Venture Capital puzzles, solved step by step

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

Showing 71–80 of 100

071Business A costs 10x earnings, grows earnings 5% a year, and still trades at 10x in ten years. Business B costs 25x earnings, grows earnings 20% a year, and trades at only 15x in ten years. Ignoring dividends, which makes you more money over the ten years, and at what annual rate?Valuation riddlesCoreCoatue ManagementNew York · 2023
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Which business returns more over ten years?
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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.
Rs 1 in the 10x business growing 5% becomes 1.63x in ten years, while Rs 1 in the 25x business growing 20% becomes 3.72x even after its multiple falls to 15x, because earnings growth of 6.19x outruns the 0.6 multiple factor.
The relationship
$M = (1+g)^{10} \times \frac{P/E_{\text{exit}}}{P/E_{\text{entry}}}: \quad A = 1.05^{10} \times \tfrac{10}{10} = 1.63, \quad B = 1.2^{10} \times \tfrac{15}{25} = 3.72$
g annual earnings growth
P/E entry, exit the price-to-earnings multiple paid and received
M money multiple over ten years
What it says in wordsMultiply how much earnings grew by how much the multiple changed; the annual rate is the tenth root of the result.
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.
Where candidates lose it
The common error is choosing A because 10x feels safe and the 40% multiple drop on B sounds painful. Candidates react to the entry price and never multiply out the earnings growth.
The second trap is the opposite: declaring that quality always wins. Say the condition, growth that lasts, and give the break-even multiple, so the answer reads as a judgement rather than a slogan.
What the interviewer asks next
If A pays out all its earnings as dividends and you reinvest them at 10%, does the answer change?
At what growth rate does B only match A?
Which of B's numbers would you test hardest in diligence?
Asked at Coatue Management, Technology, Media and Telecom, New York, 2023 (Wall Street Oasis): Would you rather buy a low quality business at a great price or a high quality business at an ok price?
072A venture fund has called Rs 200 crore from its LPs, distributed Rs 150 crore back to them, and holds a portfolio marked at Rs 350 crore. What are its DPI, RVPI and TVPI, and which one should an LP trust most?Fund economics riddlesWarm upFund of funds and LPsMulti-stage VC
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What are DPI, RVPI and TVPI?
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DPI 0.75x, RVPI 1.75x, TVPI 2.5x. All three divide by the Rs 200 crore paid in. Distributions of Rs 150 crore give a DPI of 0.75x; the Rs 350 crore still held gives an RVPI of 1.75x; the total value of Rs 500 crore gives a TVPI of 2.5x. An LP trusts DPI most, because it is cash already returned. The fund looks like a 2.5x but has not yet given back the money it called.
What does each ratio measure?
A friend who borrowed Rs 2,000, has repaid Rs 1,500 and promises the rest plus Rs 3,500 more from a deal still underway has given you back 0.75 times your money in cash and promised 1.75 times more. DPIDistributions to paid-in capital: cash returned to LPs divided by capital called. counts cash returned, RVPIResidual value to paid-in capital: the portfolio's current marked value divided by capital called. counts value still on paper, and TVPI is the two added together, all divided by the capital called. The fund called Rs 200 crore, so DPI is 150 / 200 = {dpi72:.2f}x, RVPI is 350 / 200 = {rvpi72:.2f}x and TVPI is {tvpi72:.1f}x.
Against Rs 200 crore paid in, the fund has returned Rs 150 crore of cash and holds Rs 350 crore on paper, so its TVPI is 2.5x but only 0.75x of it is money LPs have actually received.
The relationship
$\text{TVPI} = \text{DPI} + \text{RVPI} = \frac{150}{200} + \frac{350}{200} = 0.75 + 1.75 = 2.5$
DPI distributions over paid-in capital
RVPI remaining marked value over paid-in capital
TVPI total value, cash plus marks, over paid-in capital
What it says in wordsDivide cash returned and value still held by the capital called; the two add up to the fund's total multiple.
Why does an LP care more about the 0.75x than the 2.5x?
Because the marks can move and the cash cannot. RVPI rests on the fund's own valuation of private companies, often at the last round's price, so 70% of this fund's reported value is an estimate that has not been tested by a sale. If those marks fell 40%, RVPI would drop to 1.05x and TVPI to 1.8x, while DPI would stay at 0.75x. That is why many LPs judge older funds on DPI, and why the phrase 'DPI is the only metric that counts' gets repeated in fundraising conversations.
The fair limit is timing. A young fund naturally has low DPI because its companies have not been sold yet; judging a three-year-old venture fund on DPI alone would punish every fund for being young. Read DPI against the fund's age, and ask how the marks were set, last round, a comparable, or a recent offer, before trusting the RVPI.
Where candidates lose it
The usual slip is dividing by the wrong base: by the fund's total commitments instead of capital called, or dividing distributions by the NAV. Every ratio here shares the same denominator, the money LPs have actually paid in.
The second loss is stopping at the three numbers. The interviewer wants the judgement: 2.5x is mostly paper, and the fund has not yet returned the capital it called.
What the interviewer asks next
If the portfolio marks fell 40%, what would TVPI be?
Why might a young fund show a high TVPI and a DPI near zero?
How would you test whether the Rs 350 crore of marks are fair?
073Five funds bid for the same deal. Each estimates its value with an independent error spread evenly between minus 20% and plus 20%, and bids exactly its estimate. On average, by how much does the winning fund overpay?Decision and game theoryHardGrowth equityMulti-stage VC
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The winner's average overpayment is about:
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About 13.3%. Every fund's estimate is right on average, but the deal goes to the fund with the highest estimate, and the highest of five errors is not average. For errors spread evenly from minus 20% to plus 20%, the highest of n sits on average (n - 1)/(n + 1) of the way to the top: with five bidders that is 4/6 of 20%, or 13.3%. Winning is itself evidence that you guessed high.
Why does winning tell you that you overestimated?
Ask five friends to guess the weight of a goat at a fair and give the goat to whoever guesses highest; the winner will nearly always have guessed too much. When the prize goes to the highest estimate, the winner is selected for being too optimistic, so even unbiased bidders overpay on average. This is the winner's curseThe tendency for the winner of an auction with uncertain value to have overestimated that value, because winning selects the most optimistic estimate.. No one bid foolishly here; each estimate was fair. The overpayment comes entirely from the selection.
One fund's valuation error is spread flat around zero, but the highest of five errors bunches near the top of the range and averages plus 13.3%, which is how much the winning fund overpays on average.
The relationship
$E[\max \text{ of } n] = \frac{n-1}{n+1} \times 20\% = \frac{4}{6} \times 20\% = 13.3\%$
n number of bidders, 5
20% the widest error any one estimate can have
(n - 1)/(n + 1) how far toward the top the highest of n evenly spread values sits on average
What it says in wordsThe best of five evenly spread errors sits on average two-thirds of the way to the top, so the winner is about 13% too high.
What should a disciplined bidder do about it?
Bid below your estimate, and more so when there are more bidders or more uncertainty. The curse grows with the number of bidders and with the width of the errors: two bidders overpay by 6.7% on average, five by 13.3%, ten by 16.4%. So a fund in a crowded growth round should shade its bid by roughly the expected overestimate, about 12% of its estimate here, and a fund that keeps winning contested deals should ask whether it is better informed or just more optimistic. A simulation of 200,000 auctions in the source file gives 13.3%, matching the formula.
Bidders Winner's average overpayment
1 0.0%
2 6.7%
5 13.3%
10 16.4%
With errors spread evenly up to 20% either way, the winning bid's average overestimate rises with the number of bidders; one bidder has no curse at all.
The limit: the model assumes every fund's estimate is equally noisy and that all bid their raw estimate. Better-informed funds face a smaller curse, and in practice everyone shades, so real overpayment is smaller than the raw figure. The direction of the effect is what to carry into the room.
Where candidates lose it
The fast wrong answer is zero, from the true fact that each estimate is unbiased. The mistake is forgetting that the winner is not a random fund; it is the one with the highest estimate.
The second trap is answering 20%, as if the winner always has the maximum error. The highest of five sits two-thirds of the way up on average, not at the edge; give the (n - 1)/(n + 1) rule and the 13.3%.
What the interviewer asks next
How does the overpayment change with ten bidders?
If one fund has half the error of the others, how should it bid?
Why do proprietary deals, with one bidder, avoid the winner's curse?
074Venture returns follow an 80/20 rule that repeats inside itself: 20% of companies make 80% of returns, and within that top 20% the same split holds again, and again within that. What share of returns comes from the top 4% of companies, and from the top 0.8%?Power law and portfolio mathsCoreMulti-stage VCFund of funds and LPs
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What share of returns comes from the top 0.8% of companies?
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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.
Drawn to scale by share of companies, each nested square keeps a fifth of the companies and four-fifths of the returns, so the top 4% earn 64% and the tiny top 0.8% square earns 51.2%, more than half of everything.
The relationship
$\text{share of returns}(p) = p^{\ln 0.8 / \ln 0.2} = p^{0.139}: \quad 0.04^{0.139} = 0.64, \quad 0.008^{0.139} = 0.512$
p top fraction of companies
ln 0.8 / ln 0.2 the exponent that makes a fifth of any group earn four-fifths of its returns
What it says in wordsOne formula reproduces every level of the repeating 80/20 rule, and it can be read at any fraction, not just the three steps.
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.
Where candidates lose it
The common error is miscounting steps, giving 64% for the top 0.8%. Write the fractions out: 20%, then 4%, then 0.8% is three steps, so the share is 0.8 cubed.
The second trap is adding instead of multiplying, or answering 80% minus 16% and similar. Each step takes four-fifths of the previous share, not of the whole, so keep multiplying.
What the interviewer asks next
What share of companies earns 90% of returns under this rule?
With 30 companies in a fund, what is the chance of holding at least one top-0.8% company?
How should this shape change a fund's reserve policy?
075A startup sells bus-tracking to schools at Rs 400 per bus per month. Estimate the number of school buses in a city of 1.4 crore people, and the startup's annual revenue there if it wins 10% of them.Market sizing and estimationCoreSeed and early-stage VCIndia VC
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Roughly what annual revenue does 10% of the city's buses bring?
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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.
Four stated assumptions take 1.4 crore people to about 7,000 school buses, within a range of roughly 4,667 to 10,500, and winning 10% of them at Rs 400 a month brings only about Rs 34 lakh a year.
The relationship
$\text{Buses} = \frac{1.4\text{ cr} \times 0.20 \times 0.50 \times 0.25}{50} = 7{,}000$
0.20 share of the population of school age
0.50 share of those children in schools that run buses
0.25 share of those who ride the school bus
50 riders served by one bus, counting two routes a day
What it says in wordsShrink the population to the children who actually ride, then divide by how many each bus carries.
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.
Where candidates lose it
The first way to lose this is a zero slip at the end: buses times share times price times twelve has enough steps to drop a factor of ten, and Rs 3.4 crore instead of Rs 34 lakh changes the conclusion entirely.
The second is presenting the bus count and stopping. The interviewer wants to hear that the answer is small, and what would have to be true for the business to matter.
What the interviewer asks next
How many cities would the startup need to reach Rs 50 crore of revenue at this price?
What would a parent subscription at Rs 50 a month add per city?
How would you size the same market from the supply side, starting with schools?
076A Rs 400 crore seed fund buys 15% of a company. It does not take its pro rata in the next two rounds, and each of those rounds sells 20% of the company to new investors. How large must that one company's exit be for the fund's stake to return the whole Rs 400 crore fund?Power law and portfolio mathsCoreSeed and early-stage VCIndia VC
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Before you calculate the exit: what stake does the fund hold after the two rounds?
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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.
The fund's stake falls from 15% to 12% to 9.6% as two rounds each sell 20% of the company, so the exit needed to return the Rs 400 crore fund rises from Rs 2,667 crore to Rs 4,167 crore, about 56% higher.
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.
The relationship
$E = \frac{F}{s_0(1-d)^k} = \frac{400}{0.15 \times 0.8^2} \approx 4{,}167$
E exit value needed, Rs crore
F fund size, Rs 400 crore
s_0 ownership bought at seed, 15%
d share of the company each later round sells, 20%
k rounds skipped, 2
What it says in wordsShrink the stake by 80% for each skipped round, then divide the fund size by what is left.
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.
Where candidates lose it
The common slip is subtracting instead of multiplying: 15 minus 3 minus 3 gives 9%, and a nervous candidate sometimes even writes 15% minus 40%. The second round dilutes the already smaller 12%, so the stake is 9.6%, and the gap matters once you divide the fund size by it.
The second loss is stopping at the percentage. The interviewer asked for an exit value, so convert ownership into the fund-returner number and say it is a floor, because every later round only dilutes further.
What the interviewer asks next
What share of each round would the fund have to buy to hold exactly 15%?
If two more rounds each sell 15% before the exit, what is the fund-returner exit now?
Why might a fund rationally skip its pro rata in a company that is doing well?
077A founder tells you: we raised Rs 50 crore at Rs 200 crore. What share of the company did the investors buy if Rs 200 crore is the pre-money valuation, and what if it is the post-money?Valuation riddlesWarm upSeed and early-stage VCSeries A to C VC
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Answer in five seconds: what share did the investors buy?
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20% if Rs 200 crore is the pre-money, 25% if it is the post-money. Pre-money is the value before the cheque, so post-money is 200 plus 50, Rs 250 crore, and 50 over 250 is 20%. If Rs 200 crore already includes the cheque, investors own 50 over 200, which is 25%, and the pre-money was only Rs 150 crore. One unstated word moves five points of the company.
What do pre-money and post-money actually mean?
Two friends own a food stall worth Rs 2,00,000. A third friend puts in Rs 50,000 for a new oven. The stall is now worth Rs 2,50,000 and the new friend owns a fifth of it, because the Rs 50,000 is inside the new total. Pre-money is the value before the new money arrives, post-money is pre-money plus the new money, and the investor's share is always the cheque divided by post-money. The only question is which of the two numbers the founder quoted.
The same Rs 50 crore cheque buys 20% when Rs 200 crore is the pre-money, because post-money is then Rs 250 crore, and 25% when Rs 200 crore is the post-money, because the company was valued at only Rs 150 crore before the cheque.
The relationship
$s = \frac{I}{\text{Pre} + I} = \frac{I}{\text{Post}} \qquad \frac{50}{250} = 20\% \qquad \frac{50}{200} = 25\%$
s share of the company the new investors own
I the new money, Rs 50 crore
Pre, Post valuation before and after the new money
What it says in wordsThe investor's share is the cheque over the value of the company with the cheque inside it.
Why does the post-money reading cost the founder more?
If Rs 200 crore is post-money, the pre-money is only Rs 150 crore: the founder sold Rs 50 crore of shares against a company valued Rs 50 crore lower. 25% against 20% is a quarter more of the company for the same cheque, and that extra slice is diluted alongside everything else in every later round. Check both readings the same way: 20% of 250 is 50, and 25% of 200 is 50. Both pass, which is exactly why the sentence alone cannot settle it.
Headlines tend to quote whichever number sounds bigger, and a term sheet settles it by stating the pre-money and the fully diluted share count. Some convertible instruments quote a post-money cap instead, so the habit of asking which one carries straight into later work.
Where candidates lose it
Candidates divide 50 by 200 instantly and say 25%. That is right for only one of the two readings, and the interviewer chose an ambiguous sentence to see whether you ask which valuation the Rs 200 crore is.
The second slip is treating the difference as small. Five points of a Rs 250 crore company is Rs 12.5 crore of value handed over, and the extra slice keeps its weight through every later round.
What the interviewer asks next
The company had 1 crore shares before the round. What price per share does each reading imply?
The round also creates a 10% option pool before the money comes in. What is the effective pre-money?
Why might a founder prefer to quote the post-money figure in a press release?
078A customer pays Rs 10,000 a month at an 80% gross margin, churns at 2% a month, and costs Rs 1.2 lakh to acquire. What are lifetime value and LTV to CAC, first undiscounted and then discounting at 2% a month?SaaS and unit economics riddlesCoreSaaS-focused VCSeries A to C VC
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Before working it: what does discounting at 2% a month do to the LTV here?
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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.
Monthly gross profit per customer starts at Rs 8,000 and decays by 2% a month, an area of Rs 4.0 lakh; discounting each month at 2% as well makes the stream decay twice as fast, halving the area to Rs 2.0 lakh and taking LTV to CAC from 3.3 to 1.7.
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.
The relationship
$\text{LTV} = \frac{m}{c + r} = \frac{8{,}000}{0.02 + 0.02} = 2{,}00{,}000$
m monthly gross profit per customer, Rs 8,000
c monthly churn, 2%
r monthly discount rate, 2%
What it says in wordsDivide monthly gross profit by the total rate at which it leaks away, through customers leaving and through time.
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.
Where candidates lose it
The first loss is using revenue: Rs 10,000 x 50 gives Rs 5 lakh and a ratio of 4.2, which flatters the business by a quarter. A customer's value to the company is the gross profit they leave behind, not the invoice.
The second is saying churn already handles time. Churn counts customers who leave; discounting prices the wait for the ones who stay. Leave either out and the ratio looks healthier than the cash.
What the interviewer asks next
What monthly churn gives an undiscounted LTV to CAC of exactly 3?
Price rises 10% a year for customers who stay. How does that change the formula?
Why might a seed investor care more about payback months than LTV to CAC?
079Estimate the annual market for refurbished smartphones in India. Build it from the number of phones replaced each year, the share of those that are resold, the share of resold phones that are refurbished, and the average refurbished selling price.Market sizing and estimationHardConsumer internet VCIndia VC
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Which input moves this market size the most per unit of doubt?
Show the worked solution
About Rs 24,000 crore a year on these assumptions, and supply, not demand, sets the ceiling. Take 60 crore phones in use on a 4-year cycle: 15 crore retire each year. If 40% are resold and half of those are refurbished, 3 crore units reach the refurbished channel at about Rs 8,000 each. Every input is an assumption to check against market data.
Why start from the phones people retire rather than the people who want one?
A second-hand bookshop can only sell the books people give up. However many students want cheap textbooks, the shop's shelves fill at the rate books are released. A refurbished market is a used-goods market, so its volume is capped by the flow of retired devices before anyone asks about demand. Size the flow first, then check demand only to see whether it is large enough to absorb that flow.
On the stated assumptions, 60 crore phones in use retire 15 crore a year; 6 crore are resold and 3 crore of those are refurbished, a Rs 24,000 crore market, and because demand of about 4.5 crore units exceeds the 3 crore supply, supply sets the size.
How do you build and defend each step?
Say every number is an assumption and give the reason for its size. Installed base over replacement cycle gives the retiring flow: 60 crore over 4 years is 15 crore phones a year. Of those, many stay in a drawer or pass to a parent, so assume 40% are sold or traded in, 6 crore. Some go straight to a buyer as they are; assume half pass through a grading, repair and warranty channel, 3 crore units. At an average Rs 8,000, that is Rs 24,000 crore.
Step Assumption Units, crore
Smartphones in use Round starting base 60
Retired each year 4-year replacement cycle 15
Resold or traded in 40% of retired 6
Refurbished 50% of resold 3
Market, Rs crore x Rs 8,000 24,000
Each row multiplies the one above by a stated assumption, so the Rs 24,000 crore estimate can be challenged one line at a time; none of these figures is market data.
How do you show that supply rather than demand caps it?
Run a quick demand check. If about 18 crore phones are bought each year, new and used, and a quarter of buyers would take a refurbished phone at a fair price, demand is about 4.5 crore units, more than the 3 crore supply. When demand exceeds supply, the market is the supply, so the replacement cycle is the lever to test. A 3-year cycle lifts the flow to 20 crore retired phones and the market to Rs 32,000 crore, a bigger swing than any plausible change in price. That also tells an investor where the business is won: in sourcing devices, not in finding buyers.
Where candidates lose it
Most candidates size demand: population, smartphone owners, share who would buy refurbished. That gives a huge number the market can never reach, because nobody can sell phones that have not been retired. The interviewer is waiting to see whether you notice which side binds.
The second loss is stating inputs as facts. Say that each number is a round assumption to be checked, and name the one you would check first: the replacement cycle.
What the interviewer asks next
How would you estimate the replacement cycle without any industry report?
What would change if exports of used phones took a quarter of the resold flow?
Would you rather back a refurbisher that sources through trade-ins or one that buys from individuals, and why?
080A Rs 100 crore venture fund will pay Rs 17.5 crore of management fees over its life, and its terms let it recycle up to Rs 15 crore of early exit proceeds into new investments. How much does it invest with and without recycling, and what gross multiple on invested capital does each case need to hand investors 2.5x their commitments?Fund economics riddlesCoreFund of funds and LPsSeed and early-stage VC
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With full recycling, what gross multiple does the invested capital need?
Show the worked solution
Without recycling the fund invests Rs 82.5 crore and needs 3.03x; with full recycling it invests Rs 97.5 crore and needs about 2.72x. Investors want Rs 250 crore back. Without recycling, Rs 82.5 crore must produce all of it. With recycling, Rs 97.5 crore must produce Rs 265 crore, because Rs 15 crore of proceeds was put back to work rather than paid out. Carry is ignored throughout.
Why can a fund not invest all of its commitments?
A family sets aside Rs 1,00,000 for a wedding and the planner's fee is Rs 17,500 of it: only Rs 82,500 buys food and flowers. Management fees are paid out of the same commitments that fund the investments, so a fund that never recycles invests less than its headline size. Rs 17.5 crore could be 2% a year for five years and 1.5% for five more; schedules vary, but the arithmetic is the same. To turn Rs 100 crore into Rs 250 crore for investors, Rs 82.5 crore has to earn 250 / 82.5 = 3.03x.
Without recycling, Rs 82.5 crore of invested capital must produce Rs 250 crore, 3.03x; recycling Rs 15 crore of early proceeds lifts invested capital to Rs 97.5 crore, which must produce Rs 265 crore, about 2.72x, not the 2.56x that dividing 250 by 97.5 suggests.
How does recycling change the multiple, and why is it not 250 over 97.5?
RecyclingReinvesting proceeds from early exits into new companies instead of distributing them, usually capped at about the amount of fees paid. lets the fund put Rs 15 crore of early exit money back into new companies, so invested capital reaches Rs 97.5 crore. But that Rs 15 crore was proceeds the portfolio had already earned and did not pay out, so the portfolio must produce Rs 250 crore for investors plus the Rs 15 crore it recycled. Gross proceeds of 265 on cost of 97.5 is 2.72x: still well below 3.03x, but not the 2.56x a quick division gives.
The relationship
$M = \frac{2.5\,C + R}{C - F + R} = \frac{250 + 15}{100 - 17.5 + 15} \approx 2.72\times$
C commitments, Rs 100 crore
F lifetime fees, Rs 17.5 crore
R proceeds recycled, Rs 15 crore
M gross multiple needed on invested capital
What it says in wordsThe portfolio must earn what investors receive plus what was reinvested, on everything that was invested.
Say the limit too. Recycling delays money investors would have received early, so it can lift the multiple while lowering the IRR, and it works only if the fund finds good companies for the recycled rupees. It is a tool for getting fees back to work, not free return.
Where candidates lose it
The quick slip is 250 over 97.5, which gives 2.56x and looks like the natural answer. It counts the recycled Rs 15 crore as money invested but forgets it was also money the portfolio had to earn first, so it understates the bar.
The other loss is forgetting fees entirely and saying 2.5x on Rs 100 crore. The interviewer set the fee number so you would take it out of the investable pool before anything else.
What the interviewer asks next
If the fund recycles only Rs 7.5 crore, what multiple does the invested capital need?
Carry is 20% of profits above commitments. What gross proceeds give investors 2.5x net?
Why might an investor in the fund prefer no recycling even though it lowers the bar?

g	annual earnings growth
P/E entry, exit	the price-to-earnings multiple paid and received
M	money multiple over ten years

DPI	distributions over paid-in capital
RVPI	remaining marked value over paid-in capital
TVPI	total value, cash plus marks, over paid-in capital

n	number of bidders, 5
20%	the widest error any one estimate can have
(n - 1)/(n + 1)	how far toward the top the highest of n evenly spread values sits on average

Bidders	Winner's average overpayment
1	0.0%
2	6.7%
5	13.3%
10	16.4%

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

When does the cheap business win instead?

Where candidates lose it

What the interviewer asks next

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What does each ratio measure?

Why does an LP care more about the 0.75x than the 2.5x?

Where candidates lose it

What the interviewer asks next

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Why does winning tell you that you overestimated?

What should a disciplined bidder do about it?

Where candidates lose it

What the interviewer asks next

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Why do the shares multiply at each step?

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

Where candidates lose it

What the interviewer asks next

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

What does the number tell an investor?

Where candidates lose it

What the interviewer asks next

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

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

What should you say about the assumptions?

Where candidates lose it

What the interviewer asks next

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What do pre-money and post-money actually mean?

Why does the post-money reading cost the founder more?

Where candidates lose it

What the interviewer asks next

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

What does discounting do to a stream that already decays?

Where candidates lose it

What the interviewer asks next

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Why start from the phones people retire rather than the people who want one?

How do you build and defend each step?

How do you show that supply rather than demand caps it?

Where candidates lose it

What the interviewer asks next

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Why can a fund not invest all of its commitments?

How does recycling change the multiple, and why is it not 250 over 97.5?

Where candidates lose it

What the interviewer asks next