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

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Any Reported at a firm Standard

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024A portfolio rises 25% and then falls 20%. Where does it end, compared with where it started?Returns and compoundingWarm upLong-only asset managementSell-side equity research
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Answer in five seconds.
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Exactly where it started. Rs 100 rises 25% to Rs 125. The 20% fall is taken on Rs 125, which is Rs 25, bringing it back to Rs 100. As factors, 1.25 x 0.80 = 1.00. The rupee gain and the rupee loss are the same Rs 25; they look different as percentages because they are measured on different bases.
Why does a 20% fall cancel a 25% gain?
Think of a shop that marks a Rs 100 item up to Rs 125, then offers 20% off the new price. The customer pays Rs 100: the discount is taken on Rs 125, so it is worth Rs 25, the same as the mark-up. A percentage change is always measured against the level just before it, so a fall taken on a higher base removes more rupees per point than the rise added.
Rs 100 rises by Rs 25 to Rs 125, a 25% gain on a base of 100, and then falls by the same Rs 25 back to Rs 100, which is only a 20% fall because it is measured on a base of 125.
What is the general rule?
Multiply the growth factors, never add the percentages. 1.25 x 0.80 is exactly 1.00. To undo a rise of r you need a fall of r divided by (1 plus r): 0.25 / 1.25 = 20%; to undo a fall of r you need a rise of r divided by (1 minus r). That asymmetry is why a 50% loss needs a 100% gain to recover, while a 50% gain is undone by a 33% loss.
The relationship
$(1 + 0.25)(1 - 0.20) = 1.25 \times 0.80 = 1.00$
1.25 the growth factor for a 25% rise
0.80 the growth factor for a 20% fall
What it says in wordsChain the growth factors by multiplying; the product is where you end relative to the start.
For an analyst this matters when reading reported returns. A fund that was up 25% last year and down 20% this year shows an average annual return of 2.5% but has made nothing. The average of percentage returns is not the return an investor earned; the compound return, here 0%, is.
Where candidates lose it
The fast wrong answer is up 5%, from adding 25 and minus 20. The question is built so that the rupee amounts match exactly, to see whether you check the base.
Say the rupee path, 100 to 125 to 100, then the factors 1.25 x 0.8 = 1. That takes five seconds and shows method.
What the interviewer asks next
What if the order is reversed: down 20% then up 25%?
A stock falls 40%. What gain does it need to get back to where it started?
Why do fund fact sheets show compound annual growth rather than an average of yearly returns?
065Money grows at 12% a year, compounded. Roughly how long does it take to double, and how long to grow eightfold?Returns and compoundingWarm upLong-only asset managementBuy-side equity research
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Answer inside ten seconds.
Show the worked solution
About 6 years to double and about 18 years to grow eightfold. The rule of 72 says the doubling time is roughly 72 divided by the percentage rate, so 72 / 12 = 6 years; the exact figure is 6.1 years. Eightfold is three doublings, 2 x 2 x 2, so it takes three doubling times, about 18 years, and exactly 18.3. It is not eight times six.
Why does 72 divided by the rate work?
Think of a savings pot that earns 12% a year and never has anything taken out. Each year's interest earns interest the next year, so the pot grows faster in rupees every year while growing at the same percentage. The time to double depends only on the rate, and for everyday rates it is close to 72 divided by the rate in percent. The exact answer is the log of 2 over the log of 1.12, 6.12 years. The number 72 is used because it sits near the exact constant and divides neatly by 2, 3, 4, 6, 8, 9 and 12.
One rupee at 12% is worth 1.97 after 6 years, 3.90 after 12 and 7.69 after 18, so each six-year stretch roughly doubles the pot and eightfold takes about 18 years.
Why is eightfold 18 years and not 48?
Because growth multiplies. Eight is 2 x 2 x 2, so eightfold is three doublings, and each doubling takes the same six years whatever the size of the pot. The same counting gives the other useful anchors: fourfold is two doublings, 12 years; a thousandfold is about ten doublings, since 2 to the power 10 is 1,024, so about 60 years. Tenfold is a bit more than three doublings, exactly 20.3 years.
The relationship
$t_{2\times} \approx \frac{72}{12} = 6 \qquad t_{8\times} = 3 \times t_{2\times} \approx 18$
t_2x years to double
t_8x years to grow eightfold
72 the rule of 72 constant
12 the growth rate in percent
What it says in wordsDivide 72 by the rate for one doubling, then count how many doublings the target needs.
Rate Rule of 72 Exact doubling time
4% 18.0 years 17.67 years
8% 9.0 years 9.01 years
12% 6.0 years 6.12 years
18% 4.0 years 4.19 years
24% 3.0 years 3.22 years
The rule of 72 is almost exact near 8%, a little generous below it and increasingly short of the true doubling time as rates climb above 15%.
Where does an analyst use this?
Anywhere a growth rate needs to be turned into a sense of scale, fast. A company promising 24% growth is promising to double every three years; a fund charging 2% a year in fees gives up about a third of the pot over twenty years; inflation at 6% halves the value of money in about 12 years. Turning rates into doubling times is the quickest way to check whether a claim is plausible before any spreadsheet is opened.
Where candidates lose it
Eight times six, 48 years, is the whole trap. It treats compounding growth as if it added the same amount each year, which is exactly the straight-line instinct the question is designed to catch.
The quieter slip is presenting the rule as exact. Say about 6, a touch over 6 exactly, and you have shown you know the rule is an approximation that works best near 8%.
What the interviewer asks next
How long does it take to grow tenfold at 12%? (about 20 years)
At what rate does money double in five years?
Prices rise 6% a year. How long until the same basket costs twice as much?

1.25	the growth factor for a 25% rise
0.80	the growth factor for a 20% fall

t_2x	years to double
t_8x	years to grow eightfold
72	the rule of 72 constant
12	the growth rate in percent

Rate	Rule of 72	Exact doubling time
4%	18.0 years	17.67 years
8%	9.0 years	9.01 years
12%	6.0 years	6.12 years
18%	4.0 years	4.19 years
24%	3.0 years	3.22 years

Try it first

Why does a 20% fall cancel a 25% gain?

What is the general rule?

Where candidates lose it

What the interviewer asks next

Try it first

Why does 72 divided by the rate work?

Why is eightfold 18 years and not 48?

Where does an analyst use this?

Where candidates lose it

What the interviewer asks next