Portfolio Management puzzles, solved step by step

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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
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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?
020A stock trades at 60 times earnings and pays no dividend, and its investors require 12% a year. If it should trade at 20 times earnings in ten years, what annual earnings growth does today's price require?Valuation riddlesHardFundamental asset managementAsset management
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Which growth rate does the 60x multiple imply?
Show the worked solution
About 25% a year for ten years. With no dividend, all of the 12% return must come from price, so the price in ten years must be 60 x 1.12 to the tenth, about 186 times today's earnings. If the stock then trades at 20 times, earnings must be 186 divided by 20, or 9.32 times today's. That is growth of 25.0% a year. At 15% growth the investor would earn only about 3% a year.
How do you turn a multiple into a growth forecast?
Think of paying 60 years of a shop's current profit for the shop. That only makes sense if the profit is going to be far bigger soon. A high multiple is a forecast you can read: work forward from the return investors want, and back from the multiple the stock should end at, and the growth in between is what the price assumes. Here the investor wants 12% a year with no dividend, so the price must grow 3.11 times in ten years, to about 186 times today's earnings. At a mature 20 times, earnings must reach 9.32 times today's level.
To justify 60 times earnings today and 20 times in ten years at a 12% return, earnings must grow 25% a year to 9.32 times today's level, while a 15% path reaches only 4.05 and would leave the investor with about 3% a year.
The relationship
$g=\left(\frac{PE_0(1+r)^{10}}{PE_{10}}\right)^{1/10}-1=\left(\frac{60\times 3.106}{20}\right)^{1/10}-1\approx 25\%$
PE_0, PE_{10} the multiple today, 60, and in ten years, 20
r the required return, 12%, all from price since there is no dividend
g the annual earnings growth the price requires
What it says in wordsThe earnings growth needed is the required price growth, adjusted for the multiple shrinking from 60 to 20.
What do you do with the number once you have it?
You ask how often a company sustains that growth for a decade, which is rarely, and you say so. The question the interviewer wants answered is not whether the company is good but whether the price has already paid for more than the company is likely to deliver. At a still strong 15% a year, earnings reach 4.05 times today's, the price at 20 times is about 81, and the investor earns about 3.0% a year rather than 12%. Say the limitations: the exit multiple of 20 is an assumption, and buybacks, dividends or a higher exit multiple would lower the growth needed.
Where candidates lose it
The common error is saying the stock needs to grow earnings at 12%, the required return. That ignores the multiple falling from 60 to 20, which on its own costs about 10% a year of price.
The other trap is stopping at the arithmetic. On a fundamental desk the interviewer wants the judgement: 25% for a decade is a demanding assumption, and a reverse calculation like this is how you show a price is stretched without claiming to know the future.
What the interviewer asks next
What growth is needed if the stock still trades at 40 times in ten years?
How does paying a 2% dividend change the answer?
What required return does today's price imply if earnings grow at 15%?
028Rs 10 a year forever, starting next year, is worth Rs 100 at a 10% discount rate. What is the same stream worth today if the first payment arrives only in year 4?Valuation riddlesWarm upAsset managementMutual funds
Try it first
Pick the answer before you work it.
Show the worked solution
About Rs 75.1. The formula C over r gives the value one year before the first payment. With the first payment in year 4, the stream is worth 10 over 0.10, or Rs 100, at year 3. Discount that back three years: 100 over 1.1 cubed is Rs 75.1. As a check, Rs 100 less the present value of the three missing payments, Rs 24.9, gives the same answer.
Where does the perpetuity formula put its answer in time?
Think of a pension that starts paying at retirement. Its value on the day before the first cheque is one number; its value to a 30-year-old is that number shrunk by three decades of waiting. C over r values a level perpetuity exactly one period before its first payment, so a delayed stream is the ordinary perpetuity discounted from that point. Here the first payment is in year 4, so C over r lands at year 3, where it reads Rs 100. One more discounting step gets it to today.
The stream of Rs 10 a year from year 4 is worth Rs 100 at year 3, and three years of discounting at 10% brings that to Rs 75.1 today. Subtracting the three missing payments, worth Rs 24.87 today, from an immediate perpetuity of Rs 100 gives the same Rs 75.1.
How do you check it without the formula?
Start from what you know: Rs 10 a year from year 1 is worth Rs 100. The delayed stream is that same stream with the first three payments cut out. Their present values are 9.09, 8.26 and 7.51, which add to 24.87, and Rs 100 less 24.87 is 75.13. Two methods agreeing to the paisa is what makes the answer safe to say out loud.
The relationship
$PV = \frac{C/r}{(1+r)^{3}} = \frac{100}{1.1^{3}} = \frac{100}{1.331} \approx 75.13$
C the yearly payment, Rs 10
r the discount rate, 10%
3 the years between today and one period before the first payment
What it says in wordsValue the perpetuity where it begins, then discount that single sum back to today.
This is the same step as a terminal value in a discounted cash flow. The Gordon growth value at the end of year 5 is a year 5 number and must be discounted five years, not six. Counting the periods wrong by one is the most frequent error in valuation models, and this puzzle is the cleanest place to catch it.
Where candidates lose it
The two wrong answers come from the timeline. Discounting by 1.1 to the fourth assumes the formula lands at the first payment date rather than one year before it, which gives about Rs 68.3. Subtracting Rs 30 at face value ignores that early money is worth more than later money.
Draw the timeline, even in the air with your finger. Say where C over r lands before you discount, and offer the subtraction check.
What the interviewer asks next
What is the stream worth if the payments grow at 3% a year from year 4?
How much is the delay costing, as a share of the undelayed value?
Where does the same off-by-one error show up in a DCF terminal value?
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?
097A stock trades at 25 times trailing earnings, pays out 40% of earnings, earns a 20% return on equity, and has a 12% cost of equity. What growth rate does the price imply, and how does it compare with the growth its reinvestment could support?Valuation riddlesHardFundamental asset managementIndian equity research
Try it first
What perpetual growth does the 25x multiple imply?
Show the worked solution
The price implies about 10.2% growth for ever, below the 12% that reinvestment could support today. Setting 25 equal to 0.4 x (1 + g) / (0.12 - g) gives g of 2.6 / 25.4. Retaining 60% at a 20% return supports 12%, but 12% for ever equals the cost of equity and would make the stock infinitely valuable, so the market is pricing in returns fading to about 17%.
What does it mean to work backwards from a multiple?
If a flat rents for Rs 30,000 a month and sells for Rs 1.2 crore, the price tells you what buyers assume about future rents, whether or not anyone says it. A P/E is the same kind of compressed forecast: fix the payout and the cost of equity, and the multiple pins down the one perpetual growth rate the price assumes. Reverse engineering the price like this is safer than forecasting growth and seeing what multiple falls out.
The relationship
$25 = \frac{0.4\,(1+g)}{0.12 - g} \;\Rightarrow\; 3 - 25g = 0.4 + 0.4g \;\Rightarrow\; g = \frac{2.6}{25.4} \approx 10.2\%$
0.4 payout ratio
0.12 cost of equity
g the perpetual growth the price implies
What it says in wordsThe multiple equals the payout grown one year, divided by the gap between the cost of equity and growth; solve for growth.
The 25 times multiple implies 10.2% perpetual growth, below the 12% that 60% retention at a 20% return supports, and the justified P/E rises from 22 times at 10% growth to 89 times at 11.5%, which is why growth near the cost of equity cannot last for ever.
Is the stock cheap because it can grow faster than the price assumes?
Not so fast. Sustainable growth is return on equity times retention, 20% x 60% = 12%. But 12% growth for ever equals the cost of equity, which would make the stock worth an infinite multiple, so no price could reflect it; the gap tells you the market expects the 20% return on new investment to fade. At the same 60% retention, the price is consistent with a long-run return on equity of about 17.1%.
That turns the question into a sharper one: will this business keep earning well above 17% on new money for a long time? That is a question about competition and moats, and it is the right place to spend the next five minutes of the interview. Say also that one-stage models are crude; a two-stage model with high growth fading to a lower rate is the usual next step.
Where candidates lose it
The common slip is declaring the stock undervalued because 12% beats 10.2%. Plugging 12% into the model gives division by zero, and a candidate who does not notice has not understood the formula.
The second slip is using the earnings yield shortcut, 12% minus 4%, which ignores the payout ratio and lands near 8%. Set up the equation and solve it.
What the interviewer asks next
What P/E would be justified if long-run growth were 9%?
Rebuild this with a two-stage model: 12% for five years, then 6%.
How would a lower payout ratio change the implied growth at the same 25x?

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%

PE_0, PE_{10}	the multiple today, 60, and in ten years, 20
r	the required return, 12%, all from price since there is no dividend
g	the annual earnings growth the price requires

C	the yearly payment, Rs 10
r	the discount rate, 10%
3	the years between today and one period before the first payment

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

How do you turn a multiple into a growth forecast?

What do you do with the number once you have it?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the perpetuity formula put its answer in time?

How do you check it without the formula?

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

Try it first

What does it mean to work backwards from a multiple?

Is the stock cheap because it can grow faster than the price assumes?

Where candidates lose it

What the interviewer asks next