Venture Capital puzzles, solved step by step

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.

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.

About 9.1x, not 7.5x. Take a Rs 100 crore fund. Fees take Rs 17.5 crore, leaving Rs 82.5 crore to invest. 60% of that, Rs 49.5 crore, goes to companies that return nothing, so only Rs 33 crore survives. To return Rs 300 crore, those survivors must earn 300 divided by 33, about 9.1x. Forgetting fees gives 300 over 40, or 7.5x, which understates the bar by a fifth.

Why does the base shrink twice before you divide?

A farmer who must sell 3 tonnes of grain for every tonne of seed bought, after setting aside some seed for the birds and losing most fields to flood, needs the surviving fields to yield far more than 3 tonnes each. A fund's target is set on every rupee committed, but only the rupees that are invested and survive can earn it, so the required survivor multiple is the target divided by the surviving share of commitments. Fees remove 17.5%; failures remove 60% of what is left. The surviving share is 0.825 x 0.4, which is 33% of commitments.

Is 3x on commitments the number LPs actually receive?

No, and saying so earns credit. The 3x here is before the manager's carried interest. If LPs want 3x after a 20% carry on profits, the fund must return about Rs 350 crore gross, and the survivors must earn about 10.6x. Recycling, where a fund reinvests early proceeds to put more than 82.5% of commitments to work, pushes the bar back down. Either way, the shape of the answer is the point: venture survivors need near ten-times outcomes as a group, which is why a fund cannot be built from companies that can only triple.

One honest limit: the 60% that returns nothing is a round assumption. Real portfolios have a middle band of companies returning 0.5x to 2x, which lowers the bar for the top performers, and the fee load varies by fund size and terms, so confirm both before using the figure.

30% together, 10% each, and the founder keeps 70%. A post-money SAFE converts into its amount divided by its cap, as a share of the company measured after all the SAFEs: Rs 2 crore over Rs 20 crore, Rs 3 crore over Rs 30 crore and Rs 4 crore over Rs 40 crore are each 10%. Because each percentage is fixed against that full capitalisation, a later SAFE cannot dilute an earlier one; only the founder is diluted, so the order does not matter.

What does 'post-money' fix that older SAFEs did not?

Three friends each promised a tenth of a home-cooked biryani know exactly what they get, however many friends were promised a share before them; the cook's portion is what shrinks. A post-money SAFEA simple agreement for future equity whose conversion stake is fixed as the amount invested divided by the post-money valuation cap, measured after all SAFEs convert. promises its holder a fixed percentage, amount over cap, of a company that already counts every SAFE, so SAFE holders never dilute one another and the founder carries all of it. 2/20, 3/30 and 4/40 are each 10%, and the founder goes from 100% to {founder67*100:.0f}%. Under the older pre-money SAFE, each holder's stake was measured before the other SAFEs converted, so they diluted each other and no one could read their percentage until the round.

What happens at the priced round, and when does the 10% change?

The priced round then dilutes everyone, SAFE holders included. If the Series A sells 20% of the company, each SAFE holder goes from 10% to 8% and the founder from 70% to 56%. The cap is a ceiling on the price a SAFE converts at, so if the round is priced below a holder's cap, that holder converts at the round price and gets more than 10%, at the founder's expense again. That is why the question fixes the round above all three caps.

Two limits worth saying. The figures leave out an option pool, which is usually added before the round and also comes out of the founder's side. And the SAFE is a US instrument; Indian startups raise early money through structures such as compulsorily convertible preference shares or notes whose conversion terms vary, so read the actual conversion clause rather than assuming post-money SAFE mechanics.

About 31%, not 80%. Picture 1,000 companies. 100 are good and your screen passes 80 of them. 900 are bad and it passes 20% of those, 180 companies. So 260 pass, and only 80 of them are good: 80 divided by 260 is 30.8%. The screen raises the odds from 10% to about 31%, which is useful, but most companies that pass are still bad because bad ones are so common.

Why is the answer so far below 80%?

A smoke alarm that rings for every real fire and for one in five burnt toasts will mostly ring for toast, because toast is far more common than fire. When the thing you are looking for is rare, even a small false alarm rate applied to the large crowd of negatives produces more false passes than true ones. Here 20% of 900 bad companies is 180, against 80% of 100 good companies, 80. The screen's 80% describes how it treats good companies; the question asks what a pass tells you, which also depends on how many bad companies get in.

What would actually improve the screen?

Run the alternatives. Halving the false pass rate to 10% lifts the answer to 80 out of 170, about 47%, while raising the hit rate from 80% to 95% only takes it to 95 out of 275, about 35%. When good companies are rare, cutting false passes matters more than catching every good one. A second, independent check helps the same way: if a passing company goes through another screen with the same rates, the base rate is now 31%, and a second pass takes it to about 64%.

The limit is that the two checks must be independent; a second partner who looks at the same deck with the same biases is not a second screen. And the 10% base rate is an assumption about your deal flow; a fund with better sourcing starts from a higher base and every pass means more.

Ask either founder, 'Which room would your co-founder say is real?', then take the other room. If you asked the truth-teller, she honestly reports the liar's false answer. If you asked the liar, he falsely reports the truth-teller's true answer. Either way the answer has passed through exactly one lie, so it always names the fake room. You do not need to know who answered.

Why does a direct question fail?

If you ask a stranger for directions and know only that they are either always honest or always lying, their answer alone is worth nothing: you cannot tell which kind of answer you got. A direct question returns the truth from one founder and its opposite from the other, so without knowing who answered, the reply carries no information. 'Are you the truth-teller?' is worse: both say yes. The question has to be built so the answer comes out the same whoever gives it.

Why does routing the question through the other founder fix it?

Count the lies along each path. Truth then lie, or lie then truth, both contain exactly one lie, so the indirect question always returns a false answer, and a reliably false answer is as useful as a true one: you simply take the opposite. There is a second classic: 'If I asked you whether room 1 is real, would you say yes?' The truth-teller answers honestly; the liar would say no to the inner question and then lies about that, so he says yes. That version uses two lies or none, which gives the truth directly.

The limit is the setup itself. The trick relies on people who lie with perfect consistency, which nobody does; real diligence on two founders who tell different stories relies on documents, not on clever questions. What the puzzle tests is whether you can design one check whose answer does not depend on who answers it, which is the logic behind cross-referencing a claim against two sources with opposite incentives.

B, about 3.72x or 14% a year, against 1.63x or 5% a year for A. Your return is earnings growth times the change in multiple. A's earnings grow 1.05 to the tenth, 1.63x, and its multiple stays at 10x, so you make 1.63x. B's earnings grow 1.2 to the tenth, 6.19x, and its multiple falls from 25x to 15x, a factor of 0.6, so you make 3.72x. B's multiple could fall to 6.6x and still match A.

How do you split a ten-year return into its parts?

A flat bought cheap in a stagnant town can earn less than one bought dear in a fast-growing suburb, even if the suburb's prices later cool. Ignoring dividends, the return on a business is the growth in its earnings times the change in the multiple the market pays for them, so a high starting multiple only hurts if earnings growth cannot outrun the fall in that multiple. A gets 1.63 x 1.0. B gets 6.19 x 0.6. Ten years of 20% growth multiplies earnings by over six, which swamps the 40% multiple compression.

When does the cheap business win instead?

Three cases. If B's growth fades early, the answer flips: at 10% a year, B's earnings grow only 2.6x and 2.6 x 0.6 is about 1.55x, below A. If A pays out its earnings as dividends, a 10x multiple means a 10% earnings yield, and reinvesting that cash adds a lot that this sum leaves out; B, growing at 20%, likely reinvests everything. And if the growth never arrives, B's price has nothing to fall back on. The honest answer to a 'great price or great business' question is that it turns on how long and how surely the better business keeps growing.

Give the interviewer a number that frames the judgement: B's exit multiple could fall all the way to 6.6x and still match A's return. That margin, not the entry multiple, is the measure of how much you are overpaying.

64% from the top 4%, and 51.2% from the top 0.8%. Each application of the rule keeps a fifth of the companies and four-fifths of the returns. The top 20% earn 80%. A fifth of those, the top 4%, earn 80% of 80%, or 64%. A fifth of those, the top 0.8%, earn 80% of 64%, or 51.2%. So fewer than one company in a hundred produces more than half of all returns.

Why do the shares multiply at each step?

If a fifth of a city's shops make four-fifths of its sales, and among those top shops a fifth again make four-fifths of their group's sales, then the very top shops make four-fifths of four-fifths. A rule that repeats inside itself compounds: k steps leave 0.2 to the power k of the companies with 0.8 to the power k of the returns. Three steps give 0.8% of companies and 51.2% of returns. The pattern is the signature of a power lawA distribution where a small number of very large outcomes account for most of the total, and the same shape repeats at every scale., and it is why venture is described as a business of outliers.

What does this mean for how a venture fund is built?

Half the returns come from the top 0.67% of companies, fewer than one in a hundred. A fund of 30 companies, picked at random from this population, has only about a 21% chance of owning even one company from that top 0.8%, so portfolio size and access to the best deals matter more than average picking skill. That is the logic behind funds holding more companies, reserving money to double down on the winners, and refusing to cap the upside of any single investment. A fund cannot diversify its way to the median here; the median company returns little.

State the limit. A clean, repeating 80/20 rule is a stylised model, not a measured fact about any dataset; real venture returns are lumpy and vary by vintage and stage. The value of the puzzle is the shape: concentration at the top is far more extreme than one application of 80/20 suggests.

About 7,000 buses, and about Rs 34 lakh a year at 10% share. Take 1.4 crore people; about 20% are of school age, 28 lakh children. Assume half attend schools that run buses, 14 lakh, and a quarter of those ride, 3.5 lakh. At about 50 riders a bus, that is 7,000 buses, in a range of roughly 4,667 to 10,500. Winning 700 of them at Rs 4,800 a year each is about Rs 33.6 lakh.

How do you build the count so each step can be challenged?

Planning a wedding, you do not guess the number of plates; you count families, then members per family, then how many will turn up. A sizing is only as good as its weakest stated assumption, so each step should be a single number with a reason the interviewer can push on. Here the chain is people, school-age children, children in schools that run buses, children who ride, and riders per bus. The 20% school-age share is the firmest; whether half of children attend schools that run buses is the least certain and depends on the city's mix of private and government schools.

What does the number tell an investor?

Give the range, then the judgement. Riders between 20% and 30% and buses carrying 40 to 60 riders give 4,667 to 10,500 buses, and revenue between about Rs 22 lakh and Rs 50 lakh a year. Even at the top of the range, one city at 10% share is a small business, so the case for investing rests on winning many cities, a higher share, or more revenue per bus, not on this city alone. A parent subscription for live tracking, or selling route planning and fuel tracking to the school, could change the revenue per bus more than any share assumption.

To check it outside the room, ask the transport department how many vehicles are registered as school buses in the city, since many states register them under a separate category, and call three or four large schools to ask how many buses they run per thousand students. Confirm the 20% school-age share against the latest census or survey for that city rather than taking it from memory.

About Rs 4,167 crore, against Rs 2,667 crore if the fund had kept 15%. A round that sells 20% leaves existing holders 80% of what they had, so 15% becomes 12% and then 9.6%. Returning Rs 400 crore from 9.6% needs an exit of 400 divided by 0.096. Skipping pro rata raised the bar by about 56%, and later rounds would raise it further.

Why does a round that sells 20% not cut your stake by 20 points?

Think of a pizza shared by four friends. A fifth friend arrives and is given a fifth of the whole pizza; every original slice shrinks by a fifth of itself, not by a fixed number of slices. Dilution is multiplicative: a round that sells 20% of the company leaves every existing holder with 80% of their previous stake. So 15% becomes 15 x 0.8 = 12%, and the next round makes it 12 x 0.8 = 9.6%. The fund has lost 5.4 points, 36% of what it bought, without selling a single share.

How do you turn a stake into a fund-returner exit?

A fund returnerA single investment whose proceeds alone equal the whole fund size, the bar many seed funds use when they decide whether a company could matter. is one company whose proceeds alone pay back the whole fund. The exit it needs is the fund size divided by the ownership held at exit. At 15%, Rs 400 crore needs 400 / 0.15 = Rs 2,667 crore. At 9.6%, it needs 400 / 0.096 = Rs 4,167 crore. Same company, same fund, and the bar moved up by Rs 1,500 crore because of two decisions not to write a cheque.

What should you say about the assumptions?

Say three out loud: no further rounds before exit, no option pool top-up, and a clean sale in which preferences do not change the split. Each of those would push the number higher, never lower. Rs 4,167 crore is therefore a floor, not an estimate. Then add the reason the question exists: a seed fund keeps reserves precisely so it can defend its stake in the companies that are working, because the seed cheque buys the option and the pro rata cheque keeps it.

Undiscounted, LTV is Rs 4 lakh and LTV to CAC is 3.3; discounted at 2% a month, LTV is Rs 2 lakh and the ratio is 1.7. Gross profit is Rs 8,000 a month and 2% churn means an average life of 50 months, so 8,000 x 50 is Rs 4 lakh. Discounting adds the 2% rate to the 2% churn, so LTV becomes 8,000 over 0.04, which is Rs 2 lakh.

Why is the average customer life one over the churn rate?

A gym that loses 2 of every 100 members each month, month after month, keeps the average member for 50 months: some leave in the first month, some stay ten years, and the steady 2% leak averages out to one over 0.02. With a constant monthly churn c, the expected customer life is 1 / c months, and lifetime value is the monthly gross profit times that life. Use gross profit, not revenue: Rs 10,000 at 80% margin is Rs 8,000 a month, and 8,000 x 50 is Rs 4 lakh, against a Rs 1.2 lakh acquisition cost, a ratio of 3.3.

What does discounting do to a stream that already decays?

A rupee of gross profit in month 40 is worth about 0.45 today at 2% a month. When churn and the discount rate are both monthly rates, discounted LTV is monthly gross profit divided by churn plus the discount rate, because each month the stream shrinks by survival and by time together. Here that is 8,000 / (0.02 + 0.02), exactly Rs 2 lakh if each month's profit arrives at the month end.

Is 2% a month a fair rate? It compounds to about 27% a year, a venture-style hurdle rather than a bank rate, so the halving here is at the harsh end. The point survives a gentler rate: a ratio that looks comfortable at 3.3 can sit much closer to 1 once time is priced, and the 15-month payback on gross profit is often the more honest single number to quote alongside it.