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

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?
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?

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

g	price growth, 8% a year
y	dividend yield on the start of year price, 3%
n	years held, 10

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 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