Equity Research puzzles, solved step by step

Puzzles: 100
Traced to a firm: 19
Topics: 11
Hard: 30

Topic

All topics Probability and brainteasers12 Expected value and decisions8 Market sizing and estimation12 Returns and compounding9 Valuation riddles12 Three statement riddles10 EPS and share count9 Cost of capital and rates8 Growth, mix and unit economics8 Mental maths6 Data and reasoning traps6

Level

Any Warm up Core Hard

Source

Any Reported at a firm Standard

Showing 11–12 of 12 · filtered from 100Clear filters

091A stock trades at 20x forward earnings, pays out 40% of its earnings as dividends, and its cost of equity is 12%. What growth rate is priced in, and what return on equity does that growth need?Valuation riddlesCoreSell-side equity researchBuy-side equity research
Try it first
What perpetual growth does 20x imply here?
Show the worked solution
The price implies 10% growth a year, which needs a return on equity of about 16.7%. A forward P/E equals the payout ratio over cost of equity minus growth, so 20 equals 0.4 over 12% minus g, and g is 10%. Growth is funded by retained profit, 60% here, so ROE must be 10% divided by 60%, about 16.7%, well above the 12% cost of equity.
How do you pull growth out of a P/E?
Start from the dividend model and divide both sides by earnings. Price is next year's dividend over cost of equity minus growth, and dividend over earnings is the payout ratio. So a forward P/E is the payout ratio divided by the gap between cost of equity and growth: 20 equals 0.4 over 12% minus g. The gap must be 2%, which puts growth at 10%. Think of it like working out a car's speed from the distance it covered and the time it took: the multiple and the payout are the readings, growth is what they imply.
The relationship
$\frac{P}{E_1} = \frac{b}{k - g} \;\Rightarrow\; 20 = \frac{0.40}{0.12 - g} \;\Rightarrow\; g = 10\% \qquad ROE = \frac{g}{1 - b} = \frac{10\%}{60\%} \approx 16.7\%$
P / E1 price over next year's earnings, the forward P/E
b the payout ratio, 40%
k the cost of equity, 12%
1 - b the share of profit retained to fund growth, 60%
What it says in wordsThe multiple tells you the growth, and the growth, divided by what is retained, tells you the return on equity needed to fund it.
Why does the growth need a particular ROE?
Growth is not free; it is paid for with retained profit. A family that saves 60% of its income and wants its wealth to grow 10% a year must earn 10% divided by 60% on its savings, about 16.7%. A company is the same: sustainable growthThe growth a company can fund from its own retained profit: return on equity times the share of profit retained. is ROE times the share retained, so 10% growth on 60% retention needs an ROE of 16.7%. That is the number to test against the company's history, not the P/E itself.
At a 40% payout and a 12% cost of equity, a P/E of 20x implies 10% perpetual growth, which needs a 16.7% return on equity; at 7.2% growth the needed ROE equals the cost of equity and the P/E is only 8.3x.
What does the curve tell you that the formula does not?
Two things. First, when ROE equals the cost of equity, growth adds nothing: the P/E is 8.3x, one over 12%, whatever the growth. Growth only earns a higher multiple when each retained rupee earns more than shareholders demand. Second, the curve is steep near 20x, so a small change in the growth assumption moves the justified multiple a lot. The limit is the single perpetual rate: a real company grows fast for some years and then slows, and a two-stage model would put less weight on the far future.
Where candidates lose it
Candidates reach 10% and stop. The question asked for two numbers on purpose: growth that is not backed by an ROE the company can earn is a story, not a valuation.
The second loss is using trailing earnings in a formula built for forward earnings, or mixing up payout and retention. Say out loud which is which: 40% is paid, 60% is kept, and the 60% funds the growth.
What the interviewer asks next
If the company's ROE has averaged 12%, what P/E would you justify?
The cost of equity rises to 13%. What growth does 20x now imply?
Why does a company with ROE below its cost of equity destroy value by growing?
099A stock has a dividend yield of 4% and pays out 60% of its earnings as dividends. What is its P/E?Valuation riddlesCoreLong-only asset managementIndian brokerage research
Try it first
Pick the P/E.
Show the worked solution
15x. Dividend yield is dividend over price, and payout is dividend over earnings. Dividing the first by the second cancels the dividend and leaves earnings over price, the earnings yield: 4% over 60% is 6.67%. The P/E is the inverse, 1 over 0.0667, which is 15x. On a Rs 100 share, the dividend is Rs 4 and earnings are Rs 6.67.
How do the two ratios combine?
Suppose a friend tells you her rent is 30% of her salary and her rent is Rs 15,000. You know her salary without asking: Rs 15,000 over 30%. The dividend works the same way: if the dividend is 4% of the price and 60% of earnings, earnings must be 4% over 60% of the price, an earnings yieldEarnings per share divided by the share price, the inverse of the P/E. of 6.67%. Once you have earnings as a share of price, the P/E is just that number turned upside down.
On a Rs 100 share the Rs 4 dividend is 60% of earnings, so earnings are Rs 6.67 with Rs 2.67 retained, an earnings yield of 6.67% and a P/E of 15x.
The relationship
$\frac{D/P}{D/E} = \frac{E}{P} = \frac{4\%}{60\%} \approx 6.67\% \qquad \frac{P}{E} = \frac{0.60}{0.04} = 15\times$
D/P dividend yield, 4%
D/E payout ratio, dividend over earnings, 60%
E/P earnings yield, the inverse of the P/E
What it says in wordsDividend yield divided by payout is the earnings yield, so the P/E is payout divided by dividend yield.
How do you check it on a single share?
Pick a price of Rs 100. A 4% yield means a dividend of Rs 4. A 60% payout means that Rs 4 is 60% of earnings, so EPS is Rs 6.67. Price over EPS, Rs 100 over Rs 6.67, is 15x, which agrees. Doing it on one share is the fastest way to catch an inverted ratio, and it is worth saying out loud because the interviewer is listening for whether you can move between the three numbers.
Where does this shortcut mislead?
It assumes the payout ratio is stable and measured on the same earnings as the P/E. A company paying a special dividend, or keeping its dividend flat while profits fall, will show a payout ratio that does not describe normal earnings, and the implied P/E will be wrong. Check whether the yield is trailing or forward, and whether the payout is on reported or adjusted earnings, before you trust the answer.
Where candidates lose it
The fast wrong answer is 25x, one over the dividend yield. It treats the dividend as if it were all the earnings. Candidates who divide the payout by the yield the wrong way round get 0.067 and call it a P/E.
Work it on a Rs 100 share and the error cannot survive: a Rs 4 dividend at a 60% payout means Rs 6.67 of earnings.
What the interviewer asks next
The payout rises to 80% with the same yield. What happens to the P/E?
If the company grows at 5% and the cost of equity is 9%, is 4% a sensible yield?
Why might a high dividend yield signal trouble rather than value?

P / E1	price over next year's earnings, the forward P/E
b	the payout ratio, 40%
k	the cost of equity, 12%
1 - b	the share of profit retained to fund growth, 60%

D/P	dividend yield, 4%
D/E	payout ratio, dividend over earnings, 60%
E/P	earnings yield, the inverse of the P/E

Try it first

How do you pull growth out of a P/E?

Why does the growth need a particular ROE?

What does the curve tell you that the formula does not?

Where candidates lose it

What the interviewer asks next

Try it first

How do the two ratios combine?

How do you check it on a single share?

Where does this shortcut mislead?

Where candidates lose it

What the interviewer asks next