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 31–40 of 100

031A company charges Rs 10 crore of depreciation in its books, but the tax rules let it claim Rs 25 crore of depreciation this year. The tax rate is 25%. What happens to the tax it pays, the tax it reports and its deferred tax liability?Three statement riddlesHardSell-side equity researchIndian brokerage research
Try it first
Compared with a year where both depreciation numbers were Rs 10 crore, what changes?
Show the worked solution
Cash tax falls by Rs 3.75 crore, reported tax does not change, and the deferred tax liability rises by Rs 3.75 crore. Tax is paid on taxable profit, which uses the Rs 25 crore tax depreciation. The P&L charges tax on book profit, which uses Rs 10 crore. The Rs 15 crore gap at 25% is Rs 3.75 crore of tax postponed, not saved: it comes back when tax depreciation later falls below book.
Why can two depreciation numbers exist for the same machine?
Think of a salaried employee who is allowed to claim a deduction early in the year for an expense she will actually use over three years. Her tax bill falls now, but she cannot claim it again later. Tax rules often allow depreciation faster than the books show it, so the same asset produces a larger deduction early and a smaller one later. The total over the asset's life is the same; only the timing differs. Confirm the current depreciation rates in the tax rules before using real numbers.
The books charge tax of Rs 25.00 crore on profit of Rs 100 crore while the tax return pays Rs 21.25 crore on Rs 85 crore, and the Rs 3.75 crore gap is parked in the deferred tax liability until the timing difference reverses.
Where does each number land in the three statements?
The income statement shows a tax expense of Rs 25.00 crore, split into current tax of Rs 21.25 crore and deferred tax of Rs 3.75 crore, so net income is Rs 75 crore either way. The cash flow statement adds back the Rs 3.75 crore of deferred tax as a non-cash charge, so operating cash flow is Rs 3.75 crore higher than it would be if both depreciation numbers matched. On the balance sheet the deferred tax liability rises by Rs 3.75 crore, matched by the extra cash.
Year Book depreciation Tax depreciation Liability movement Liability at year end
1 10 25 +3.75 3.75
2 10 15 +1.25 5.00
3 10 10 +0.00 5.00
4 10 0 -2.50 2.50
5 10 0 -2.50 0.00
Over a five year asset life both methods deduct Rs 50 crore, so the liability builds early and unwinds to zero by the end.
The analyst's point: a growing company that keeps buying assets keeps adding new early-year gaps, so its deferred tax liability can grow for years and behave almost like permanent free funding. When capex slows, the reversals arrive and cash tax rises above reported tax. That is worth one sentence in the room.
Where candidates lose it
The common loss is saying depreciation is non-cash, so nothing happens to cash. The tax saved by depreciation is cash, and here the tax return claims more of it than the books.
The second loss is lowering the reported tax charge. The P&L follows book profit; the difference is recorded as deferred tax, not as lower expense. Say that the saving is a postponement, and show when it reverses.
What the interviewer asks next
What happens in year four, when tax depreciation is zero and book depreciation is still Rs 10 crore?
Why might an analyst treat a steadily growing deferred tax liability as closer to equity than to debt?
Give an example of a timing difference that creates a deferred tax asset instead.
032A company announces a 1-for-4 rights issue at Rs 80 a share when its shares trade at Rs 120. What is the theoretical ex-rights price?EPS and share countCoreIndian brokerage researchSell-side equity research
Try it first
Pick the ex-rights price before you calculate.
Show the worked solution
Rs 112. For every four shares worth Rs 120, a holder pays Rs 80 for one new share. The five shares then hold Rs 480 of old value plus Rs 80 of new cash, Rs 560 in all, so each is worth Rs 560 divided by 5, which is Rs 112. The right to buy one new share is worth Rs 112 minus Rs 80, or Rs 32.
Why does the share price fall when nobody has lost anything?
Picture four friends who each own a Rs 120 share in a shared tiffin business. A fifth friend joins by paying only Rs 80. The business is now worth Rs 560 and there are five equal owners, so each stake is worth Rs 112. After a rights issue the old and new shares are identical, so the price has to settle at the weighted average of the old value and the new cash per share. The fall from Rs 120 to Rs 112 is not a loss to existing holders, because they were offered the cheap share too.
The relationship
$\text{TERP} = \frac{N_{old} \times P + N_{new} \times S}{N_{old} + N_{new}} = \frac{4 \times 120 + 1 \times 80}{5} = 112$
N_old, N_new old shares and new shares in the ratio, 4 and 1
P the price before the issue, Rs 120
S the subscription price, Rs 80
What it says in wordsThe theoretical ex-rights price is total value after the issue divided by total shares after the issue.
Four old shares at Rs 120 and one new share bought at Rs 80 pool to Rs 560 across five shares, so each share is worth Rs 112 after the issue: a weighted average, not the old price.
How do you prove an existing holder is no worse off?
Take a holder of four shares. If she subscribes, she had Rs 480 of shares and Rs 80 of cash; now she has five shares at Rs 112, still Rs 560. If she sells her right instead, she keeps four shares worth Rs 448 and receives about Rs 32 for the right, still Rs 480. Her wealth is unchanged either way; only a holder who ignores the right loses its value.
The analyst's follow-through: because the issue is priced below market, it carries a bonus element. Historical EPS is restated by dividing by the factor 120 over 112, or 1.0714, so per-share figures before and after the issue compare like with like.
Where candidates lose it
The fast wrong answer is Rs 100, halfway between Rs 120 and Rs 80. It treats the ratio as one for one. The weights are the share counts, four old to one new.
The second loss is calling the fall to Rs 112 a loss to shareholders. The interviewer wants to hear that the holder who takes up or sells the right is exactly where she started.
What the interviewer asks next
What is the value of the right attached to each old share?
How would you restate last year's EPS of Rs 12 for this issue?
Why might a company price a rights issue far below the market price?
033You have a US dollar cost of equity of 9% for a company. Expected inflation is 5% in India and 2.5% in the US. What is the equivalent rupee cost of equity for discounting rupee cash flows?Cost of capital and ratesHardBuy-side equity researchLong-only asset management
Try it first
Which conversion keeps the valuation the same in both currencies?
Show the worked solution
About 11.7%. Scale the dollar rate by relative inflation: (1 + 9%) x (1 + 5%) / (1 + 2.5%) = 1.1166, so the rupee cost of equity is 11.66%. Adding the 2.5 point gap gives 11.5%, close but slightly low. The conversion keeps the value the same in both currencies, because rupee cash flows grow faster by exactly the same factor.
Why must the rate change when the business has not?
Think of a salary quoted two ways. A Rs 10 lakh salary rising with 5% inflation and the same salary quoted in dollars rising with 2.5% inflation are one job. If you discount the faster-rising rupee salary at the slower dollar rate, you make the same job look more valuable. A discount rate carries the inflation of its currency, so moving between currencies means moving the rate by the inflation gap, in the same way the cash flows move.
The relationship
$1 + k_{Rs} = (1 + k_{\$}) \times \frac{1 + \pi_{India}}{1 + \pi_{US}} = 1.09 \times \frac{1.05}{1.025}$
k_$ the dollar cost of equity, 9%
k_Rs the rupee cost of equity
pi expected inflation in each country, 5% and 2.5%
What it says in wordsOne plus the rupee rate equals one plus the dollar rate, scaled up by the ratio of the two inflation factors.
A 9.00% dollar rate plus the 2.5 point inflation gap reaches 11.50%, and the compounding term adds 0.16 points more, so the rupee cost of equity is 11.66% rather than a spread added by feel.
How do you prove the value is the same in both currencies?
Take a cash flow worth 100 in today's money, received in year five. In dollars it grows at 2.5% inflation and is discounted at 9%: 1.025 to the fifth over 1.09 to the fifth gives a discount factor of 0.7353. In rupees it grows at 5% and is discounted at 11.66%: the factor is 0.7353. The two factors match to the fourth decimal in every year, which is the proof that the conversion is right.
Say the limitation. This converts the currency, nothing else. If the dollar rate was built for a US listed peer, it may not carry any premium for country risk, and whether to add one is a separate judgement you should state and defend, not a number to fold silently into the conversion.
Where candidates lose it
The common loss is keeping 9% for rupee cash flows, which values Indian inflation at a US discount rate and inflates the answer. The second is adding a few points for India by feel, which mixes two questions, currency and country risk, into one unexplained number.
Give the exact formula, the number, and the 11.5% approximation, then say the value check in one sentence.
What the interviewer asks next
If the rupee is expected to depreciate 3% a year against the dollar, what does that imply about the inflation gap?
How would you convert a dollar risk-free rate to a rupee one?
When would you add a country risk premium, and where in the build would it go?
034A subscription app spends Rs 900 to acquire a user. Each user earns Rs 60 a month of contribution after direct costs, and 4% of users cancel every month. What are the lifetime value, the LTV to CAC ratio and the payback period?Growth, mix and unit economicsCoreSell-side equity researchBuy-side equity research
Try it first
What is the lifetime contribution of an average user?
Show the worked solution
LTV is Rs 1,500, LTV to CAC is 1.67x, and payback is 15 months per retained user, about 22 months for the cohort. Lifetime value is Rs 60 divided by 4% churn. CAC over monthly contribution, Rs 900 over Rs 60, gives the usual 15 month payback. Counting users who leave along the way, the cohort's cumulative contribution only reaches Rs 900 after about 22.4 months.
Why does churn set the lifetime value?
Think of a leaking water tank. If 4% of the water drains out every hour, a litre poured in stays, on average, 25 hours. With a constant monthly churn rate, the average customer life is one divided by churn, so lifetime value is monthly contribution divided by churn. Rs 60 over 0.04 is Rs 1,500. Halve churn to 2% and LTV doubles to Rs 3,000; that is why subscription analysts watch churn more closely than price.
A user who never cancels repays the Rs 900 acquisition cost in 15 months, but with 4% monthly churn the cohort's contribution bends towards a Rs 1,500 ceiling and only crosses Rs 900 at about 22.4 months, so churn caps value and stretches payback.
Which payback number do you give the interviewer?
Give both and say which is which. The common convention is CAC divided by monthly contribution, 15 months, which describes a user who stays. A cohort of acquired users earns less than that each month because some have left, so the money spent on the cohort comes back later, after about 22.4 months. By month 15 only 54% of the cohort is still paying.
Measure Working Result
Average life 1 / 4% 25 months
LTV Rs 60 / 4% Rs 1,500
LTV / CAC 1,500 / 900 1.67x
Payback, retained user 900 / 60 15 months
Payback, cohort 1,500 x (1 - 0.96^n) = 900 22.4 months
Every figure uses contribution after direct costs and is undiscounted; discounting would lower LTV and stretch both paybacks.
Then give a view. A ratio of 1.67x is thin: the business keeps Rs 600 per user over the user's life before any overhead, and a common rule of thumb looks for about three times. The two levers are churn and acquisition cost, and the numbers show churn is the stronger one.
Where candidates lose it
The common loss is multiplying Rs 60 by some chosen number of months instead of letting churn set the life. The second is quoting 15 months as if every acquired user repays it; with churn the cohort takes about half as long again.
Say that the figures are undiscounted and use contribution, not revenue. LTV built on revenue flatters the ratio because it ignores the cost of serving the user.
What the interviewer asks next
If churn falls to 3%, what are LTV and LTV to CAC?
How would a 1% monthly discount rate change the lifetime value?
Why might churn be higher in the first three months than later, and what does that do to the formula?
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?
036One hundred equity funds launched ten years ago. Sixty survive today and have averaged 14% a year. The forty that closed averaged 3% a year before closing. What return did the average fund launched ten years ago earn?Data and reasoning trapsCoreLong-only asset managementBuy-side equity research
Try it first
A database of today's funds shows a 14% average. What was the average across all 100 launched?
Show the worked solution
About 9.6% a year, not 14%. The 14% average only covers funds that survived, and funds usually close because they did badly. Weight both groups by their count: 60 funds at 14% and 40 at 3% gives 0.6 x 14% plus 0.4 x 3%, or 9.6%. A database that drops closed funds overstates the average investor's experience by 4.4 points a year.
Why does the survivor average mislead?
Ask the toppers of a coaching class how hard the entrance exam was, and they will say it was manageable. The students who failed have gone home and nobody asked them. Survivorship bias is judging a group by the members still standing, when the ones that dropped out left because their results were poor. A fund database that lists only live funds is the coaching class that only asks its toppers.
The 60 surviving funds average 14% and the 40 closed funds average 3%, so the average across all 100 funds launched is 9.6%, and studying only survivors overstates the typical result by 4.4 points a year.
How big is the gap once it compounds?
Over ten years, 14% a year turns Rs 100 into Rs 371, while 9.6% turns it into Rs 250. A gap of 4.4 points a year looks modest, but it is the difference between multiplying money 3.7 times and 2.5 times. That is why performance studies that ignore closed and merged funds tend to make active management look better than it was.
The relationship
$\bar r = \frac{60 \times 14\% + 40 \times 3\%}{100} = 9.6\%$
60, 40 the number of funds that survived and that closed
14%, 3% the average annual return of each group
What it says in wordsThe true average weights each group by how many funds it holds, including the ones no longer listed.
Say the limitation too. Averaging annual returns across funds of different lives is a simplification; a careful study would weight by assets and by years in existence. The direction of the bias does not change, though: leaving out the losers always flatters the average.
Where candidates lose it
The lazy answer is 14%, taking the database at face value. The second is 8.5%, averaging the two group averages without weighting them by the number of funds in each.
The interviewer wants to hear the name of the bias, the weighted number and one sentence on where it bites: fund league tables, backtests on today's index members, and studies of successful founders.
What the interviewer asks next
Where does the same bias show up when you backtest a strategy on today's index constituents?
If the 40 closed funds were merged into other funds rather than shut, does the bias still exist?
How would you weight the average if the surviving funds were much larger than the closed ones?
037You toss a fair coin until you see two heads in a row. How many tosses do you expect to need?Probability and brainteasersCoreSquarepoint CapitalLondon · 2025
Try it first
Give your instinct before setting up the algebra.
Show the worked solution
6 tosses. Call S the expected tosses from the start and H the expected tosses once you have one head. From the start, one toss either gives a head (move to H) or a tail (stay at S): S = 1 + H/2 + S/2. From one head, one toss either finishes or sends you back: H = 1 + S/2. Substituting, S = 6 and H = 4.
Why is the answer not four?
Think of climbing two steps on a slippery stair where any slip sends you back to the bottom. Reaching the second step is not just two climbs; every slip costs you what you had gained. The chance of heads-heads on a single pair is one in four, but a tail after a head wipes out your progress, so the expected wait is longer than four. The clean way to count the cost of those slips is to name each position you can be in.
From Start a head moves you to One head and a tail keeps you at Start; from One head a head finishes and a tail sends you back, so one equation per state gives 6 expected tosses from the start and 4 from one head.
How do you check 6 another way?
Count sequences directly. The first HH lands on toss n in exactly F(n-1) of the 2ⁿ equally likely sequences, where F is the Fibonacci sequence: 1 way on toss 2, 1 on toss 3, 2 on toss 4, 3 on toss 5. Summing n times F(n-1) over 2ⁿ across all n gives 6.0000, the same 6. The same series shows the spread: you finish within six tosses 67% of the time, and the other third of runs can go on for a long while, which is what pulls the average up to 6.
The relationship
$S = 1 + \tfrac{1}{2}H + \tfrac{1}{2}S, \qquad H = 1 + \tfrac{1}{2}\cdot 0 + \tfrac{1}{2}S \;\Rightarrow\; S = 6$
S expected tosses still needed from the start, or after a tail
H expected tosses still needed after one head
1 the toss you are about to make
What it says in wordsEach state's expected wait is one toss plus the chance-weighted wait from wherever that toss sends you.
Where candidates lose it
The instant answer is 4, from one in four. It treats the tosses as separate pairs and ignores the reset a tail causes. Another common slip is writing H = 1 + H/2 for the one-head state, sending a tail to the wrong place: after a tail you are back at the start, not still holding a head.
Name the states out loud, draw the arrows in the air, then solve. That is what the interviewer is marking.
What the interviewer asks next
How many tosses do you expect before seeing heads then tails, HT? Why is it fewer?
How many tosses to see three heads in a row?
If the coin lands heads with probability p, what is the expected wait for two heads in a row?
Asked at Squarepoint Capital, Quantitative Research, London, 2025 (Wall Street Oasis): a few siimple questions on statistical problems e.g. # of throws expected to get 2 heads in a row
038A stock trades at Rs 500 and reports results tomorrow. You think there is a 60% chance it moves to Rs 560 and a 40% chance it moves to Rs 440. What is the expected price, and what does it say about today's price?Expected value and decisionsWarm upSell-side equity researchHedge fund long/short
Try it first
What is the expected price after results?
Show the worked solution
The expected price is Rs 512, 2.4% above today. Weight each outcome by its chance: 0.6 x 560 plus 0.4 x 440 gives 336 plus 176. Today's Rs 500 sits exactly halfway between Rs 440 and Rs 560, so the market is pricing roughly a 50% chance of good results. Your 60% view is what separates Rs 512 from Rs 500, and that probability is what you would have to defend.
Why is the expected price not the likely price?
A cricket fan who thinks her team wins 60% of the time does not expect the team to win exactly 0.6 of a match; she expects the win most often and a loss sometimes. Expected value is the probability-weighted average of every outcome, and it can be a price that never actually trades. Here the stock will be at Rs 560 or Rs 440 tomorrow, never at Rs 512, yet Rs 512 is the right number to compare with today's price.
Weighting Rs 560 by 0.6 and Rs 440 by 0.4 gives an expected price of Rs 512, while today's Rs 500 sits halfway between the outcomes and implies a 50% chance of good results.
What does today's price tell you about the market's view?
Run the calculation backwards. If Rs 500 is the market's expected price, the chance q of the good outcome solves 560q + 440(1 - q) = 500, so q = 60 / 120 = 50%. The interesting number is not Rs 512 but the gap between your 60% and the market's 50%: that gap is the whole of your view. An analyst would next ask what evidence justifies seeing more upside than the market does.
The relationship
$E[P] = 0.6 \times 560 + 0.4 \times 440 = 512 \qquad q = \frac{500 - 440}{560 - 440} = 50\%$
E[P] the expected price after results
q the chance of good results that makes today's price fair
What it says in wordsWeight the outcomes by your probabilities to get your expected price, and solve for the probability that makes today's price the expected one.
Say the limitations. Two outcomes are a simplification of a whole spread of possible moves, and a 2.4% expected gain on one event is small next to the Rs 60 swing either way. The expected value is a way to state a view precisely, not a reason on its own to act on it.
Where candidates lose it
The common slip is answering Rs 560 because it is the more likely outcome. The expected value averages both branches.
The second loss is stopping at Rs 512. The interviewer wants the implied probability too: today's price already carries a view, and saying 50% shows you know your edge is the difference between two probabilities, not a price.
What the interviewer asks next
What probability of good results would make Rs 500 fair if the upside were Rs 580?
Options on the stock imply a move of plus or minus 12%. Is that consistent with your tree?
How would you size a position when the expected gain is 2.4% but the swing is 12%?
039Estimate the size of the global 5G smartphone market in 2022, in units and in value. Build it from the pool of phones replaced each year, the 5G share of new sales by price band and the average selling price, stating each assumption.Market sizing and estimationCoreAllianceBernsteinNew York · 2022
Try it first
Which assumption swings the answer the most?
Show the worked solution
Roughly 607 million 5G phones worth about US$ 351 billion, on stated assumptions. Assume 4 billion smartphones in use, replaced every 3 years, so about 1.33 billion are sold in 2022. Assume all premium phones, 60% of mid-range and 15% of entry phones are 5G, a blended 45.5%, which gives 607 million units. At a blended price near $579, the value is about US$ 351 billion.
Where do you start: people, phones or networks?
Think of estimating how many pairs of school shoes a town buys. You do not count shops; you count children and ask how often a pair wears out. Annual phone sales are the phones in use divided by how many years a phone lasts, so start from the installed base and the replacement cycle, then split the year's sales by price band. Say every number as an assumption: an installed base of 4 billion and a 3 year cycle are round figures for the method, to be checked against an industry tracker before they go into a note.
Price band Share of 2022 sales 5G share 5G units, m Average price, US$ Value, US$ bn
Premium, above $600 20% 100% 267 900 240
Mid, $250 to $600 30% 60% 240 380 91
Entry, below $250 50% 15% 100 200 20
Total 100% 45.5% 607 579 351
All inputs are assumptions for the estimate; premium phones are a fifth of units but most of the 5G value.
Four billion phones in use and a three year cycle give 1.33 billion sales, of which 607 million are 5G, worth about US$ 351 billion; the answer is only as good as the replacement cycle, which is the widest step.
How do you show the interviewer you know where the estimate is weak?
Run the sensitivity on the widest step. A cycle of 2.5 years gives 728 million 5G units and 3.5 years gives 520 million, a swing larger than any plausible error in the price assumptions. Then sanity check the output: 45% of phones sold being 5G in 2022 should sit sensibly against what you know about network rollouts, and a blended price near $579 says premium phones carry most of the value. The limitation: first-time buyers are folded into the cycle, which slightly understates sales in markets still adding users.
Where candidates lose it
Candidates lose this by starting top-down from the world population with a chain of shares, which stacks five guesses before touching the phone. Others give one number with no split by price band, so the interviewer cannot see where the 5G share comes from.
Say the structure first, then each assumption as a round number, then the sensitivity on the replacement cycle. That order is what the interviewer marks.
What the interviewer asks next
How would the estimate change for India alone, where more sales sit in the entry band?
Why might the replacement cycle lengthen in a year when 5G phones become common?
How would you split the value between handset makers and chip suppliers?
Asked at AllianceBernstein, Equity Research, New York, 2022 (Wall Street Oasis): Estimate the market size of 5G smartphone sales in 2022.
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?

Year	Book depreciation	Tax depreciation	Liability movement	Liability at year end
1	10	25	+3.75	3.75
2	10	15	+1.25	5.00
3	10	10	+0.00	5.00
4	10	0	-2.50	2.50
5	10	0	-2.50	0.00

N_old, N_new	old shares and new shares in the ratio, 4 and 1
P	the price before the issue, Rs 120
S	the subscription price, Rs 80

k_$	the dollar cost of equity, 9%
k_Rs	the rupee cost of equity
pi	expected inflation in each country, 5% and 2.5%

Measure	Working	Result
Average life	1 / 4%	25 months
LTV	Rs 60 / 4%	Rs 1,500
LTV / CAC	1,500 / 900	1.67x
Payback, retained user	900 / 60	15 months
Payback, cohort	1,500 x (1 - 0.96^n) = 900	22.4 months

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Why can two depreciation numbers exist for the same machine?

Where does each number land in the three statements?

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Why does the share price fall when nobody has lost anything?

How do you prove an existing holder is no worse off?

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Why must the rate change when the business has not?

How do you prove the value is the same in both currencies?

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Why does churn set the lifetime value?

Which payback number do you give the interviewer?

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

How does the answer change with the margin?

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Why does the survivor average mislead?

How big is the gap once it compounds?

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Why is the answer not four?

How do you check 6 another way?

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Why is the expected price not the likely price?

What does today's price tell you about the market's view?

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Where do you start: people, phones or networks?

How do you show the interviewer you know where the estimate is weak?

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Why split a return into earnings and multiple at all?

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

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