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 1–3 of 3 · filtered from 100Clear filters

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?
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?
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?

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

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

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

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

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 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 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