Portfolio Management puzzles, solved step by step

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

Showing 1–4 of 4 · filtered from 100Clear filters

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
Try it first
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%?
041Two stocks both have an 11% cost of equity. The value stock's dividends grow 3% a year forever and the growth stock's grow 8%. Using a constant-growth model, what is each stock's equity duration, and roughly how much does each fall if the cost of equity rises by 50 basis points?Valuation riddlesHardBlackRockNew York · 2026BlackRockNew York · 2026
Try it first
Which stock is more sensitive to a rise in the discount rate, and by roughly how much?
Show the worked solution
Durations of 12.5 and 33.3 years; the value stock falls about 6% and the growth stock about 14%. With price equal to D over (r minus g), duration is 1 over (r minus g): 1 over 0.08 and 1 over 0.03. A 50 basis point rise takes the spreads to 8.5 and 3.5 points, so prices fall by 1 minus 8/8.5, which is 5.9%, and 1 minus 3/3.5, which is 14.3%.
Why does a stock have a duration at all?
Think of two people valuing a lottery ticket: one pays out a small sum every year starting now, the other pays little now and a lot decades from now. If the interest rate rises, the second ticket loses far more value, because its money is further away and gets discounted for longer. An equity is a stream of future cash flows, so like a bond it has a duration: the weighted distance to its cash, and the growth stock's cash sits much further out. In the constant-growth model that distance comes out as 1 over (r minus g).
At an 11% cost of equity, the value stock growing 3% has a duration of 12.5 years and loses 5.9% if the rate rises half a point, while the growth stock growing 8% has a duration of 33.3 years and loses 14.3%. The dashed lines show that the straight duration estimate overstates both falls.
How do you get the price changes exactly, and why is the duration estimate too big?
The price is proportional to 1 over (r minus g), so compare spreads before and after. For the value stock the spread moves from 8 to 8.5 points, a 5.9% price fall; for the growth stock it moves from 3 to 3.5, a 14.3% fall. Duration times the rate change gives 6.25% and 16.7%, a little larger, because price is a curved function of the rate: like a bond with convexity, the stock loses less than the straight-line estimate. The curvature matters more the longer the duration.
The relationship
$P = \frac{D_1}{r-g} \qquad D_{eq} = -\frac{1}{P}\frac{dP}{dr} = \frac{1}{r-g} \qquad \frac{P_{new}}{P_{old}} = \frac{r-g}{r + \Delta r - g}$
D_1 next year's dividend
r the cost of equity, 11%
g the perpetual growth rate, 3% or 8%
D_{eq} equity duration, the percentage price change per point of rate change
What it says in wordsIn a constant-growth model, sensitivity to the discount rate is one over the gap between the rate and the growth rate.
This is the arithmetic behind a familiar market pattern: when real yields rise sharply, long-duration growth stocks usually fall more than value stocks. Say the limitation too. The model assumes growth is fixed while the rate moves; in practice rates often rise because growth is strong, which can offset part of the fall. And no real company grows at 8% forever, so the growth stock's duration is a stylised upper figure.
Where candidates lose it
Candidates often say both stocks move the same because they share a cost of equity, or they compute the percentage change of r, about 4.5%, and apply it to both prices. Neither uses the spread between r and g, which is the whole mechanism.
State the formula, name the spread, and give the two durations first. Then offer the exact falls and say why they are smaller than the duration estimate.
What the interviewer asks next
What happens to the growth stock's duration if its growth rate rises to 10%?
Why might a rate rise driven by stronger growth hurt growth stocks less than this model suggests?
How would you hedge the rate sensitivity of a growth-heavy portfolio?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis): Which equities have duration ? multiple stocks vs value stocks
Asked at BlackRock, Risk and Quantitative Analysis, New York, 2026 (Wall Street Oasis): Which equities have duration? VaR, market views, stock valuation.
072An equity index trades at 22 times forward earnings, pays out 40% of earnings as dividends, and its long-run earnings growth is expected to be 10% a year. The 10-year government bond yields 7%. Which asset is cheaper?Valuation riddlesHardPIMCOSan Diego · 2026
Try it first
The earnings yield is 4.5% and the bond yields 7%. What does that comparison tell you?
Show the worked solution
On expected return, equities offer about 11.8% against the bond's 7%, a premium of about 4.8 points. The earnings yield, 1 over 22 or 4.5%, looks worse than 7%, but it ignores growth. The dividend yield is 40% of 4.5%, or 1.82%, and adding 10% growth gives about 11.8%. Whether equities are cheaper depends on whether a 4.8-point premium pays enough for equity risk.
Why is 4.5% against 7% the wrong comparison?
Compare a fixed-rent lease with a shop whose profits grow each year. The lease might pay more in year one, but the shop's income keeps rising. A bond's yield is everything it will ever pay, while an earnings yield is only the starting point of a stream expected to grow, so the two cannot be compared directly. The earnings yield of 4.5% sits 2.5 points below the bond, and that gap says almost nothing about which is cheaper.
The earnings yield of 4.5% looks worse than the bond's 7%, but the index's expected return, a 1.8% dividend yield plus 10% growth, is about 11.8%, a premium of about 4.8 points over the bond.
How do you put them on the same footing?
Estimate the equity's expected return the way you would a bond's. For a stock or an index, that is roughly the dividend yield plus long-run growth, the logic of the Gordon growth model. The dividend yield is the payout ratio times the earnings yield, and the growth rate does the rest. Here 40% of 4.55% is 1.82%, and 10% growth takes the total to 11.82%. Against 7% on the bond, equities offer 4.82 extra points a year.
The relationship
$E[R] \approx \frac{D}{P} + g = 0.40 \times \frac{1}{22} + 10\% = 1.82\% + 10\% = 11.82\%$
D/P the dividend yield, the payout ratio times earnings over price
g long-run growth in earnings and dividends, 10%
What it says in wordsAn equity's expected return is roughly its dividend yield plus the rate at which its dividends grow.
Then test the growth number, because the answer rests on it. Growing earnings 10% while paying out 40% means reinvesting 60% at a return on equity of about 16.7%, which is demanding for a whole market. If growth were 8%, the premium would shrink to about 2.8 points. A view on which asset is cheaper is really a view on whether that premium, after testing growth, pays enough for the extra risk of equities. That is the judgement to state, with the numbers that drive it.
Where candidates lose it
The trap is comparing the earnings yield with the bond yield and declaring bonds cheaper. That comparison ignores growth and treats a rising income stream as if it were fixed.
The second loss is taking the 10% growth at face value. Check it against the payout ratio: growth needs reinvestment, and the implied return on equity tells you whether the number is plausible.
What the interviewer asks next
What equity risk premium would you need to call equities and bonds fairly valued here?
How does inflation change the comparison between an earnings yield and a nominal bond yield?
What growth rate makes the index's expected return exactly equal to 7%?
Asked at PIMCO, Debt Capital Markets, San Diego, 2026 (Wall Street Oasis): Which is cheaper us bonds or us equities
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?

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

D_1	next year's dividend
r	the cost of equity, 11%
g	the perpetual growth rate, 3% or 8%
D_{eq}	equity duration, the percentage price change per point of rate change

D/P	the dividend yield, the payout ratio times earnings over price
g	long-run growth in earnings and dividends, 10%

0.4	payout ratio
0.12	cost of equity
g	the perpetual growth the price implies

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

Why does a stock have a duration at all?

How do you get the price changes exactly, and why is the duration estimate too big?

Where candidates lose it

What the interviewer asks next

Try it first

Why is 4.5% against 7% the wrong comparison?

How do you put them on the same footing?

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