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 21–30 of 81 · filtered from 100Clear filters

025Stock A trades at 30 times earnings and is expected to grow earnings 25% a year. Stock B trades at 15 times with 8% expected growth. Which is cheaper on a PEG basis, and what does the PEG ratio leave out?Valuation riddlesWarm upSell-side equity researchLong-only asset management
Try it first
Which has the lower PEG?
Show the worked solution
A is cheaper on PEG, 1.2 against 1.9, but PEG ignores how long the growth lasts and what it costs. PEG divides the P/E by the growth rate, so it rewards fast growth whatever its duration. If A's 25% lasts only three years, today's price is 15.4 times year-three earnings for A against 11.9 times for B, and A is the dearer stock.
What does PEG measure?
Think of two rented flats, one expensive in a fast-improving area and one cheap in a quiet one. Comparing rent per square foot is not enough; you would also ask how quickly each area is improving. PEG does that for stocks. It divides the P/E by the growth rate, so it expresses price per unit of growth: A pays 1.2 times per point of growth, B pays 1.88. A PEG near 1 is often used as a rough marker of fair value, which is a convention, not a law.
Stock A at 30x and 25% growth has a PEG of 1.2 against 1.9 for stock B at 15x and 8%, but if A's fast growth lasts only three years, today's price is 15.4 times A's year-three earnings against 11.9 times for B.
What does PEG leave out?
Two things. First, duration: a growth rate is a speed, and PEG ignores how long the speed lasts. If A grows 25% for three years and then slows to B's pace, you are paying 15.4 times its year-three earnings for a company then growing like B, which costs 11.9 times. Second, the cost of growth: a company that must reinvest most of its earnings to grow is worth less than one that grows with little capital, and PEG cannot see the difference.
P/E Growth PEG Price / year-3 earnings
Stock A 30x 25% 1.2 15.4x
Stock B 15x 8% 1.88 11.9x
Year-3 earnings assume each stock grows at its stated rate for three years, with today's price held fixed.
How a research analyst would use it: as a quick screen across a sector, then replaced by something that sees duration and reinvestment, such as a DCF or a return on capital comparison. The limitation in one line: PEG compares price to the speed of growth, not to its length or its cost.
Where candidates lose it
The common loss is picking B because 15x looks cheaper than 30x. The question is about price relative to growth, and on that measure A wins.
The second, bigger loss is stopping at the PEG. The interviewer asked what it ignores; name duration and the capital growth needs, and give one number that shows duration changing the answer.
What the interviewer asks next
How many years must A grow at 25% before its P/E on those earnings falls to B's current 15x?
Why does PEG break down for companies with very low growth?
How would return on capital change your view of the two stocks?
027You can repeat a bet that wins 60% of the time and pays even money, as often as you like. What fraction of your capital should you stake each time to grow your money fastest, and what happens if you stake double that fraction?Expected value and decisionsHardHedge fund long/shortMulti-manager pod
Try it first
Stake double the growth-maximising fraction. What happens to your money over many bets?
Show the worked solution
Stake 20% of capital each time; at 40% your money slowly shrinks. The Kelly fraction for an even-money bet is the win chance minus the loss chance, 0.6 minus 0.4. At 20% the typical path grows about 2.0% a bet. At 40% each bet still has a positive expected gain, but growth is about -0.24% a bet, below zero, because big losses compound harder than big wins.
Why is the bet with the highest expected value not the one that grows fastest?
Imagine a shopkeeper who puts half the till into stock every morning. A good day lifts the till by half; a bad day cuts it in half. One of each leaves 1.5 x 0.5 = 0.75 of where she started, even though the good and bad days were the same size. Wealth compounds, so what matters over many bets is the average of the logarithm of each outcome, not the average outcome. Expected value per bet rises in a straight line with the stake; growth rises, peaks and then falls.
The relationship
$g(f) = 0.6\,\ln(1+f) + 0.4\,\ln(1-f) \qquad f^{*} = p - q = 0.2$
f the fraction of capital staked on each bet
g(f) growth of capital per bet on the typical path, in log terms
p - q the edge: win chance less loss chance
What it says in wordsGrowth per bet is the chance-weighted log of what each outcome does to your capital, and it peaks when you stake your edge.
Growth per bet peaks at 2.01% when 20% of capital is staked, falls to zero at about 38.9%, and turns negative at 40%, so overbetting a real edge can make you poorer over many bets.
What do the numbers look like at each stake?
Stake Expected gain per bet Growth per bet, typical path
10% +2.0% +1.50%
20% +4.0% +2.01%
30% +6.0% +1.47%
40% +8.0% -0.24%
50% +10.0% -3.40%
Expected gain keeps rising with the stake while growth peaks at 20% and turns negative near 40%.
Read the two columns against each other. Every row has a positive expected gain, yet the 40% and 50% rows lose money on the path you will actually live through. Half Kelly at 10% keeps about 75% of the maximum growth with far smaller swings, which is why many desks size below full Kelly. Say the limitation too: the formula assumes you know the 60% exactly. Real edges are estimates, and overestimating one pushes you towards the overbetting side of the curve.
Where candidates lose it
Candidates maximise expected value and conclude you should stake everything, because every bet is favourable. That answer goes broke on the first loss. The interviewer is testing whether you know that repeated bets compound, so the log of wealth is what you should maximise.
The second loss is saying 60%, the win probability, as the stake. The Kelly fraction for even money is the edge, 0.6 minus 0.4, not the win chance.
What the interviewer asks next
The bet now pays 2 to 1 with a 40% win chance. What is the Kelly fraction?
Why might a portfolio manager size positions at half Kelly?
How does position sizing on a stock idea resemble this bet, and where does the analogy break?
028Estimate the annual premium pool for two-wheeler insurance in India. Build it from the fleet on the road, the share of vehicles that stay insured once the upfront cover runs out, and the average premium for third-party and own-damage cover.Market sizing and estimationHardIndian brokerage researchSell-side equity research
Try it first
Which single assumption moves this estimate the most?
Show the worked solution
About Rs 19,000 crore a year on these assumptions. Take an assumed fleet of 25 crore two-wheelers, 1.8 crore in each of the last five years and 1.6 crore in each older year. All young vehicles are insured; after year five the insured share falls from 60% to 20%. That leaves 14.6 crore insured vehicles paying Rs 800 to Rs 1,500 a year, a pool near Rs 18,700 crore.
Where do you start, the vehicles or the policies?
Think of a gym. Counting everyone who ever signed up tells you little; the revenue comes from the members who still renew. Start from the fleet, but the number that decides the pool is how many vehicles are still insured, not how many are on the road. In India third-party cover is compulsory by law and new two-wheelers are sold with a multi-year third-party policy, so the young fleet is close to fully insured. Confirm the current rules before quoting them. Once that upfront cover runs out, many owners of older, cheaper bikes let it lapse.
Vehicle age Fleet, crore Share insured Insured, crore Premium, Rs a year Pool, Rs crore
1 to 5 years 9.0 100% 9.0 1,500 13,500
6 to 10 years 8.0 45% 3.6 1,000 3,600
11 to 15 years 8.0 25% 2.0 800 1,600
Total 25.0 58% 14.6 18,700
Every input here is an assumption for the estimate, not a reported figure. Premium per insured vehicle falls with age because the own-damage part is priced on the vehicle's value.
Every vehicle in its first five years is insured, but after year five the insured share falls from 60% to 20%, so only 14.6 crore of 25 crore vehicles pay a premium and the pool is about Rs 18,700 crore rather than Rs 27,900 crore.
How do you show the interviewer which assumption matters?
Run one sensitivity out loud. If ten more vehicles in every hundred older than five years renewed their cover, the pool would rise by about Rs 1,440 crore, roughly 8% of the total. That is the lever an insurer or a regulator can pull: enforcement of the third-party requirement at the roadside. The premium per vehicle matters less because it is set in narrow bands for third-party cover. The limitation to say: premiums, fleet and lapse rates here are assumptions for the method, and an analyst would replace each with a sourced number before writing it into a note.
Where candidates lose it
The costly mistake is multiplying the whole fleet by an average premium. That quietly assumes every old bike is insured and overstates the pool by about 49% on these numbers. The interviewer is waiting to see whether you ask how many of those vehicles actually carry cover.
The second loss is presenting assumed inputs as facts. Say each one as an assumption, give a round number, and move on; the structure is what is being marked.
What the interviewer asks next
How would the pool change if the upfront third-party period were shortened from five years to one?
Which part of the pool, third-party or own-damage, is more exposed to price competition between insurers?
How would you check your fleet assumption against registration data?
030Two companies each have an enterprise value of 1,000, EBIT of 80 and a 25% tax rate. A has no debt. B has 500 of debt at 6% interest. Which trades on the lower P/E, and at what cost of debt would their P/Es match?Valuation riddlesHardBuy-side equity researchHedge fund long/short
Try it first
Same business, same enterprise value. What does B's debt do to its P/E?
Show the worked solution
B trades lower, at 13.3x against A's 16.7x, and they match when B's debt costs 8%. A earns 80 x 0.75 = 60 on equity of 1,000. B pays 30 of interest, earns 37.5 on equity of 500. The P/Es match when the after-tax cost of debt equals A's earnings yield of 6%: r x 0.75 = 6%, so r = 8%. Cheaper debt lowers B's P/E; dearer debt raises it.
Why would the same business carry two different P/Es?
Suppose a flat earns rent of Rs 6 for every Rs 100 of its price. Buy it with half cash and half a loan at 4.5% after tax, and your cash earns more than 6%, because the borrowed half costs less than it yields. P/E divides the equity's value by the equity's earnings, and debt changes both, by different amounts. Swapping equity for debt that costs less than the equity's earnings yield lowers the P/E; swapping for dearer debt raises it.
A, no debt B, 500 of debt at 6%
EBIT 80.0 80.0
Interest 0.0 30.0
Tax at 25% 20.0 12.5
Net income 60.0 37.5
Equity value 1,000 500
P/E 16.7x 13.3x
The same EBIT and the same enterprise value give two different P/Es once half of B is funded with debt.
B's P/E sits below A's 16.7x while its debt costs less than 8%, is 13.3x at 6%, and climbs above A once debt costs more than 8%, because leverage lowers P/E only while after-tax debt is cheaper than the earnings yield.
Why is 8% the crossing point?
Swapping 500 of equity for 500 of debt removes equity that was earning its share of A's 6% earnings yield, 30 of net income, and adds an after-tax interest cost of 500 x r x 0.75. If that cost is exactly 30, net income halves along with equity and the P/E does not move; 30 over 375 is 8% before tax. The general rule: leverage cuts P/E when the after-tax cost of debt is below the earnings yield, E/P. The limitation worth saying: B's lower P/E is not a sign it is cheaper. Its equity is riskier, so investors should demand a lower multiple for the same business.
Where candidates lose it
Candidates say B must trade on a higher P/E because leverage is risky, or on the same P/E because the business is identical. Both skip the arithmetic. Work the net income and the equity value and the answer drops out.
The second loss is calling B cheaper because its P/E is lower. The interviewer wants to hear that a low P/E caused by leverage is a capital structure effect, which is why analysts compare levered companies on EV/EBIT instead.
What the interviewer asks next
What are the two companies' EV/EBIT multiples, and why is that the fairer comparison?
If B's tax rate were zero, where would the crossing point be?
B is in a sector where peers carry no debt. How do you adjust its P/E before comparing?
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?

	P/E	Growth	PEG	Price / year-3 earnings
Stock A	30x	25%	1.2	15.4x
Stock B	15x	8%	1.88	11.9x

f	the fraction of capital staked on each bet
g(f)	growth of capital per bet on the typical path, in log terms
p - q	the edge: win chance less loss chance

Stake	Expected gain per bet	Growth per bet, typical path
10%	+2.0%	+1.50%
20%	+4.0%	+2.01%
30%	+6.0%	+1.47%
40%	+8.0%	-0.24%
50%	+10.0%	-3.40%

Vehicle age	Fleet, crore	Share insured	Insured, crore	Premium, Rs a year	Pool, Rs crore
1 to 5 years	9.0	100%	9.0	1,500	13,500
6 to 10 years	8.0	45%	3.6	1,000	3,600
11 to 15 years	8.0	25%	2.0	800	1,600
Total	25.0	58%	14.6		18,700

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What does PEG measure?

What does PEG leave out?

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Why is the bet with the highest expected value not the one that grows fastest?

What do the numbers look like at each stake?

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Where do you start, the vehicles or the policies?

How do you show the interviewer which assumption matters?

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Why would the same business carry two different P/Es?

Why is 8% the crossing point?

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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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