Venture Capital puzzles, solved step by step

Puzzles: 100
Traced to a firm: 7
Topics: 12
Hard: 30

Showing 51–60 of 100

051A startup sells a point-of-sale billing app to kirana stores for Rs 500 a month. Assume India has 1.2 crore kirana stores, 15% of them have daily sales above Rs 20,000, and 30% of those would pay for the app. What is the serviceable market, in rupees a year?Market sizing and estimationCoreSaaS-focused VCIndia VC
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Before you multiply: which figure is the serviceable market?
Show the worked solution
About Rs 324 crore a year. Start with 1.2 crore stores. 15% sell more than Rs 20,000 a day, which leaves 18 lakh stores busy enough to need billing software. 30% of those will pay, which leaves 5.4 lakh paying stores. At Rs 500 a month, or Rs 6,000 a year, that is Rs 324 crore. The Rs 7,200 crore you get by charging every store is the total market, not the one this app can serve.
Why not multiply every kirana store by the price?
A tailor opening a shop in a colony of 2,000 homes does not have 2,000 customers. He has the homes that buy stitched clothes, at his prices, from a shop, rather than from a relative with a machine. A serviceable market counts only the buyers who feel the problem badly enough to pay for the fix, so every filter must describe a real reason a store would or would not buy. The Rs 20,000 daily sales cut is that reason here: a store taking in that much has enough bills, credit customers and stock lines that paper billing starts to hurt. Multiply all 1.2 crore stores by Rs 6,000 and you get Rs 7,200 crore, the total addressable marketRevenue if every possible buyer in the category bought the product at its price. A ceiling, rarely a plan., which describes the country rather than this app.
Drawn to scale, the 1.2 crore kirana stores shrink to 18 lakh with daily sales above Rs 20,000 and then to 5.4 lakh willing to pay, and at Rs 6,000 a year each those paying stores make a serviceable market of Rs 324 crore.
The relationship
$\text{SAM} = N \times s \times w \times p \times 12 = 1.2\text{ cr} \times 0.15 \times 0.30 \times 500 \times 12 = \text{Rs } 324 \text{ crore}$
N all kirana stores, 1.2 crore
s share with daily sales above Rs 20,000, 15%
w share of those willing to pay, 30%
p price a month, Rs 500
What it says in wordsShrink the store count by each real reason a store would not buy, then turn the monthly price into a yearly one.
How do you say the answer so it survives the follow-up?
Say the chain as three multiplications and carry the unit at every step: stores, then stores, then rupees. The most common slip in this question is mixing a monthly price with an annual market, so say both numbers out loud: Rs 27 crore a month, Rs 324 crore a year. Then name the weakest link yourself. The 30% willingness to pay is the least grounded figure, and the answer moves one for one with it: at 20% the market is Rs 216 crore, at 40% it is Rs 432 crore.
Finish with what the number does and does not tell an investor. Rs 324 crore is a ceiling on this product's revenue, not a forecast; the app would win some share of the 5.4 lakh stores, and competitors are chasing the same ones. A partner hearing this will ask two things: what share is realistic in five years, and whether the stores can be sold a second product, such as working capital credit, that makes the market bigger than billing alone. The 1.2 crore store count is the question's premise; outside the room, confirm it against a published retail survey before building on it.
Where candidates lose it
The first way to lose this is to announce the Rs 7,200 crore figure with pride. It tells the interviewer you have not separated the stores that could buy from the stores that would, which is the whole skill being tested.
The second is a unit slip. Rs 27 crore and Rs 324 crore are both correct numbers for different periods; say which is which, and say that willingness to pay is the assumption you trust least.
What the interviewer asks next
How would you test the 30% willingness to pay before investing?
What second product would make this market larger, and by roughly how much?
If a competitor offers billing free and charges for payments, what happens to your sizing?
052You bought into a software company at 50x ARR, while mature peers trade at 10x. ARR grows 60% a year, and each year a new funding round dilutes your stake by 15%. Valuing the company at 10x ARR, how many years until your stake is worth what you paid?Growth and compoundingHardSaaS-focused VCGrowth equity
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Your first guess for the payback year?
Show the worked solution
About 5.2 years, not 3.4. Paying 50x when the exit multiple is 10x means ARR must grow 5x before you get your money back. At 60% a year that takes 3.4 years. But each round leaves you 85% of your previous stake, so your share of value grows by 1.6 x 0.85, or 36% a year. Solving 1.36 to the power n equal to 5 gives about 5.2 years, and that is only break-even.
What does paying 50x when peers trade at 10x actually commit you to?
Buying a mango sapling at the price of a grown tree is not a mistake if the sapling grows; it is a bet on how fast. When you pay five times the multiple a mature business earns, the company's revenue has to grow fivefold just for you to break even, because the multiple will fall back to the mature level by the time you sell. Growth first pays back the premium; only growth beyond that makes money. At 60% a year, 1.6 to the power n equals 5 gives 3.4 years.
Why does dilution stretch the answer by almost two years?
Think of a family shop that takes in a new partner every year, each time giving away 15% of everyone's share. The shop can double and your slice can still barely move. Your stake's value grows at the company's growth rate times what you keep after each round, so 60% growth with 15% dilution is 1.6 x 0.85, or 36% a year for you, not 60%. The bar the company must clear keeps rising: 5x becomes 5 divided by 0.85 after one year, then again after two, which is the red curve in the figure.
ARR growing 60% a year crosses the flat 5x break-even line at 3.4 years, but once each round dilutes the stake 15% the bar rises every year, and the ARR curve only catches it at 5.2 years.
The relationship
$\big[(1+g)(1-d)\big]^n = \frac{50}{10} \;\Rightarrow\; n = \frac{\ln 5}{\ln (1.6 \times 0.85)} = \frac{1.609}{0.308} = 5.2$
g ARR growth a year, 60%
d dilution from each year's round, 15%
50/10 entry multiple over the exit multiple, the growth needed
n years to break even
What it says in wordsYour stake compounds at growth times retention, and it has to compound until it covers the gap between the entry and exit multiples.
Say the limits before the interviewer does. The model holds 60% growth flat for five years, which few companies manage; growth that fades makes the wait longer. It treats each round's new cash as spent on the growth already assumed, rather than sitting on the balance sheet adding to value. And break-even after 5.2 years is a 0% return on money that a venture fund needs to multiply several times, so the honest conclusion is that the entry price has already spent most of the upside.
Where candidates lose it
The common answer is 3.4 years: candidates handle the multiple compression correctly and then forget that they own less of the company every year. The interviewer put dilution in the question precisely to see whether you apply it to the growth rate.
The second trap is subtracting, 60% minus 15% giving 45% a year. Growth and retention multiply: 1.6 x 0.85 is 1.36. At 45% you would answer 4.3 years and be wrong by nearly a year.
What the interviewer asks next
What ARR growth rate would get you back in three years with the same dilution?
How does a pro rata right change this calculation?
If growth falls by a fifth of itself each year, is break-even ever reached?
053A company's valuation rises 150% at its Series B, falls 60% at its Series C and rises 50% at its Series D, one round a year. What is the net change from the Series A valuation, and what steady annual rate over the three years gives the same result?Mental maths and speed testsCoreSeries A to C VCGrowth equity
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Net change from Series A to Series D?
Show the worked solution
Up 50% in all, or about 14.5% a year. Turn each round into a multiplier: up 150% is x 2.5, down 60% is x 0.4, up 50% is x 1.5. The shortcut is that 2.5 x 0.4 is exactly 1, so the Series C fall wiped out the Series B gain, and the net is just Series D's x 1.5. The annual rate is the cube root of 1.5, which sits between 14% and 15%.
Why can the three percentages not simply be added?
A shopkeeper who marks a shirt up 150% and then runs a 60% off sale is not ahead by 90%. The markup took Rs 100 to Rs 250, and the sale took 60% of Rs 250, landing back at Rs 100. Each percentage is measured against whatever the value was just before it, so moves in a chain multiply rather than add. Adding gives +140%, a number that describes nothing in this company's history. Multiplying gives 2.5 x 0.4 x 1.5, which is 1.5: a valuation of 100 went to 250, back to 100, then to 150.
From an index of 100 the valuation rises to 250 at Series B, falls back to 100 at Series C and ends at 150 after Series D, so the three rounds multiply to 1.5x, about 14.5% a year, not the +140% that adding the percentages suggests.
How do you find the annual rate without a calculator?
You need the number that, cubed, gives 1.5. Bracket it. 1.14 cubed is about 1.48 and 1.15 cubed is about 1.52, so the rate sits just below 14.5%, and saying 'about 14.5% a year' is the right precision for the room. The exact figure is 14.47%. Check it the other way: half of 50% is 25%, far too high, which is the arithmetic average and the error the question is fishing for.
The relationship
$(1+r)^3 = 2.5 \times 0.4 \times 1.5 = 1.5 \quad\Rightarrow\quad r = 1.5^{1/3} - 1 = 14.5\%$
2.5, 0.4, 1.5 the three rounds written as multipliers
r the steady annual rate with the same end result
What it says in wordsMultiply the rounds to get the total, then take the root for the number of years to get the steady rate.
One sentence of judgement helps. A company that fell 60% in one round and still ended up 50% ahead over three years has had a volatile path, and the steady 14.5% hides that volatility completely. Investors who entered at the Series B price are down 40% at Series D, which is why the entry round matters as much as the company's overall path.
Where candidates lose it
The fast wrong answer is +140%, from adding the three percentages. The second is averaging them to about 47% a round. Both treat percentages as if they shared a base, which is the exact mistake the question is built to catch.
The slower loss is spotting 2.5 x 0.4 = 1 but then stumbling on the cube root. Bracket it between 1.14 and 1.15 out loud; an interviewer wants to see the method, not four decimal places.
What the interviewer asks next
An investor came in at Series B. What is their multiple at Series D?
What single fall at Series C would have left the company flat over the three rounds?
Why can a flat overall path still leave some investors well below their entry price?
054The cap table: seed paid Rs 5 crore for 10%, Series A Rs 20 crore for 20% and Series B Rs 50 crore for 20%, each with a 1x non-participating preference; founders and staff hold the other 50%. Above roughly what exit value does every class convert to common, and which class sets that line?Preferences, payouts and protectionsCoreSeries A to C VCMulti-stage VC
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Which exit value is the line above which every class converts?
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Above about Rs 250 crore, and Series B sets that line. A non-participating holder converts when its share of the exit beats its preference, so each class's line is preference divided by stake: seed Rs 5 crore over 10% is Rs 50 crore, Series A Rs 20 crore over 20% is Rs 100 crore, Series B Rs 50 crore over 20% is Rs 250 crore. The class that paid the most per percentage point is the last to convert.
Why is the line for each class its preference divided by its stake?
Suppose a friend offers you a choice: your Rs 500 back, or a tenth of whatever the raffle raises. You take the tenth only once the pot is above Rs 5,000. A 1x non-participating preference is exactly that choice, so a class converts once its stake times the exit value beats its money back, and the crossover is preference divided by stake. Preference over stake is also the price the class paid for the whole company, which is why the class that bought at the highest valuation, here Series B at an implied Rs 250 crore, holds out longest.
Divided alone, the preferences give conversion lines of Rs 50, 100 and 250 crore; once seniors holding their preference shrink the pool, seed and Series A convert later, at Rs 100 and Rs 130 crore, but Series B's line stays at Rs 250 crore, above which every class converts.
Why does the question say 'roughly', and do the lower lines move?
The simple division assumes everyone else has converted. Below Rs 250 crore Series B keeps its Rs 50 crore, and that comes out of the pot before the others share it. When a senior class holds its preference, the pool left for everyone else shrinks, so the junior classes need a bigger exit before converting pays. Series A with B holding gets 20/80 of the exit less Rs 50 crore, which beats Rs 20 crore only above Rs 130 crore. Seed, with both A and B holding, needs Rs 100 crore. The top line does not move, because by the time Series B is deciding, everyone below it has already converted.
Class Preference Stake Preference / stake Line with seniors holding
Seed Rs 5 crore 10% Rs 50 crore Rs 100 crore
Series A Rs 20 crore 20% Rs 100 crore Rs 130 crore
Series B Rs 50 crore 20% Rs 250 crore Rs 250 crore
Each class's conversion line, first on its own and then allowing for the preferences still held above it; only the top line is unchanged.
In the room, give Rs 250 crore and Series B first, then offer the refinement as a check. It shows you understand that preferences interact, which matters when you are the junior investor negotiating behind a large late round.
Where candidates lose it
The common wrong answer is Rs 75 crore, the sum of the preferences. At that exit every class is repaid, but converting is still worse than taking the money for all three, so nobody converts yet. Covering the preferences and making conversion pay are two different lines.
The second loss is naming the right number without the reason. Say that the class with the highest price per percentage point converts last; the interviewer is checking whether you can spot that class on any cap table.
What the interviewer asks next
If Series B had a 2x preference, where would its line move?
Between Rs 100 and Rs 130 crore, who gets what?
How would participation change whether Series B ever needs to convert?
055A founder owns 40% of a company worth Rs 100 crore. It raises money at Rs 300 crore post-money, selling 20% of the company. A year later it raises again at Rs 250 crore post-money, selling another 20%. What is her stake worth after each round?Dilution and ownership riddlesCoreSeed and early-stage VCSeries A to C VC
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What is her stake worth after the second round?
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Rs 40 crore today, Rs 96 crore after the first round, Rs 64 crore after the second. Each round sells 20% of the company, so she keeps 80% of her stake: 40% becomes 32%, then 25.6%. Her rupees are that stake times the post-money value: 32% of Rs 300 crore is Rs 96 crore, and 25.6% of Rs 250 crore is Rs 64 crore. Dilution cut her percentage both times; only the down round cut her value.
Why does selling 20% leave her with 32%, not 20%?
Cut a pizza into ten slices and give four to a friend. If the host then takes a fifth of every plate for a late guest, the friend loses a fifth of her four slices, not two of them. A round that sells 20% of the company shrinks every existing holder by the same fifth, so her stake is multiplied by 0.8, not reduced by 20 points. 40% x 0.8 is 32%. The second round does it again: 32% x 0.8 is 25.6%.
Her ownership falls from 40% to 32% to 25.6% as each round takes a fifth of it, while her stake's rupee value rises from Rs 40 crore to Rs 96 crore through the up round and falls back to Rs 64 crore through the down round.
If her percentage fell both times, why did her wealth rise the first time?
Because the first round was priced well above today's value. A Rs 300 crore post-money with Rs 60 crore raised means a pre-money of Rs 240 crore, so the company was repriced from Rs 100 crore to Rs 240 crore before the new money arrived. Dilution is a smaller slice; whether it costs you money depends entirely on the price the new investor pays for the slice. The second round sold 20% at Rs 250 crore post-money, which is Rs 50 crore raised on a pre-money of Rs 200 crore, below the Rs 300 crore she was last marked at. Her percentage fell by the same fifth, and this time her value fell too, from Rs 96 crore to Rs 64 crore.
The relationship
$\text{Value} = s_0 (1-d_1)(1-d_2) \times \text{Post}_2 = 0.40 \times 0.8 \times 0.8 \times 250 = 64$
s_0 her starting stake, 40%
d_1, d_2 the share of the company sold in each round, 20%
Post_2 the post-money value after round two, Rs 250 crore
What it says in wordsHer stake shrinks by what each round sells; her wealth is that stake times the latest price.
One limit is worth stating. The Rs 64 crore is a paper value at the last round's price, and it assumes her shares are worth the same per share as the investors' preferred shares, which carry preferences hers do not. In a sale below the preferences, her real payout would be less.
Where candidates lose it
The classic slip is subtracting points: 40% minus 20% is 20%, then zero after the second round. It sounds absurd once said, but candidates rushing through it say it anyway.
The second trap is reading the percentage drop as the whole story. Say both numbers each time, percentage and rupees, and point out that the first round made her richer and the second made her poorer, even though both cost her the same fifth.
What the interviewer asks next
What pre-money would the second round have needed for her value to stay at Rs 96 crore?
How would an option pool created in round one change her stake?
Why might she prefer a smaller round at a lower price to a bigger one at a higher price?
056Your pro rata right lets you invest Rs 5 crore in the next round, and you decide after you see how it is priced. 40% of rounds turn out good and return 5x the round price; 60% are bad and return 0.5x. Assume the pricing tells you which kind it is. What is the right worth, compared with an obligation to invest Rs 5 crore in every round?Probability and expected valueCoreSeed and early-stage VCMulti-stage VC
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How much more is the right worth than the obligation, in expected profit?
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Rs 8 crore of expected profit against Rs 6.5 crore, so the choice is worth Rs 1.5 crore. A good round turns Rs 5 crore into Rs 25 crore, a Rs 20 crore profit; a bad one leaves Rs 2.5 crore, a Rs 2.5 crore loss. Forced to invest, you expect 0.4 x 20 minus 0.6 x 2.5, or Rs 6.5 crore. With the right you pass on bad rounds, so you expect 0.4 x 20, or Rs 8 crore. The Rs 1.5 crore gap is the price of the bad branch you no longer have to take.
Why is a right worth more than an obligation to do the same thing?
A season ticket that lets you skip any match you like is worth more than one that forces you to sit through the washed-out ones, even though the matches are the same. A right is worth exactly the losses it lets you refuse, so its value is the expected loss on the branches you would walk away from. Here the bad branch costs Rs 2.5 crore 60% of the time, an expected Rs 1.5 crore, and the right lets you skip all of it. The good branch is the same Rs 20 crore profit either way, so it adds nothing to the difference.
In the left tree you pass after a bad round and keep zero, so the expected profit is Rs 8 crore; in the right tree you must also take the Rs 2.5 crore loss 60% of the time, so the expected profit falls to Rs 6.5 crore, and the Rs 1.5 crore gap is the value of the choice.
The relationship
$V_{\text{right}} - V_{\text{oblig}} = p\,W - \big[p\,W + (1-p)\,L\big] = -(1-p)\,L = 0.6 \times 2.5 = 1.5$
p chance the round is good, 40%
W profit in a good round, Rs 20 crore
L profit in a bad round, minus Rs 2.5 crore
What it says in wordsThe right and the obligation share the good branch, so the right's extra value is just the expected loss on the bad branch it lets you skip.
Where does this simple answer overstate the right in real life?
The question lets the round's pricing tell you for certain whether it is good. In practice the signal is noisy, so a pro rata right is worth less than Rs 1.5 crore here: you will sometimes pass on a good round and sometimes take a bad one. If your read were no better than a coin, the choice would be worth nothing, because you could not tell the branches apart. Two more limits: a hot round may leave you no allocation even with the right, and a fund must hold reserves to exercise it, and that cash earns nothing while it waits.
This is why seed funds fight for pro rata rights and why the best later investors try to cut them back. The value sits with whoever gets to look before deciding, and it grows with how far apart the good and bad outcomes are. Say that last point in the room: a wider spread between good and bad rounds makes the right worth more, exactly as a wider spread makes any option worth more.
Where candidates lose it
The usual slip is to value the right at the Rs 8 crore it earns and stop there. The question asks for the right compared with the obligation, and the answer is the Rs 1.5 crore difference, the expected loss you get to avoid.
The second trap is ignoring the assumption that pricing reveals the quality of the round. Say it out loud and add that a noisier signal shrinks the value toward zero; that sentence is what separates a mechanical answer from an investor's one.
What the interviewer asks next
If your read on round quality were right only 75% of the time, what would the right be worth?
Why might a lead investor in the next round want to cap your pro rata?
How does holding reserves to exercise pro rata rights affect the fund's overall return?
057A software company grows revenue 70% a year with a free cash flow margin of minus 45%. What is its Rule of 40 score, and what margin would it need to reach 40 if growth falls to 50%?SaaS and unit economics riddlesWarm upSaaS-focused VCGrowth equity
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If growth falls to 50%, what free cash flow margin reaches 40?
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A score of 25 today, and a margin of minus 10% to reach 40 at 50% growth. The Rule of 40 adds revenue growth and free cash flow margin: 70 plus minus 45 is 25. If growth slows to 50 and the burn stays at minus 45, the score falls to 5. To get back to 40 the margin must rise to minus 10, a 35 point improvement in a year when growth is also slowing.
What does adding growth to margin actually measure?
A cricket team can win on runs scored or on runs saved; a selector looks at the margin of victory, not either number alone. The Rule of 40A rough test for software companies: revenue growth % plus free cash flow margin % should reach at least 40. treats growth and cash generation as substitutes, so a company may burn cash only to the extent that its growth pays for the burn. Today this company grows 70 and burns 45, for a score of 25: it is spending more than its growth justifies. A company growing 20 with a 20% margin also scores 40 and passes, which is the point: the test does not care which mix you choose.
Growth of 70 less a burn of 45 scores 25; if growth slows to 50 with the burn unchanged the score sinks to 5, and only cutting the margin to minus 10 lifts it back to exactly 40.
The relationship
$g + m \ge 40 \;\Rightarrow\; m \ge 40 - 50 = -10\%%$
g revenue growth, % a year
m free cash flow margin, % of revenue
What it says in wordsSubtract the growth you have from 40 and what remains is the worst margin you can run.
Why is the slowdown the dangerous part of this question?
Growth rarely falls while the cost base shrinks on its own. When growth drops 20 points with the burn unchanged, the score falls from 25 to 5, and the company has to find 35 points of margin to make up for it. Those points come from cutting sales hiring and marketing, which is often what was producing the growth, so the cut can slow growth further. Say this in the room: the score is easy to compute, the hard part is whether the company can move margin that far without breaking the growth engine.
State the limits too. The rule is a rule of thumb from listed software companies; applied to an early company growing 200% it says little, because small numbers make growth rates swing. Which margin you use matters as well, free cash flow or EBITDA, and companies quote whichever flatters them, so ask which one is in the deck.
Where candidates lose it
The common error is answering plus 10%, as if the company must turn profitable. The margin is negative and can stay negative: 50 plus minus 10 is 40. Getting the sign wrong tells the interviewer you computed without looking at the setup.
The second loss is stopping at the number. The real point is that a 35 point margin swing in a slowing year is a large ask; say whether it looks achievable and what it would cost in growth.
What the interviewer asks next
Which margin would you use, free cash flow or EBITDA, and why does it matter here?
Is the Rule of 40 useful for a company growing 200% from a small base?
What spending would you cut first to move the margin from minus 45 to minus 10?
058One of 1,000 bank statements in a data room is forged. A forensic test run on a pooled sample of statements shows only whether any statement in the pool is forged, and every result takes a week. You have exactly one week. What is the fewest tests that identify the forged statement?Logic and brainteasersHardMulti-stage VCFintech VC
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How many tests, all started on day one?
Show the worked solution
Ten tests, run at once. Number the statements and write each number in binary, ten digits long. Test pool 1 holds every statement whose first digit is 1, pool 2 every statement whose second digit is 1, and so on. A week later the ten results, read as positive equals 1, spell out the forged statement's number. Ten is the minimum because nine tests give only 512 possible result patterns, fewer than the 1,000 statements.
Why can you not just halve the pile and test again?
Halving is the right instinct when you can see each answer before asking the next question, like guessing a number with higher or lower clues. Here every result takes the full week, so all the testing has to be designed up front. When questions must be asked in parallel, each statement needs its own unique pattern of answers, and the puzzle becomes how many yes or no answers it takes to give 1,000 statements a distinct pattern each. Ten answers give 2 to the 10, or 1,024 patterns; that is enough.
Writing each statement number in binary assigns it to the pools whose digits are 1, so a forged statement #357 lights up pools T2, T4, T5, T8 and T10, and those positives read back as 0101100101, which is 357.
How do you prove ten is the fewest?
Count the outcomes. Each test has two results, so n tests have at most 2 to the n distinct result patterns, and you need at least one pattern per possible forgery. Nine tests give 512 patterns for 1,000 suspects, so at least two statements would share a pattern and you could not tell them apart. Ten give 1,024, which is why 10 is both achievable and the minimum. One small detail: if you number the statements 1 to 1,000, the all-negative pattern is never used, which is fine because one statement is definitely forged.
The relationship
$2^{n} \ge 1000 \;\Rightarrow\; n \ge \log_2 1000 = 9.97 \;\Rightarrow\; n = 10$
n number of pooled tests run in parallel
2^n number of distinct positive and negative patterns n tests can produce
What it says in wordsYou need as many result patterns as suspects, and each extra test doubles the patterns.
Say where the trick breaks. It relies on exactly one forgery; with two forged statements, the positives are the union of two binary numbers and no longer name either one, so you would need more tests and a different design. It also assumes the test is perfectly sensitive in a pool of 500 statements. In real diligence a forger rarely fakes only one document, so the useful habit is the reasoning, counting outcomes before designing the checks.
Where candidates lose it
The common answer is a halving strategy that needs ten rounds, ten weeks. It is the right count for the wrong reason: the deadline rules out sequential testing, and the interviewer wants to hear you notice that before you start.
The second trap is giving 10 without the proof of the minimum. Say the counting argument in one sentence: nine tests give 512 patterns, fewer than 1,000 statements, so two statements would look the same.
What the interviewer asks next
What if two statements were forged?
If you had two weeks, could you do it with fewer tests in total?
How would the design change if a pool larger than 100 statements made the test unreliable?
059A company's assets will be worth either Rs 200 crore or Rs 1,400 crore next year, with equal odds. It owes Rs 1,000 crore of debt, all due then, so the expected asset value of Rs 800 crore is below the debt. Why is the equity still worth something, and how much? Ignore discounting.Valuation riddlesHardSilver LakeSan Francisco · 2022
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What is the equity worth?
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About Rs 200 crore, because equity is a call option on the assets. Shareholders get whatever is left after the debt, but never less than zero. In the bad case the assets of Rs 200 crore all go to lenders and equity gets nothing; in the good case equity keeps Rs 1,400 crore less Rs 1,000 crore, or Rs 400 crore. Half of Rs 400 crore is Rs 200 crore. Lenders, expecting Rs 600 crore on Rs 1,000 crore owed, carry the downside.
If expected assets are below the debt, where does the equity value come from?
Think of a lottery ticket bought with borrowed money where the lender can only take the ticket back. If it loses you owe nothing more; if it wins you keep the winnings above the loan. Limited liability makes equity a call optionThe right, not the obligation, to buy an asset at a fixed price. Its payoff is zero below that price and rises one for one above it. on the company's assets with a strike equal to the debt, so equity is worth the average of its payoffs, not the payoff at the average assets. The payoff at Rs 800 crore of assets is zero. The average of the payoffs, 0 and Rs 400 crore, is Rs {ev_eq59:.0f} crore.
The equity payoff is flat at zero below the Rs 1,000 crore debt and rises above it, so the chord joining the two outcomes passes above the curve: at expected assets of Rs 800 crore the payoff is zero, but expected equity is Rs 200 crore.
The relationship
$E[\max(A - D, 0)] = \tfrac{1}{2}\max(200-1000,0) + \tfrac{1}{2}\max(1400-1000,0) = 0 + 200 = 200$
A asset value next year, Rs 200 or Rs 1,400 crore
D debt due, Rs 1,000 crore
max(A - D, 0) what shareholders receive; never negative
What it says in wordsTake what equity gets in each outcome, floored at zero, and average those, rather than averaging the assets first.
Who pays for the equity's value, and why does it matter in practice?
Assets are worth Rs 800 crore on average, and equity takes Rs 200 crore of that, so the debt is worth Rs 600 crore against Rs 1,000 crore owed. Equity's option value is paid for by the lenders, which is why shareholders of a distressed company prefer riskier bets: more spread raises the option's value and pushes more of the downside onto the debt. If the outcomes were Rs 0 or Rs 1,600 crore, the same Rs 800 crore average would give equity Rs 300 crore. That is the answer to why a distressed company's stock can trade well above zero, and why lenders write covenants to stop management gambling.
State the limits. Ignoring discounting, and treating the odds as known, keeps the arithmetic clean; a real valuation would use risk-neutral probabilities and a time value, which is what an option pricing model does. And the debt is assumed to be one bullet due next year; covenants that let lenders take control earlier would cut the equity's option short.
Where candidates lose it
The fast wrong answer is zero, or a negative number, from netting Rs 1,000 crore of debt against Rs 800 crore of average assets. It forgets that shareholders cannot owe more than they put in, and that floor is the entire source of value.
The second trap is getting Rs 200 crore but not naming the mechanism. Say the words call option, struck at the debt, and add who pays for it: the lenders. That is the sentence the interviewer is listening for.
What the interviewer asks next
If the outcomes were Rs 0 or Rs 1,600 crore, what would the equity be worth?
Why might management of this company take on a riskier project with a lower expected value?
What covenant would you write into the debt to protect against that?
Asked at Silver Lake, Technology, Media and Telecom, San Francisco, 2022 (Wall Street Oasis): Why would a distressed company have a high equity value?
060A fund reports a 32% IRR on a deal it exited at 1.15x after six months, and a 12% IRR on a deal that made 3.1x over ten years, each on Rs 10 crore. Which made its LPs more money, and what would the six-month deal's IRR be if the proceeds then sat idle at 0% for the rest of a ten-year fund life?Fund economics riddlesHardFund of funds and LPsSeed and early-stage VC
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Measured over the full ten years with the cash idle, the six-month deal's IRR is about:
Show the worked solution
The 12% deal made Rs 21 crore against Rs 1.5 crore, fourteen times as much. The quick flip shows a 32% IRR because 1.15x in six months annualises to 1.15 squared, about 1.32. But LPs spend rupees, not rates. If the Rs 11.5 crore then sat idle for the remaining nine and a half years, the deal is 1.15x over ten years, an IRR of about 1.4%. The ten-year deal's 3.1x is about 12% a year on the whole Rs 10 crore for the whole time.
How can a 32% IRR make less money than a 12% IRR?
A cab that charges a high rate per minute for a two-minute ride earns less than a modest-rate cab hired for the whole day. IRR is a rate per year while the money is out, so a short deal can post a huge IRR on a tiny rupee profit, and the rate says nothing about how long the capital earned it. On Rs 10 crore the flip made Rs 1.5 crore. The ten-year deal made Rs 21 crore, with a lower rate sustained over twenty times as long.
On the same Rs 10 crore the six-month flip made Rs 1.5 crore and the ten-year hold Rs 21 crore, so the higher 32% IRR made one-fourteenth of the money, and measured over the fund's ten years with the cash idle it is only 1.4%.
The relationship
$\text{IRR} = M^{1/t} - 1: \quad 1.15^{1/0.5} - 1 = 32.2\%, \quad 3.1^{1/10} - 1 = 12.0\%, \quad 1.15^{1/10} - 1 = 1.4\%$
M money multiple, what came back over what went in
t years the money was out
What it says in wordsFor a single cash out and a single cash back, the IRR is the multiple spread evenly across the years, so the same multiple over more years is a lower rate.
Why does it matter what the cash does after the exit?
The 32% assumes the Rs 11.5 crore can be put straight back to work at a similar rate. For an LP whose money was committed for ten years, cash returned early earns only what the LP can do with it next, and if that is nothing, the deal's true rate over the commitment is about 1.4%. This is why LPs read IRR beside the money multiple, and why a fund boasting a top-quartile IRR on a 1.3x fund gets hard questions. Some funds also time capital calls with credit lines to shorten the period the money counts as out, which flatters IRR without adding a rupee.
The fair limit: early cash is worth something if the LP really can reinvest it, and the reinvestment rate decides by how much. In the room, say both numbers for every deal, multiple and IRR, and say which one an LP banks.
Where candidates lose it
The trap is to rank by IRR and pick the 32% deal. The question asks which made more money, and the answer is in rupees: Rs 21 crore against Rs 1.5 crore.
The second loss is the follow-on. Candidates who see the issue still fumble the ten-year IRR; keep it to one step, the tenth root of 1.15, and bracket it near 1.4% rather than guessing.
What the interviewer asks next
At what reinvestment rate after exit would the quick flip match the ten-year deal's profit?
How does a subscription credit line change a fund's reported IRR?
Which would you report to LPs first, IRR or TVPI, and why?

N	all kirana stores, 1.2 crore
s	share with daily sales above Rs 20,000, 15%
w	share of those willing to pay, 30%
p	price a month, Rs 500

g	ARR growth a year, 60%
d	dilution from each year's round, 15%
50/10	entry multiple over the exit multiple, the growth needed
n	years to break even

2.5, 0.4, 1.5	the three rounds written as multipliers
r	the steady annual rate with the same end result

Class	Preference	Stake	Preference / stake	Line with seniors holding
Seed	Rs 5 crore	10%	Rs 50 crore	Rs 100 crore
Series A	Rs 20 crore	20%	Rs 100 crore	Rs 130 crore
Series B	Rs 50 crore	20%	Rs 250 crore	Rs 250 crore

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Why not multiply every kirana store by the price?

How do you say the answer so it survives the follow-up?

Where candidates lose it

What the interviewer asks next

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What does paying 50x when peers trade at 10x actually commit you to?

Why does dilution stretch the answer by almost two years?

Where candidates lose it

What the interviewer asks next

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Why can the three percentages not simply be added?

How do you find the annual rate without a calculator?

Where candidates lose it

What the interviewer asks next

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Why is the line for each class its preference divided by its stake?

Why does the question say 'roughly', and do the lower lines move?

Where candidates lose it

What the interviewer asks next

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Why does selling 20% leave her with 32%, not 20%?

If her percentage fell both times, why did her wealth rise the first time?

Where candidates lose it

What the interviewer asks next

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Why is a right worth more than an obligation to do the same thing?

Where does this simple answer overstate the right in real life?

Where candidates lose it

What the interviewer asks next

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What does adding growth to margin actually measure?

Why is the slowdown the dangerous part of this question?

Where candidates lose it

What the interviewer asks next

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Why can you not just halve the pile and test again?

How do you prove ten is the fewest?

Where candidates lose it

What the interviewer asks next

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If expected assets are below the debt, where does the equity value come from?

Who pays for the equity's value, and why does it matter in practice?

Where candidates lose it

What the interviewer asks next

Try it first

How can a 32% IRR make less money than a 12% IRR?

Why does it matter what the cash does after the exit?

Where candidates lose it

What the interviewer asks next