, because value depends on the gap between return and growth and that gap is small.
Now say what the number is sensitive to. One point on the required return moves the value by Rs 100 crore up or Rs 75 crore down, because a 7-point gap becoming 6 or 8 is a large change in proportion. The land and buildings may be worth more than the operating surplus, so a seller would also ask what the campus fetches as property. And check the structure before promising anyone the surplus: many colleges, in India among other places, are run by trusts or societies, and whether an owner can take surplus out at all is a legal question to confirm.
Where candidates lose it
The trap is diving into a formula without answering why. The question starts with why should I buy, and a candidate who opens with a number has skipped the half the interviewer cares about most.
The arithmetic trap is dividing Rs 40 crore by 12%, which treats a growing surplus as flat and values it at Rs 333 crore. Name the growth, use next year's surplus, and divide by the gap.
What the interviewer asks next
What growth rate is the seller implicitly assuming if he asks Rs 800 crore?
How would you value the land separately, and when would it exceed the value of the operating business?
What would make you use a higher required return for this college than for a listed education company?
Asked at Wellington Management, Investment Research, Boston, 2024 (Wall Street Oasis): Why should I buy your College and how much would you sell it for?
084A bank earns a 16% return on equity, grows at 8% a year and has a 13% cost of equity. What price to book is justified? What happens if its return on equity falls to 13%?Valuation riddlesCoreFundamental asset managementIndian equity research
Try it first
At a 13% return on equity, what is the justified price to book?
Show the worked solution
1.6 times book at a 16% return on equity, falling to 1.0 times at 13%. The justified multiple is the return on equity less growth, over the cost of equity less growth: 8 over 5 at 16%, and 5 over 5 at 13%. When a bank earns exactly its cost of equity, a rupee of book is worth a rupee, and growth adds nothing.
Why does a bank earning its cost of equity trade at book?
Suppose a friend offers to take Rs 1 lakh and pay you exactly the return you could get anywhere else with the same risk. The deal is worth Rs 1 lakh; no more, however large your friend makes it. A bank reinvesting at a return equal to its cost of equity creates no value on the money it keeps, so its book is worth exactly book. Only the spread between the two rates earns a premium.
With growth at 8% and a 13% cost of equity, the justified price to book is 1.0 times where return on equity equals 13% and rises to 1.6 times at 16%; below 13% the line falls under book, because growth then destroys value.
Where does the formula come from?
Start from the dividend discount model. Next year's dividend is book times return on equity times the share paid out. To grow at 8% with a 16% return, the bank must retain half its earnings, since growth is return on equity times retention; that leaves a payout of 1 minus g over ROE. Substituting gives price over book equal to (ROE minus g) over (r minus g). At 16%, retention is 50%, payout 50%, and the multiple 1.6 times.
The relationship
$\frac{P}{B} = \frac{ROE - g}{r - g} = \frac{0.16 - 0.08}{0.13 - 0.08} = 1.6$
ROE return on equity, 16%
g long-run growth, 8%
r cost of equity, 13%
What it says in wordsThe premium to book is the return spread over the cost spread, both measured from the growth rate.
The limit is that every input is a long-run steady state. A bank with a cyclical credit cost earns 20% in good years and 8% in bad ones, and the formula wants the through-cycle figure. Say so, then say which input you would argue about first.
Where candidates lose it
The common slip is scaling: 13 is less than 16, so the multiple falls in proportion to about 1.3 times. The relationship is not proportional; it is anchored at 1.0 times where the two rates meet.
The deeper miss is not seeing that growth is only good when return on equity beats the cost of equity. Below 13%, faster growth lowers the multiple, and saying that out loud is what the question is testing.
What the interviewer asks next
The bank earns 11% on equity. What happens to the multiple if growth rises from 8% to 10%?
What return on equity does a price to book of 2.0 times imply at the same growth and cost of equity?
Why do lenders with similar returns on equity trade at very different multiples?

f	the fraction of shares bought back, 10%
PE	the price to earnings multiple, 20
y_c	the after-tax return on the cash spent, 3%

S_1	next year's surplus, Rs crore
r	the buyer's required return
g	the permanent growth rate of the surplus

ROE	return on equity, 16%
g	long-run growth, 8%
r	cost of equity, 13%

Try it first

What is the company actually swapping?

How do you get the exact number quickly?

Does accretive mean the buyback was a good idea?

Where candidates lose it

What the interviewer asks next

Try it first

What is the question really asking?

Why divide by 12% less 5%?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a bank earning its cost of equity trade at book?

Where does the formula come from?

Where candidates lose it

What the interviewer asks next