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

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Any Warm up Core Hard

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013You roll a fair die and receive Rs 100 times the face shown. Before you are paid you may reroll once, but then you must keep the second roll. What is the right to reroll worth?Expected value and decisionsCoreHedge fund long/shortMulti-manager pod
Try it first
Which first rolls should you keep?
Show the worked solution
The reroll right is worth Rs 75. A single roll is worth Rs 350 on average. With the reroll, keep 4, 5 and 6, which average Rs 500, and reroll 1, 2 and 3 for an expected Rs 350. Half the time you get Rs 500 and half the time Rs 350, so the game is worth Rs 425, Rs 75 more than without the right.
What is the rule for keeping or rerolling?
Think of a job offer in hand while you wait on another interview. You take the offer if it beats what you expect the other process to give you, not if it beats your dream job. Keep any result worth more than the expected value of the alternative; here the alternative is a fresh roll worth Rs 350. So 4, 5 and 6 are kept, and 1, 2 and 3 are rerolled.
The value of a fresh roll, Rs 350, cuts between faces 3 and 4, so you keep 4, 5 and 6 and reroll 1, 2 and 3, which makes the game worth Rs 425 and the reroll right worth Rs 75.
How do you value the right itself?
Value the game with the right, then subtract the game without it. With the right: half the time you hold 4, 5 or 6, averaging Rs 500; half the time you reroll and expect Rs 350. That is Rs 425. Without it the game is worth Rs 350. The right is worth the difference, Rs 75, and all of it comes from the three bad faces being replaced.
The relationship
$V = \tfrac{1}{2}\times 500 + \tfrac{1}{2}\times 350 = 425 \qquad 425 - 350 = 75$
500 average of the kept faces 4, 5 and 6, in rupees
350 expected value of a fresh roll, in rupees
What it says in wordsThe game with the reroll right is worth the average of the kept outcomes and the expected reroll, each half the time.
This is an option, and a research interviewer asks it to see whether you value flexibility correctly. The right has value only because you can refuse it when the first roll is good. Its value is the gain in the bad states, Rs 250, 150 and 50 on faces 1, 2 and 3, averaged over six faces: Rs 75. The same logic prices an option to expand a project or to delay an investment.
Where candidates lose it
The common loss is keeping only 5 and 6, or only 6, because a 4 feels ordinary. The cut-off is the expected reroll, Rs 350, and a sure Rs 400 beats it.
The second is answering Rs 425 when asked what the right is worth. That is the game with the right; the right is the difference, Rs 75.
What the interviewer asks next
What if you may reroll twice?
What would you pay to play if you had to announce keep or reroll before seeing the first roll?
How does the answer change for a 20-sided die paying Rs 100 a face?

014Estimate the annual fare revenue of one metro rail line in a large Indian city. Work from the number of stations, peak and off-peak ridership, and the average fare.Market sizing and estimationCoreIndian brokerage researchConsulting style estimation

Try it first

You know the line carries about 25,000 boardings in a peak hour and runs 17 hours. What goes wrong if you multiply the two?

Show the worked solution

About Rs 271 crore a year, on these assumptions. A weekday carries 240,000 boardings: six peak hours at 25,000, four shoulder hours at 12,000 and seven quiet hours at 6,000. Weekends and holidays run at 60% of a weekday, giving 77.5 million trips a year. At an average fare of Rs 35, that is about Rs 271 crore.

How do you estimate a weekday's ridership?

Think about any busy road near an office district: jammed from 8 to 11 and from 5 to 8, calm in between. A metro line has the same shape. Ridership is concentrated in two peaks, so build the day hour by hour; a flat hourly average taken from the peak overstates it badly. Assume the line runs 6 am to 11 pm, 17 hours, with 25,000 boardings in each of six peak hours, 12,000 in four shoulder hours and 6,000 in seven quiet hours. That gives 240,000.

On these assumptions a weekday carries 240,000 boardings, with six peak hours at 25,000 and long quiet stretches at 6,000, so multiplying the peak rate by all 17 hours would give 425,000, about 77% too high.

How do you check it and turn it into revenue?

Check from the stations. A line of 25 stations at 240,000 boardings a day is about 9,600 per station, which is plausible for a mix of busy interchanges and quiet suburban stops. Two routes that land close together give you licence to use the number; if they did not, the gap would tell you which assumption to test. Then annualise: 260 weekdays at 240,000 and 105 weekend and holiday days at 60% of that give 77.52 million trips.

Step	Assumption	Result
Weekday boardings	6 x 25,000 + 4 x 12,000 + 7 x 6,000	240,000
Weekday trips a year	260 days	62.4 m
Weekend and holiday trips	105 days at 60%	15.12 m
Average fare	Distance-based, blended	Rs 35
Annual fare revenue	Rs 271 crore

Every input is an assumption for the exercise; the structure matters more than any single number.

Say which input you trust least: the average fare, since metro fares usually rise with distance and a line's trip-length mix is hard to guess. A research analyst would also note that fare revenue is only part of a metro's income, with advertising and property often material, and would check the operator's published ridership rather than rely on the estimate.

Where candidates lose it

The trap is taking a peak-hour number and multiplying by operating hours, which inflates the day by more than 75% on these assumptions. The interviewer is listening for whether you think about the shape of demand through the day.

The second is giving one number with no check. Tie the daily figure back to the station count, and name the fare as your softest assumption.

What the interviewer asks next

How would a new interchange with another line change the estimate?
What fare increase would offset a 10% fall in ridership?
How would you estimate the line's non-fare revenue?

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?
016A stock trades at Rs 1,000. It generated free cash flow of Rs 30 a share this year and its cost of equity is 11%. If the cash flow grows at a constant rate forever, what growth rate is the market assuming?Valuation riddlesCoreBuy-side equity researchHedge fund long/short
Try it first
Quick estimate first: roughly what growth is priced in?
Show the worked solution
About 7.8% a year, forever. In a growing perpetuity the price equals next year's cash flow, Rs 30 x (1 + g), divided by 11% minus g. Setting that equal to Rs 1,000 and solving gives g = (110 - 30) / 1,030 = 7.77%. The quick version: an 11% required return minus a 3% cash yield leaves about 8% for growth.
Why solve for growth instead of assuming it?
Think of a shop listed for sale at a price that looks high for its current takings. Rather than argue about the price, ask what sales growth would justify it; then the argument becomes whether that growth is believable. A price is a bundle of assumptions, and solving for the growth it needs turns a vague sense that a stock is expensive into one number you can test. This is the idea behind a reverse DCFA valuation run backwards: start from the market price and solve for the growth or margins the price implies..
With Rs 30 of free cash flow and an 11% cost of equity, value rises steeply with the growth assumed and reaches the Rs 1,000 share price at 7.8% perpetual growth; at 7% the value would be only Rs 803.
How do you solve it cleanly?
Write the growing perpetuity with next year's cash flow on top: 1,000 = 30 x (1 + g) / (0.11 - g). Multiply out: 110 - 1,000g = 30 + 30g, so 80 = 1,030g and g = 7.77%. The shortcut, cost of equity minus cash yield, gives 8% and is close because the yield is small; the exact answer is a little lower because the Rs 30 grows before it is paid.
The relationship
$g = \frac{P \cdot r - FCF_0}{P + FCF_0} = \frac{1{,}000 \times 0.11 - 30}{1{,}000 + 30} = 7.77\%$
P the share price, Rs 1,000
r the cost of equity, 11%
FCF_0 this year's free cash flow a share, Rs 30
What it says in wordsThe implied growth is the part of the required return that the current cash yield does not supply.
Then judge it. Growing at 7.8% forever means growing faster than most economies can for ever, which is a demanding assumption. Look at how steep the curve is near the answer: at 7% the stock would be worth Rs 803, at 8.5% Rs 1,302. Small changes in the growth belief move the value a lot, which is why a single-stage model is a sense check here, not a valuation.
Where candidates lose it
The common loss is dividing 30 by 1,000 and answering 3%, confusing the cash yield with growth. The required return is the yield plus growth, not the yield alone.
The second is stopping at 8% without noting it is approximate, or quoting 7.8% without judging whether perpetual growth at that rate is plausible. The interviewer wants the number and a view on it.
What the interviewer asks next
What cost of equity would make 5% growth consistent with the price?
How would you run the same test with a two-stage model?
What would you check in the accounts to see whether 7.8% growth is achievable?

017A company reported net income of Rs 100 crore, depreciation of Rs 20 crore and capex of Rs 20 crore. There were no debt or dividend movements, yet its cash fell by Rs 30 crore. Receivables rose Rs 80 crore, inventory rose Rs 60 crore and payables rose Rs 10 crore. Reconcile the numbers.Three statement riddlesCoreSell-side equity researchResearch KPO and GCC

Try it first

What was operating cash flow?

Show the worked solution

Working capital absorbed Rs 130 crore, more than all of the profit. Start at net income of 100 and add depreciation of 20. Receivables up 80 and inventory up 60 each tie up cash; payables up 10 frees some. Operating cash flow is minus 10. Capex of 20 takes the change in cash to minus 30, matching the fall.

How can a profitable company lose cash?

Think of a wholesaler who sells a lot this month, but on 90 days' credit, and stocks up for the festive season. The books show a profit, yet the bank balance falls, because the sales have not been collected and the new stock has been paid for. Profit counts sales when they are made; cash counts them when they are collected, and growing receivables and inventory are the gap between the two.

Rs 100 crore of net income plus Rs 20 crore of depreciation is more than consumed by Rs 80 crore of new receivables and Rs 60 crore of new inventory, leaving operating cash flow of Rs -10 crore and a Rs 30 crore fall in cash after Rs 20 crore of capex.

Which way does each working capital line push cash?

Read the balance sheet change and ask whether cash went out or came in. A rise in an asset such as receivables or inventory uses cash; a rise in a liability such as payables provides it. Receivables and inventory together took Rs 140 crore and payables returned Rs 10 crore, so working capital absorbed Rs 130 crore against Rs 120 crore of profit plus depreciation. That makes operating cash flow minus 10.

Line	Rs crore	Why
Net income	100	starting point
Depreciation	+20	non-cash charge added back
Receivables up	-80	sales not yet collected
Inventory up	-60	stock bought, not yet sold
Payables up	+10	suppliers not yet paid
Operating cash flow	-10
Capex	-20	investment in assets
Change in cash	-30

The indirect cash flow statement, which starts from profit and walks each balance sheet change back to cash.

What a research analyst does next: compare the rise in receivables to sales growth. If receivables grew much faster than revenue, customers are paying more slowly or sales were pushed through on easy terms, and that is a question for management. The limitation: a fast-growing company can have this pattern for healthy reasons, so the answer needs the trend, not one year.

Where candidates lose it

The common loss is getting the working capital signs backwards and adding the rise in receivables. An asset going up is cash going out; say that rule before you calculate.

The second is reconciling the arithmetic and stopping. The interviewer also wants a view: receivables up 80 on profit of 100 is a flag worth naming.

What the interviewer asks next

What would you want to know about revenue growth before judging the receivables increase?
How would days sales outstanding help here?
If the company then factored its receivables for cash, how would the statements change?

018A company's pre-tax profit is flat year on year, but its effective tax rate falls from 30% to 25%. How much does EPS grow, and should the stock's P/E multiple rise because of it?EPS and share countCoreSell-side equity researchHedge fund long/short
Try it first
By how much does EPS grow, with the share count unchanged?
Show the worked solution
EPS grows 7.1%, and the multiple should not rise on that growth. Net income goes from 70 to 75 on flat pre-tax profit of 100. The price can reasonably rise about 7.1% at the same P/E, because each share now earns more. But the growth happens once: next year, with pre-tax profit still flat, EPS growth is zero.
Where does the 7.1% come from?
Think of a salaried employee whose tax bill falls. Take-home pay jumps once, but next year's take-home pay only grows if the salary does. A lower tax rate raises net income by the tax saved, here 5 on a base of 70, which is 7.1%, even though the business earned nothing more. Say the base out loud: people who answer 5% are measuring against pre-tax profit.
Pre-tax profit stays at 100 while tax falls from 30 to 25, so net income rises from 70 to 75, a 7.1% step in year one followed by zero growth in later years because the lower rate is already in the base.
Why should the multiple stay where it is?
A P/E multiple pays for the level of earnings and for their future growth. The tax cut raises the level, so at an unchanged P/E of 20 the price rises 7.1% to match. It does not raise future growth, so a higher multiple would pay twice for the same one-off step. Re-rate to 22x on the back of 7% growth and the price would rise 17.9%, most of it paying for growth that will not repeat.
Two caveats a research analyst adds. First, ask why the rate fell: a statutory cut is durable, while a one-time credit or a shift of profit to a lower-tax region may reverse, in which case even the level change deserves a discount. Second, check whether competitors got the same cut; if they did, some of it may be passed to customers through prices. Confirm any tax rate you use against the current rules rather than memory.
Where candidates lose it
The trap is treating the EPS jump like organic growth and arguing for a higher multiple because earnings grew faster. Growth that comes from a rate change is a step, and a step is paid for once.
The second loss is answering 5%, the change in the tax rate, instead of 7.1%, the change in net income.
What the interviewer asks next
What if the lower rate comes from a one-time tax credit?
How would you show this in an EPS bridge from last year to this year?
How does the answer change if the company also buys back 5% of its shares?
020A company's revenue grew 12% and its prices rose 5%. How much did volume grow?Growth, mix and unit economicsWarm upSell-side equity researchIndian brokerage research
Try it first
Answer in five seconds.
Show the worked solution
About 6.7%. Revenue is price times volume, so the growth factors multiply: 1.12 = 1.05 x (1 + v). Dividing, 1.12 / 1.05 = 1.0667, so volume grew about 6.67%. Subtracting 5 from 12 gives 7%, which overstates volume by the small cross term, 0.33 points.
Why divide instead of subtract?
Think of a tea stall that raises the price of a cup and also sells more cups. The takings grow for both reasons, and the extra cups are also sold at the higher price, which is a little extra neither change produces alone. Revenue is price times volume, so growth factors multiply, and the way to back one out is to divide. 1.12 divided by 1.05 is 1.0667.
Drawing revenue as price times volume, a 5% wider and 6.67% taller rectangle has 12% more area: a 5% price strip, a 6.67% volume strip and a 0.33% corner where the extra volume is sold at the higher price.
When does the shortcut of subtracting matter?
The error from subtracting is the corner of the rectangle, price growth times volume growth. At single-digit rates the corner is small, about 0.33 points here, but it grows quickly with the rates, and in high-inflation markets it becomes the whole story. With 30% price rises and 40% revenue growth, subtracting says 10% volume growth; dividing says 7.7%.
The relationship
$1 + v = \frac{1 + g_{rev}}{1 + g_{price}} = \frac{1.12}{1.05} = 1.0667$
v volume growth
g_rev revenue growth, 12%
g_price price growth, 5%
What it says in wordsVolume growth is revenue growth divided by price growth, as factors.
For an analyst this split is the first question on any results call: how much of the growth was price and how much was volume. Volume growth says whether the company is winning customers; price growth says whether it has pricing power or is passing on costs. The limitation: reported price growth often mixes true price changes with shifts in product mix, so ask how the company defines it.
Where candidates lose it
The trap is subtracting and answering 7% without a second thought. It is close enough to pass as mental arithmetic but wrong in method, and the interviewer asks the question to hear which one you use.
Say the division, give 6.7%, and add that the gap is the cross term. That turns a one-line answer into evidence of method.
What the interviewer asks next
Revenue fell 3% while prices rose 8%. What happened to volume?
How would you split growth into price, volume and mix for a company selling three products?
Why might management prefer to report volume growth before or after the effect of mix?
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?

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

023Estimate how many new two-wheelers are sold in India in a year. Build it from households, ownership and how often vehicles are replaced.Market sizing and estimationWarm upIndian brokerage researchConsulting style estimation

Try it first

Which split makes an estimate like this defensible?

Show the worked solution

About 18 million a year, on these assumptions. Take 300 million households, half owning a two-wheeler, at 1.1 each: a fleet of about 165 million. Replacing each every 11 years gives 15 million a year. Ownership rising one point a year adds 3 million first-time buyers. Check the total against published industry sales before relying on it.

Where does demand for new vehicles come from?

Think of a housing society's parking lot. Each year a few old scooters are swapped for new ones, and a few families who never had one buy their first. New vehicle sales are replacement of the existing fleet plus first-time buyers, and splitting the two is what makes the estimate defensible, because each has its own driver. Replacement depends on fleet size and vehicle life; first-time buying depends on how fast ownership spreads.

On these assumptions 300 million households, half of them owning a two-wheeler at 1.1 each, give a fleet of 165 million; replacing it every 11 years gives 15 million a year and rising ownership adds 3 million first-time buyers, about 18 million in total.

How do you build each branch, and which assumption matters most?

Start with households: roughly 1,400 million people at a little under five a household gives about 300 million, stated as an assumption. Half own a two-wheeler, and owning households average 1.1, so the fleet is about 165 million. If a vehicle lasts 11 years, about one in eleven is replaced each year, 15 million, which makes replacement most of the market. First-time demand is ownership rising one point a year on 300 million households, 3 million.

Input	Assumption	Million
Households	about 1,400 m people, under 5 a home	300
Owning households	50%	150
Fleet in use	1.1 per owning household	165
Replacement a year	11-year life	15
First-time buyers	ownership up 1 point a year	3
New two-wheelers a year	18

Each line is an assumption the interviewer can push on; the vehicle life moves the answer most.

Test the most sensitive input out loud. A 9-year life instead of 11 lifts replacement to about 18 million; a 13-year life cuts it to about 13 million. That range, about 6 million, is twice the whole first-time branch, which tells you where to look first. Then say you would check the total against the industry body's published annual sales rather than quoting a figure from memory.

Where candidates lose it

The common loss is dividing the population by some ownership ratio and stopping, which estimates the fleet, not annual sales. New sales are a flow; the fleet is a stock, and the replacement life turns one into the other.

The second is leaving out first-time buyers, or making them the whole answer. Name both branches and say which one is larger.

What the interviewer asks next

How would a shift to electric two-wheelers change the replacement cycle?
What happens to sales in a year when rural incomes fall sharply?
How would you size the market for two-wheeler loans from this estimate?

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?

500	average of the kept faces 4, 5 and 6, in rupees
350	expected value of a fresh roll, in rupees

P	the share price, Rs 1,000
r	the cost of equity, 11%
FCF_0	this year's free cash flow a share, Rs 30

v	volume growth
g_rev	revenue growth, 12%
g_price	price growth, 5%

Try it first

What is the rule for keeping or rerolling?

How do you value the right itself?

Where candidates lose it

What the interviewer asks next

Try it first

How do you estimate a weekday's ridership?

How do you check it and turn it into revenue?

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 solve for growth instead of assuming it?

How do you solve it cleanly?

Where candidates lose it

What the interviewer asks next

Try it first

How can a profitable company lose cash?

Which way does each working capital line push cash?

Where candidates lose it

What the interviewer asks next

Try it first

Where does the 7.1% come from?

Why should the multiple stay where it is?

Where candidates lose it

What the interviewer asks next

Try it first

Why divide instead of subtract?

When does the shortcut of subtracting matter?

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

Where does demand for new vehicles come from?

How do you build each branch, and which assumption matters most?

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