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 1–2 of 2 · filtered from 100Clear filters

020A company's revenue grew 12% and its prices rose 5%. How much did volume grow?Growth, mix and unit economicsWarm upSell-side equity researchIndian brokerage research
Try it first
Answer in five seconds.
Show the worked solution
About 6.7%. Revenue is price times volume, so the growth factors multiply: 1.12 = 1.05 x (1 + v). Dividing, 1.12 / 1.05 = 1.0667, so volume grew about 6.67%. Subtracting 5 from 12 gives 7%, which overstates volume by the small cross term, 0.33 points.
Why divide instead of subtract?
Think of a tea stall that raises the price of a cup and also sells more cups. The takings grow for both reasons, and the extra cups are also sold at the higher price, which is a little extra neither change produces alone. Revenue is price times volume, so growth factors multiply, and the way to back one out is to divide. 1.12 divided by 1.05 is 1.0667.
Drawing revenue as price times volume, a 5% wider and 6.67% taller rectangle has 12% more area: a 5% price strip, a 6.67% volume strip and a 0.33% corner where the extra volume is sold at the higher price.
When does the shortcut of subtracting matter?
The error from subtracting is the corner of the rectangle, price growth times volume growth. At single-digit rates the corner is small, about 0.33 points here, but it grows quickly with the rates, and in high-inflation markets it becomes the whole story. With 30% price rises and 40% revenue growth, subtracting says 10% volume growth; dividing says 7.7%.
The relationship
$1 + v = \frac{1 + g_{rev}}{1 + g_{price}} = \frac{1.12}{1.05} = 1.0667$
v volume growth
g_rev revenue growth, 12%
g_price price growth, 5%
What it says in wordsVolume growth is revenue growth divided by price growth, as factors.
For an analyst this split is the first question on any results call: how much of the growth was price and how much was volume. Volume growth says whether the company is winning customers; price growth says whether it has pricing power or is passing on costs. The limitation: reported price growth often mixes true price changes with shifts in product mix, so ask how the company defines it.
Where candidates lose it
The trap is subtracting and answering 7% without a second thought. It is close enough to pass as mental arithmetic but wrong in method, and the interviewer asks the question to hear which one you use.
Say the division, give 6.7%, and add that the gap is the cross term. That turns a one-line answer into evidence of method.
What the interviewer asks next
Revenue fell 3% while prices rose 8%. What happened to volume?
How would you split growth into price, volume and mix for a company selling three products?
Why might management prefer to report volume growth before or after the effect of mix?
045A market grows 8% in a year. A company with a 20% share of it grows its sales 12%. What is its market share at the end of the year?Growth, mix and unit economicsWarm upSell-side equity researchIndian brokerage research
Try it first
Pick the new share.
Show the worked solution
About 20.7%. Set the market at 100. The company's sales go from 20 to 22.4 and the market from 100 to 108, so its share is 22.4 over 108, or 20.74%. The gain is only 0.74 of a point, because share moves with relative growth: 1.12 divided by 1.08, about 3.7% more share, on a 20% base.
Why does growing 4 points faster add less than 1 point of share?
A student who scored 20 out of 100 last term and improves her marks by 12% while the class average improves by 8% has done a little better relative to the class, not a lot. Market share is a ratio, so it changes by the ratio of the two growth rates, 1.12 over 1.08, applied to the share you started with. A 3.7% relative gain on a 20% share is about 0.74 of a point.
The company's 20% slice becomes 20.74% of a market that is itself 8% larger, because a share gain is relative growth, 1.12 over 1.08, applied to the starting share.
The relationship
$s_1 = s_0 \times \frac{1 + g_c}{1 + g_m} = 20\% \times \frac{1.12}{1.08} = 20.74\%$
s_0, s_1 market share at the start and end of the year
g_c the company's sales growth, 12%
g_m the market's growth, 8%
What it says in wordsNew share is old share times one plus company growth, divided by one plus market growth.
How do you use this the other way round in a model?
Analysts often forecast a company as market growth plus a share assumption. If you forecast 12% growth in an 8% market, you are assuming a share gain of about 0.74 points a year, and five years of that takes 20% to about 24.0%. Saying that out loud tests whether the forecast is believable: which competitor is giving up that share, and why? The limitation: this assumes the market figure and the company's sales are measured the same way, which is often not true when companies report by segment.
Where candidates lose it
The two fast wrong answers are 24%, adding growth rates to a share, and 22.4%, dividing by the old market size. Both skip the fact that the denominator grew too.
Set the market at 100 out loud. It turns the problem into 22.4 over 108, and the interviewer hears the method before the number.
What the interviewer asks next
What sales growth would the company need to reach a 22% share in one year?
If the company's share stays at 20% and the market grows 8%, what is its growth?
Why might a company gain share and still see profit fall?

v	volume growth
g_rev	revenue growth, 12%
g_price	price growth, 5%

s_0, s_1	market share at the start and end of the year
g_c	the company's sales growth, 12%
g_m	the market's growth, 8%

Try it first

Why divide instead of subtract?

When does the shortcut of subtracting matter?

Where candidates lose it

What the interviewer asks next

Try it first

Why does growing 4 points faster add less than 1 point of share?

How do you use this the other way round in a model?

Where candidates lose it

What the interviewer asks next