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 1–7 of 7 · filtered from 100Clear filters

004An index rises 10% one day and falls 10% the next, alternating for ten days. Where does it end? A leveraged product returns exactly twice the index's move each day. Where does that end?Returns and compoundingHardHedge fund long/shortLong-only asset management
Try it first
The index ends ten days down about 4.9%. Where does the 2x daily product end?
Show the worked solution
The index ends at 0.951, down 4.9%, and the 2x product ends at 0.815, down 18.5%. Each up-down pair multiplies the index by 1.1 x 0.9 = 0.99 and the product by 1.2 x 0.8 = 0.96. Five pairs give 0.99 to the fifth and 0.96 to the fifth. The product loses nearly four times as much, not twice.
Why does a flat-looking path lose money at all?
A shop marks a shirt up 10% and then runs a 10% sale: the tag ends at 99% of where it began, because the discount is taken on the higher price. A gain and an equal percentage loss never cancel; the pair always leaves you with one minus the square of the move. For 10% that is 1 minus 0.01, so each pair costs 1%. Five pairs cost a little under 5%.
Over ten alternating days the index ends at 0.951 while the 2x daily product ends at 0.815, below the 0.902 that simply doubling the index's loss would give, because each pair costs the product 4% against 1% for the index.
Why is the 2x product four times worse and not twice?
Doubling the daily move doubles the swing, and the pair loss is the square of the swing. Twice the move means four times the loss per pair: 0.2 squared is 0.04 against 0.1 squared at 0.01. Compounded over five pairs the product lands at 0.815, a 18.5% loss, about 3.8 times the index's 4.9%. Someone who expected twice the index would have looked for 0.902.
The relationship
$(1+r)(1-r) = 1 - r^2 \qquad 0.99^5 = 0.951 \qquad 0.96^5 = 0.815$
r the daily move, 0.10 for the index and 0.20 for the 2x product
1 - r^2 what one up-down pair leaves you with
What it says in wordsEach up and down pair shrinks the value by the square of the move, so doubling the move quadruples the shrinkage.
The general name is volatility dragThe gap between the average of a set of returns and the compound return they produce, roughly half the variance of the returns.. It is why a daily leveraged product can fall over a month in which its index ended flat, and why it is built for short holding periods. The limitation is honest: in a steady trend with little back and forth, daily compounding can leave the product ahead of twice the index.
Where candidates lose it
The trap is doubling the index's result and answering down 9.8%. It treats a product that resets its leverage every day as if it held a fixed position for ten days.
The second loss is getting the numbers without the reason. Say that the pair loss is the square of the move, and the four times falls out of that in one line.
What the interviewer asks next
What if the index rises 10% every day for ten days? Is the 2x product ahead of or behind twice the index's return?
What about a minus 2x daily product on the same alternating path?
How would you estimate the monthly drag on a 3x product from the index's daily volatility?
015A fund earns 12% a year before fees for 20 years and charges a 2% annual fee. How much of the final wealth does the fee take?Returns and compoundingCoreLong-only asset managementResearch KPO and GCC
Try it first
Guess first: what share of the gross ending wealth does the 2% fee take after 20 years?
Show the worked solution
About 30% of the final wealth. At 12% a rupee grows to 9.65 in 20 years; at 10% after fees it grows to 6.73. The difference, 2.92, is 30.3% of the gross result. A fee that looks like one sixth of the annual return takes close to a third of the ending wealth, because the fee compounds too.
Why is the answer so much bigger than 2%?
Think of a leaking water tank that loses a small share of its contents every day. The leak looks trivial against the day's inflow, but it runs every day on the whole tank, and over a year it drains a large share of what would have collected. A percentage fee is charged on the whole balance every year, and every rupee it removes also loses the growth it would have earned for the rest of the period.
Over 20 years a rupee grows to 9.65 at 12% gross but only 6.73 at 10% after a 2% annual fee, so the fee takes 2.92, or 30% of the gross ending wealth.
How do you work it quickly in the room?
Use the ratio rather than the two big numbers. The net path grows at 1.10 against 1.12, so each year it keeps 1.10 / 1.12 = 98.2% of the gross path. Over 20 years that ratio compounds to 0.982 to the 20th, about 0.70, so the fee takes about 30%. A mental shortcut: 20 years at roughly 1.8% a year is about 36% by simple addition, and compounding pulls it back to about 30%.
The relationship
$1 - \left(\frac{1.10}{1.12}\right)^{20} = 1 - \frac{6.727}{9.646} = 30.3\%$
1.10 one year of growth after the fee
1.12 one year of growth before the fee
20 years
What it says in wordsThe share of wealth the fee takes is one minus the net-to-gross ratio compounded over the period.
The same arithmetic applies to any recurring drag: fund expenses, trading costs, a tax on annual gains. It is also why a long-horizon investor compares costs in basis points. The limitation: the calculation assumes the gross return is the same with and without the fee; a manager who earns the fee through better returns changes the comparison.
Where candidates lose it
The common loss is answering 2% or 40%, either treating the fee as one-off or multiplying 2% by 20 years. Both skip compounding, which runs on the fee as well as on the return.
The second is calculating the two multiples correctly and then dividing the gap by the net result rather than the gross. The question asks what share of the gross result the fee takes.
What the interviewer asks next
What fee would take half the gross wealth over 30 years?
How does the answer change if the gross return is 8% instead of 12%?
What return would an active fund need before fees to match a 0.2% fee index fund returning 12% gross?
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
Try it first
Answer in five seconds.
Show the worked solution
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?
040A stock rose from Rs 200 to Rs 450 over five years, while its EPS grew 10% a year from Rs 10. How much of the return came from earnings growth and how much from a change in the multiple? Ignore dividends.Returns and compoundingCoreSell-side equity researchBuy-side equity research
Try it first
What was the P/E at the end of the five years?
Show the worked solution
EPS rose to Rs 16.1 and the P/E rose from 20x to 27.9x, so earnings explain about 59% of the gain and re-rating the rest. The price multiplied by 2.25. EPS multiplied by 1.1⁵ = 1.611, and the multiple by 1.397; the two multiply to 2.25. A year, that is 17.6% total: 10% from earnings and 6.9% from the multiple.
Why split a return into earnings and multiple at all?
A house bought for Rs 50 lakh sells for Rs 1 crore. Part of the gain came from the rent rising, part from buyers being willing to pay more years of rent for the same flat. A share price is EPS times P/E, so any change in price is a change in earnings multiplied by a change in the multiple. The split matters because earnings growth can continue while the business grows, but a multiple cannot rise for ever.
The relationship
$\frac{P_1}{P_0} = \frac{E_1}{E_0} \times \frac{PE_1}{PE_0} \qquad 2.25 = 1.611 \times 1.397$
P share price at the start and end
E earnings per share
PE the price to earnings multiple
What it says in wordsThe price change is the earnings change multiplied by the change in the multiple.
Earnings growth at the starting 20x multiple takes the price from Rs 200 to Rs 322.1, and re-rating from 20x to 27.9x adds the last Rs 127.9; the return is earnings growth times multiple change, and only the earnings part can repeat.
Does the rupee split depend on which step you take first?
Order Earnings leg, Rs Re-rating leg, Rs
Earnings first, at 20x 122.1 127.9
Re-rating first, on EPS of Rs 10 170.6 79.4
Log split, order-free 59% of the gain 41% of the gain
The rupee split changes with the order because the two effects multiply; the log split does not.
Yes, and that is worth saying before the interviewer does. Because the effects multiply, the cross term goes to whichever leg is taken second, so a rupee split is a convention, while the log split of 59% and 41% is not. The analyst's conclusion: 41% of this return came from investors paying more for each rupee of profit. If the multiple drifts back to 20x, a holder earns only the EPS growth, and the next five years look very different from the last.
Where candidates lose it
The usual loss is dividing 450 by the old EPS of 10, or growing EPS by 10% simple interest to Rs 15 or Rs 20. Compound the EPS first: Rs 16.1 is the number that makes the rest work.
The second loss is giving a rupee split as if it were unique. Say that it depends on order and give the log split as the fair answer.
What the interviewer asks next
If the P/E returns to 20x over the next five years while EPS keeps growing 10%, what is the annual return?
How would you include dividends in the split?
Why might a multiple rise legitimately, without it being a bubble?
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
Try it first
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?
079A stock's price compounds at 8% a year for ten years, and it pays a 3% dividend yield that you reinvest. What does Rs 1 lakh become on price alone, and what does it become on total return?Returns and compoundingHardLong-only asset managementBuy-side equity research
Try it first
Price alone takes Rs 1 lakh to about Rs 2.16 lakh. Where does total return land?
Show the worked solution
Rs 2.16 lakh on price alone and Rs 2.84 lakh on total return. Price compounds at 8% a year: 1.08 to the tenth is 2.16. Reinvested dividends lift the yearly return to 11%, and 1.11 to the tenth is 2.84. The gap of about Rs 68,050 is more than the Rs 30,000 that ten years of 3% seems to promise, because the dividends compound too.
Why is the answer not simply 8% growth plus 3% times ten?
Think of a bank fixed deposit with two options: interest paid out every quarter, or interest added to the deposit. The cumulative option ends with more money because each quarter's interest starts earning interest. A reinvested dividend is the cumulative option: it buys more shares, and those shares rise in price and pay dividends of their own. Adding 3% a year for ten years treats the dividend like the payout option on a fixed base, which undercounts twice: the base keeps rising and the reinvested money keeps compounding.
Rs 1 lakh reaches Rs 2.16 lakh on price alone, Rs 2.59 lakh if the 3% dividends are taken in cash and added up, and Rs 2.84 lakh if they are reinvested, and the shaded gap between price and total return widens every year.
Where does the gap come from, piece by piece?
Split it into two layers. Dividends taken in cash are 3% of a price that grows 8% a year, so they start at Rs 3,000 and end near Rs 6,000, adding to about Rs 43,460 over ten years. Reinvesting them adds a further Rs 24,590, the return earned on dividends already received. So the Rs 68,050 gap is roughly two thirds the dividends themselves and one third compounding on them, and the second layer grows fastest in the later years.
The relationship
$(1+g+y)^{n} = 1.11^{10} \approx 2.84 \qquad (1+g)^{n} = 1.08^{10} \approx 2.16$
g price growth, 8% a year
y dividend yield on the start of year price, 3%
n years held, 10
What it says in wordsWith dividends reinvested, each year's return is price growth plus yield, and that combined rate compounds.
What does this change about how you compare stocks?
Compare on total return, always. A high-yield stock with slow price growth can beat a faster grower on the number an investor keeps, and a price chart alone hides that. Index providers publish total return versions of their indices for exactly this reason. The limits are worth one sentence: tax on dividends leaks some of the gap, and reinvesting assumes you can buy at a fair price each year.
Where candidates lose it
The common wrong answer is Rs 2.46 lakh: price growth plus 30% of the starting amount. It treats the dividend as paid on a frozen Rs 1 lakh and then left idle. Both halves are wrong, and together they understate the gap by more than half.
The quieter loss is getting 2.84 without being able to split it. Saying how much is dividends and how much is compounding on dividends shows you understand why the curves pull apart.
What the interviewer asks next
Over thirty years at the same rates, what share of the total return comes from dividends?
If dividends are taxed at 20% before reinvesting, what does Rs 1 lakh become?
Why would a company with high returns on capital rather retain earnings than pay a dividend?
090A stock returns plus 50%, then minus 30%, then plus 20% over three years. What is its average annual return, and what is its compound annual return?Returns and compoundingCoreLong-only asset managementBuy-side equity research
Try it first
Rs 100 goes through all three years. What is it worth at the end?
Show the worked solution
The average return is 13.3% a year, but the compound annual return is 8.0%. The average adds 50, minus 30 and 20 and divides by three. The money multiplies: 1.5 times 0.7 times 1.2 is 1.26, so Rs 100 becomes Rs 126, and the rate that compounds to 1.26 in three years is 8.0%. The compound rate is what an investor actually earned.
Why is the simple average the wrong measure of what you earned?
A shop marks a Rs 100 shirt up 50% to Rs 150, then puts it on a 30% sale. The sale takes Rs 45 off, not Rs 30, because it is taken from the higher price. Each year's return is applied to whatever the previous year left you, so returns multiply, and a simple average of them overstates the growth of the money whenever returns vary. The loss in year two comes out of a larger base than the one the gain was earned on.
Returns of plus 50%, minus 30% and plus 20% average 13.3% a year, which would turn Rs 100 into Rs 145.6, but the money actually goes 150, 105, 126, a compound rate of 8.0% a year.
The relationship
$\bar r = \frac{50 - 30 + 20}{3} = 13.3\% \qquad g = (1.5 \times 0.7 \times 1.2)^{1/3} - 1 = 1.26^{1/3} - 1 \approx 8.0\%$
r bar the arithmetic average of the yearly returns
g the compound annual growth rate, the geometric average
1.26 the ending value of each rupee invested
What it says in wordsThe average adds the returns; the compound rate multiplies them and takes the cube root, which is what the money actually did.
How do you estimate the gap without a calculator?
Use the rule that the compound rate is roughly the average minus half the variance. The returns sit 36.7, minus 43.3 and 6.7 points from their average; squared and averaged, they give a variance of about 0.109 in decimal terms. Half of that, about 5.4 points, is the drag volatility puts on compounding, and 13.3 minus 5.4 gives about 7.9%, close to the exact 8.0%. The approximation is rougher when swings are this large, but it shows the mechanism: the more a return bounces, the further the compound rate falls below the average.
When is the average the right number to use?
When you want the best guess of a single future year's return, the arithmetic average is the unbiased estimate, which is why cost of capital work often uses it. When you want to describe what happened to money held over several years, only the compound rate is honest. Fund factsheets report compound annual growth rates for that reason, and a pitch that quotes an average return for a volatile stock is flattering it.
Where candidates lose it
The trap is quoting 13.3% as the return. It is a real number, but it is not what the investor earned, and on a volatile stock the gap is large. Interviewers ask this to see whether you know the difference.
The second miss is treating minus 30% as cancelling plus 30% somewhere. A 30% fall needs a 42.9% rise to recover, so gains and losses of the same size never net to zero.
What the interviewer asks next
What steady yearly return would have produced the same Rs 126?
A fund reports a 15% average return with high volatility. What would you ask for?
Why is the geometric average always at or below the arithmetic average?

r	the daily move, 0.10 for the index and 0.20 for the 2x product
1 - r^2	what one up-down pair leaves you with

1.10	one year of growth after the fee
1.12	one year of growth before the fee
20	years

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

P	share price at the start and end
E	earnings per share
PE	the price to earnings multiple

Try it first

Why does a flat-looking path lose money at all?

Why is the 2x product four times worse and not twice?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer so much bigger than 2%?

How do you work it quickly in the room?

Where candidates lose it

What the interviewer asks next

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 split a return into earnings and multiple at all?

Does the rupee split depend on which step you take first?

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

Try it first

Why is the answer not simply 8% growth plus 3% times ten?

Where does the gap come from, piece by piece?

What does this change about how you compare stocks?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the simple average the wrong measure of what you earned?

How do you estimate the gap without a calculator?

When is the average the right number to use?

Where candidates lose it

What the interviewer asks next