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

Showing 1–6 of 6 · filtered from 100Clear filters

010Without a calculator: which leaves you with more, 15% growth a year for five years or 20% a year for four years?Mental mathsCoreLong-only asset managementBuy-side equity research
Try it first
Instinct first: which is bigger?
Show the worked solution
20% for four years, narrowly: 2.07x against 2.01x. The rule of 72 says 20% doubles in about 3.6 years and 15% in about 4.8, so both paths just pass double, the 20% path by more. Check exactly: 1.2 squared is 1.44 and 1.44 squared is 2.0736; 1.15 squared is 1.3225, squared again is about 1.749, times 1.15 is 2.0114.
How do you get close without multiplying anything?
Think of two savers racing to double their money. The rule of 72A shortcut: dividing 72 by a growth rate in per cent gives roughly the number of years it takes to double. tells you when each one doubles, and whoever has more time left after doubling is ahead. At 20%, 72 over 20 is 3.6 years, leaving 0.4 years of 20% growth. At 15%, 72 over 15 is 4.8 years, leaving 0.2 years of 15%. The 20% saver has twice as long left at a higher rate, so 20% for four years should win.
The 20% path crosses double at 3.8 years and reaches 2.07x at year four, while the 15% path crosses double only at 4.96 years and reaches 2.01x at year five, so four years at 20% ends slightly ahead.
Why check the edge exactly, and how?
The rule of 72 is rough at high rates: 20% actually doubles in 3.8 years, not 3.6, and 15% in 4.96, not 4.8. When a shortcut says the answer is close, the gap it shows can be smaller than the shortcut's own error, so you check the edge with exact arithmetic. Here it is easy: square 1.2 twice to get 2.0736, and for 1.15 square twice and multiply once more to get 2.0114. The margin is about 3%, real but narrow.
The relationship
$1.2^4 = (1.44)^2 = 2.0736 \qquad 1.15^5 = (1.3225)^2 \times 1.15 \approx 2.0114$
1.44 1.2 squared, two years at 20%
1.3225 1.15 squared, two years at 15%
What it says in wordsSquare twice to get four years of growth, then multiply once more for the fifth.
Why a research interviewer asks it: growth comparisons of this kind turn up every day, a company growing earnings faster for fewer years against one growing slower for longer. Saying the approximation first and then checking it shows the habit they want, and the limitation is worth one line: the answer flips if the 15% path runs for six years.
Where candidates lose it
The trap is adding the rates, 75% against 80%, or trusting that more years always wins. Both skip the compounding the question is about.
The second loss is doing long multiplication in silence. Give the doubling-time approximation out loud, then check by squaring: it shows method and gets the exact answer in under a minute.
What the interviewer asks next
How many years at 15% does it take to beat four years at 20%?
What annual rate over five years matches 20% over four?
Use the same method: 10% for seven years or 7% for ten years?
021Convert USD 2.4 billion into rupees at Rs 83 to the dollar and express the answer in crore. Then say how many crore make one billion rupees.Mental mathsWarm upIndian brokerage researchResearch KPO and GCC
Try it first
How many crore make one billion?
Show the worked solution
Rs 19,920 crore, and 100 crore make a billion. USD 2.4 billion at Rs 83 is Rs 199.2 billion. A crore is ten million, 10 to the 7, and a billion is 10 to the 9, so a billion rupees is 100 crore. Rs 199.2 billion times 100 is Rs 19,920 crore, just under a fifth of a lakh crore.
Why do people get the crore conversion wrong?
Think of two rulers marked in different units, one in inches and one in centimetres. Reading a length off one and writing it in the other is easy once you line them up, and error-prone if you do it from memory. Indian and Western number names are two rulers on the same powers of ten. The Indian system groups by lakh and crore, 10 to the 5 and 10 to the 7, while the Western system groups by thousands, so the two only line up at a thousand and at a billion, which equals 100 crore.
On one ruler of powers of ten, a lakh is 10 to the 5, a crore is 10 to the 7 and a billion is 10 to the 9, so a billion is 100 crore and Rs 199.2 billion is Rs 19,920 crore.
What is the fastest safe route in the room?
Do the currency first, then the units. 2.4 x 83 is 199.2, so Rs 199.2 billion. Multiply billions by 100 to get crore, and divide crore by 1,00,000 to get lakh crore; never convert through million unless you have to. That gives Rs 19,920 crore, or about 0.2 lakh crore. If you prefer the Western route: Rs 199,200 million divided by 10 million per crore gives the same 19,920.
Indian Power of ten Western
1 lakh 10^5 100 thousand
10 lakh 10^6 1 million
1 crore 10^7 10 million
100 crore 10^9 1 billion
1 lakh crore 10^12 1 trillion
The two naming systems on the same powers of ten; the rows to remember are a crore as ten million and a billion as 100 crore.
In an Indian research role this comes up every day, because company filings report in crore or lakh while global peers and many investors think in millions and billions. The exchange rate here is the one given in the question; for real work, use the rate on the date of the numbers you are converting and state it.
Where candidates lose it
The common loss is dropping or adding a zero: answering Rs 1,992 crore or Rs 1,99,200 crore. It happens when the conversion is done through million in the head. Line up billion with 100 crore and the error disappears.
The second is mixing digit groupings when writing the number, for example writing 19,920 crore with Western commas in one place and Indian commas in another. Pick one system per number and say which.
What the interviewer asks next
Express Rs 3.5 lakh crore in US dollars at Rs 83.
A company reports revenue of Rs 8,450 crore. What is that in million rupees?
How would you present a peer table that mixes Indian and US companies?
035A product sells with a 40% gross margin. The company cuts its price by 10%. By how much must volume rise to keep gross profit the same?Mental mathsCoreSell-side equity researchIndian brokerage research
Try it first
Answer inside ten seconds.
Show the worked solution
About 33%. Take a price of 100 and a cost of 60, so profit is 40 a unit. Cut the price by 10 and profit falls to 30 a unit, because the cost does not move. To keep gross profit the same you need 40 divided by 30 as many units, 1.33 times, so volume must rise by a third. The formula is cut divided by (margin minus cut).
Why is the volume needed so much bigger than the cut?
A tea stall sells a cup for Rs 10 that costs Rs 6 to make, keeping Rs 4. Knock Re 1 off the price and the stall keeps Rs 3, a quarter less on every cup. A price cut comes straight out of the margin, because costs do not fall with price, so the percentage fall in profit per unit is the cut divided by the margin. Here a 10% price cut removes a quarter of the 40% margin, and volume has to make up the whole quarter.
Before the cut, 100 units at Rs 40 of profit each make 4,000; after a 10% price cut each unit earns Rs 30, so the business needs 133.3 units, 33.3% more, to make the same 4,000.
The relationship
$\Delta V = \frac{c}{m - c} = \frac{0.10}{0.40 - 0.10} = 33.3\%$
c the price cut, as a share of the old price
m the gross margin, as a share of the old price
Delta V the rise in volume needed to hold gross profit
What it says in wordsThe volume needed equals the cut divided by what is left of the margin after the cut.
How does the answer change with the margin?
Gross margin Volume rise needed for a 10% price cut
20% 100%
30% 50%
40% 33%
60% 20%
80% 14%
The thinner the margin, the more volume a price cut needs; at a 20% margin a 10% cut needs volume to double.
This is why analysts treat price cuts in thin-margin businesses with suspicion. A retailer on a 20% gross margin that cuts prices 10% needs twice the volume just to stand still, while a software company on 80% needs only 14% more. Say the limitation too: the puzzle holds unit cost fixed. If volume brings purchasing discounts or spreads fixed factory costs, the bar is lower.
Where candidates lose it
The instinctive answer is 10% or 11%, which treats the cut as if it hit revenue and profit equally. It hits profit harder, because cost stays where it was.
Set the price at 100 out loud. It turns percentages into rupees, and the interviewer hears the margin fall from 40 to 30 before you give 33%.
What the interviewer asks next
What volume rise is needed if the price is cut 20% instead?
If volume rises 20% after the 10% cut, what happens to gross profit?
Why might management still cut prices when the maths looks this unfavourable?
046A company's EBITDA margin moves from 12% to 15%. Is that a 3% improvement or a 25% improvement?Mental mathsWarm upSell-side equity researchIndian brokerage research
Try it first
Which sentence would you write in a results note?
Show the worked solution
Both, if you say it properly: the margin rose 3 percentage points, which is a 25% relative rise. The difference between two percentages is measured in percentage points, here 15 minus 12, or 300 basis points. The relative change is 3 divided by 12, which is 25%. On flat sales of Rs 1,000 crore, that is EBITDA rising from Rs 120 crore to Rs 150 crore, 25% more.
Why are both numbers right and one of them misleading?
If a bank's loan rate moves from 8% to 9%, nobody says the rate rose 1%; they say it rose one percentage point, even though the interest bill rose 12.5%. A percentage point is the difference between two percentages; a per cent change is that difference measured against the starting value. Saying margin improved 3% leaves the reader guessing whether the margin went to 15% or to 12.36%, which is 3% more than 12%.
The margin gap from 12% to 15% is 3 percentage points, or 300 basis points, and the same move is a 25% rise measured against the old 12%, which on flat sales of Rs 1,000 crore takes EBITDA from Rs 120 crore to Rs 150 crore.
When does the 25% matter more than the 3 points?
When you are forecasting profit. At flat sales, a margin that rises by a quarter of itself lifts EBITDA by a quarter, so the 25% is what flows into earnings and valuation. On Rs 1,000 crore of sales, EBITDA moves from Rs 120 crore to Rs 150 crore. The same 3 points on a 30% margin would be only a 10% rise in EBITDA, which is why the same points of margin matter much more for a thin-margin business.
Starting margin Up 3 points to Relative rise
6% 9% 50%
12% 15% 25%
20% 23% 15%
30% 33% 10%
The same 3 points is a 50% rise on a 6% margin and a 10% rise on a 30% margin.
In the room, give the answer in one line: 3 percentage points, 300 basis points, a 25% relative improvement. Then add the one caution that shows judgement: check whether sales were flat, because a margin can rise while EBITDA falls if revenue shrinks.
Where candidates lose it
Candidates pick one number and defend it, which misses the point of the question. The interviewer wants to hear the vocabulary: percentage points or basis points for the difference, per cent for the relative change.
The second loss is saying 3% in a note. It reads as a 3% relative change, and a portfolio manager who takes it that way will model the wrong EBITDA.
What the interviewer asks next
A bank's net interest margin moves from 3.2% to 3.5%. Say the change two ways.
Margin rises from 12% to 15% while revenue falls 20%. What happens to EBITDA?
Why do rates desks talk in basis points rather than per cent?
060Without a calculator, estimate the present value of Rs 10 crore a year for 10 years, received at the end of each year, at a discount rate of 10%.Mental mathsCoreBuy-side equity researchLong-only asset management
Try it first
Pick the closest answer before you work it.
Show the worked solution
About Rs 61 crore. A perpetuity of Rs 10 crore at 10% is worth Rs 100 crore. A perpetuity that starts only after year 10 is worth that Rs 100 crore discounted back ten years; 1.1 to the power 10 is about 2.59, giving Rs 38.6 crore. The ten-year stream is the difference, Rs 100 crore less Rs 38.6 crore, about Rs 61.4 crore, or roughly six years of undiscounted cash.
Why subtract two perpetuities?
Think of a shop lease that pays you rent for ten years. It is the same as a lease that pays you forever, minus a lease that starts in year eleven and pays you forever after. A ten-year annuity equals a perpetuity starting now less a perpetuity starting in year eleven, and a perpetuity is one division: payment over rate. That turns an awkward ten-term sum into two numbers you can hold in your head: Rs 100 crore, and the same Rs 100 crore pushed ten years into the future.
Each Rs 10 crore payment is worth less the later it arrives, from Rs 9.09 crore in year one to Rs 3.86 crore in year ten, and the ten present values add to Rs 61.4 crore, about six years of cash.
How do you get 1.1 to the power 10 in your head?
Square, don't multiply. 1.1 squared is 1.21; squared again is about 1.46 for four years; squared again is about 2.14 for eight years; one more 1.21 for years nine and ten gives about 2.59. Three squarings and one multiplication get you to 1.1^10 = 2.59, and 100 divided by 2.59 is about 38.6. A second route checks it: the rule of 72 says 10% doubles money in about 7.2 years, so ten years is a little more than one doubling, around 2.6.
The relationship
$PV = \frac{C}{r}\left(1 - \frac{1}{(1+r)^{n}}\right) = 100 \times (1 - 0.386) = 61.4$
C the yearly payment, Rs 10 crore
r the discount rate, 10%
n the number of payments, 10
C/r the value of the same payment forever, Rs 100 crore
What it says in wordsThe annuity is the perpetuity value times the share of it that is not pushed beyond year ten.
What sanity checks can you say out loud?
Bracket it first. Every payment is worth between Rs 3.86 crore, the year ten payment, and Rs 9.09 crore, the year one payment, so the total sits between Rs 38.6 crore and Rs 90.9 crore. A tighter check treats all ten payments as arriving at the midpoint, year 5.5: Rs 100 crore over 1.1^5.5 is about Rs 59 crore, close to 61.4. The midpoint shortcut always comes out a little low, because discounting curves, which is itself a useful thing to say.
Where candidates lose it
Answering Rs 100 crore ignores discounting, and answering Rs 50 crore by halving on a hunch gives the interviewer no method to follow. The question is a test of whether you have a structure that works without a calculator.
The other loss is multiplying 1.1 by itself ten times out loud and losing the thread by year six. Square three times and multiply once; it is four steps and it sounds fluent.
What the interviewer asks next
What is it worth if each payment arrives at the start of the year instead? (about Rs 67.6 crore)
What discount rate makes the same stream worth Rs 50 crore? (about 15.1%)
How does the answer change if the payments grow 5% a year?
085A company has three segments growing at 5%, 15% and 30%, which make up 60%, 30% and 10% of revenue. What is group revenue growth? Work it in your head.Mental mathsCoreSell-side equity researchResearch KPO and GCC
Try it first
Answer in ten seconds.
Show the worked solution
10.5%. Weight each segment's growth by its share of revenue: 60% of 5% is 3 points, 30% of 15% is 4.5 points and 10% of 30% is 3 points, which add to 10.5%. The simple average of 16.7% is wrong because it gives the small, fast segment as much say as the large, slow one.
Why does the big slow segment decide most of the answer?
Think of a family where one parent earns most of the income and gets a 5% raise while a teenager's pocket money rises 30%. The household is not 17% richer. Group growth is a weighted average, and the weights are each segment's share of revenue, so the largest segment pulls the answer towards its own growth rate. The fast segment only adds 3 points, because it is only a tenth of the business.
Drawn with width as revenue share and height as growth, the core contributes 3.0 points, the adjacent segment 4.5 and the new segment 3.0, so group growth is 10.5%, well below the simple average of 16.7%.
How do you do it in your head without slipping?
Turn each segment into points of group growth, one at a time, and say them out loud. Six tenths of five is three; three tenths of fifteen is four and a half; one tenth of thirty is three; three plus four and a half plus three is ten and a half. Multiplying by tenths is just moving a decimal, so every step is a small whole-number product. Then sanity check: the answer must sit between the slowest and fastest rates and closer to the slow one, and 10.5% does.
The relationship
$g = \sum_i w_i\,g_i = 0.6 \times 5 + 0.3 \times 15 + 0.1 \times 30 = 10.5\%$
w_i segment i's share of this year's revenue
g_i segment i's growth rate
What it says in wordsGroup growth is each segment's growth weighted by how much of revenue it is today.
What happens next year if each segment keeps its growth rate?
The weights move. After a year the fast segment is a bigger slice, about 11.8% instead of 10%, and the core shrinks to about 57.0%. So group growth rises to about 11.1% next year with no segment accelerating at all: mix alone lifts it. That is worth saying in the room, because management commentary often credits acceleration to execution when it is arithmetic. The limit runs the other way too: a fast segment rarely keeps 30% growth as it gets bigger.
Where candidates lose it
The trap is the simple average, 16.7%. It comes out when candidates hear three growth rates and average them before listening to the weights. The interviewer asked for mental maths precisely to see whether you weight first.
The second miss is using the wrong weights, such as profit shares or next year's revenue, when the question gives this year's revenue mix. Use the weights you were given and say so.
What the interviewer asks next
What growth would the new segment need for the group to grow 12%?
If the segments had different margins, how would you work out profit growth?
Why might a company's reported growth accelerate even if no segment speeds up?

1.44	1.2 squared, two years at 20%
1.3225	1.15 squared, two years at 15%

Indian	Power of ten	Western
1 lakh	10^5	100 thousand
10 lakh	10^6	1 million
1 crore	10^7	10 million
100 crore	10^9	1 billion
1 lakh crore	10^12	1 trillion

c	the price cut, as a share of the old price
m	the gross margin, as a share of the old price
Delta V	the rise in volume needed to hold gross profit

Gross margin	Volume rise needed for a 10% price cut
20%	100%
30%	50%
40%	33%
60%	20%
80%	14%

Try it first

How do you get close without multiplying anything?

Why check the edge exactly, and how?

Where candidates lose it

What the interviewer asks next

Try it first

Why do people get the crore conversion wrong?

What is the fastest safe route in the room?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the volume needed so much bigger than the cut?

How does the answer change with the margin?

Where candidates lose it

What the interviewer asks next

Try it first

Why are both numbers right and one of them misleading?

When does the 25% matter more than the 3 points?

Where candidates lose it

What the interviewer asks next

Try it first

Why subtract two perpetuities?

How do you get 1.1 to the power 10 in your head?

What sanity checks can you say out loud?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the big slow segment decide most of the answer?

How do you do it in your head without slipping?

What happens next year if each segment keeps its growth rate?

Where candidates lose it

What the interviewer asks next