Venture Capital puzzles, solved step by step

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

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001A Rs 300 crore fund put Rs 10 crore into each of 30 companies and returned 3.0x, Rs 900 crore. Fifteen companies were written off, and the best one returned Rs 540 crore. Had the fund passed on that best company and backed one more company earning the average of the other 29, what would the fund have returned?Power law and portfolio mathsCoreSeed and early-stage VCSeries A to C VC
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Before you work it: what does the fund return without its best company?
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About 1.24x, down from 3.0x. The other 29 companies returned Rs 360 crore, an average of Rs 12.4 crore each. Replace the Rs 540 crore winner with one more average company and the fund returns Rs 372 crore on Rs 300 crore. Missing that one company costs Rs 528 crore, more than three times what all 15 write-offs cost together.
Why does one company out of thirty move the whole fund?
Picture a cricket team that scores 900 runs in a series, 540 of them from one batter. Drop that batter for an average player and the team's total collapses, however steady the rest were. A venture fund works the same way: returns are so uneven that the best company is often worth more than all the others put together. Here the winner returned 54x its cheque. The other 29 returned Rs 360 crore between them, an average of Rs 12.4 crore on Rs 10 crore each, about 1.24x.
The relationship
$M = \frac{900 - 540 + 360/29}{300} = \frac{372.4}{300} \approx 1.24\times$
900 what the fund actually returned
540 what the best company returned
360/29 one more company earning the average of the other 29
300 the fund's committed capital
What it says in wordsTake the winner out, put an average company in its place, and divide by the money the fund invested.
The fund returned Rs 900 crore, 3.0x. Removing the Rs 540 crore winner and adding one average company worth Rs 12.4 crore leaves Rs 372 crore, 1.24x, and that single miss costs Rs 528 crore against Rs 150 crore lost on all fifteen write-offs.
How does that compare with the cost of the failures?
Fifteen companies went to zero. At Rs 10 crore each, that is Rs 150 crore of lost capital, the most a write-off can ever cost. A write-off loses at most the cheque; a missed winner loses everything the winner would have returned, which has no ceiling. In this fund the one miss costs Rs 528 crore, about 3.5 times every write-off combined. This asymmetry is why venture investors say the error that matters is the company they passed on, not the one that failed.
What do you add after the number?
Say what it implies for how a fund behaves. A partner who rejects a deal because it might fail is guarding against the smaller of the two errors. The right question at the investment committee is whether a company could return the fund if it works. The limit is worth one sentence too: this is one fund's numbers, and a portfolio with a flatter spread of outcomes would care less about any single company.
Where candidates lose it
The instinctive answer is that one company in thirty moves the fund by about a thirtieth, so 2.9x. That treats venture outcomes as if they were spread evenly, which is exactly the assumption the interviewer wants you to drop.
The second loss is doing the arithmetic but not the comparison. The point of the question is that one missed winner costs more than every failure together; say that sentence, with the Rs 150 crore beside the Rs 528 crore.
What the interviewer asks next
How many companies earning 1.24x would you need to make up for missing the winner?
If write-offs cost at most the cheque, why do funds still care about their loss ratio?
What does this imply for how a seed fund should size its follow-on reserves?
003A company has 18 months of cash and starts raising its next round once 9 months have passed. Each month of raising it has an independent 10% chance of closing. What is the chance it closes before the cash runs out, and by which month must it start raising for an 80% chance?Probability and expected valueCoreSeed and early-stage VCSeries A to C VC
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With 9 months of raising at 10% a month, what is the chance of closing?
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About 61%, and for an 80% chance it must start raising after month 2. The chance of failing all nine months is 0.9 to the power 9, about 39%, so it closes about 61% of the time. An 80% chance needs 0.9 to the power n below 20%, which first happens at 16 months (81.5%). Eighteen months of cash less 16 means raising from month 2.
Why not just add 10% a month?
Think of a friend who picks up the phone one time in ten. Call nine times and you will not reach them 90% of the time, because every call after the one they answer is wasted. With repeated independent tries, count the chance of failing every time and subtract it from one; adding the chances double counts the months after a close. Adding 10% a month would give 100% at ten months and more than 100% after that, which is the tell that the method is wrong.
The relationship
$P(\text{close within } n) = 1 - 0.9^{n} \qquad 1 - 0.9^{9} = 0.613$
0.9 the chance of not closing in any one month
n months of raising before the cash runs out
What it says in wordsThe round closes unless every single month fails, so subtract the chance of all-failure from one.
At a 10% chance each month, nine months of raising gives a 61% chance of closing, and the curve first clears 80% at sixteen months, 81.5%, while simply adding 10% a month would wrongly reach certainty at ten.
How do you find the month it must start?
Set 0.9 to the power n below 0.2 and solve. Taking logs, n is at least ln 0.2 over ln 0.9, about 15.3, so round up to 16 whole months. Fifteen months gives 79.4%, just short of 80%, and sixteen gives 81.5%. With 18 months of cash, sixteen months of raising means starting after month 2. Doubling the time spent raising only lifts the odds from 61% to about 81%, because each extra month adds 10% of a shrinking remainder.
What would you say to a founder about this?
The model is simple and its lesson is not: fundraising odds climb slowly, so a company that waits until half its runway is gone has already given up a large share of its chances. The limitation is worth naming. Real months are not independent; a round that has not closed in six months usually gets harder, not equally likely, so the true curve is flatter still.
Where candidates lose it
The fast answer is 90%, nine months at 10% each. It is wrong in a way the interviewer can prove in one line: at eleven months the same method gives 110%.
The second loss is rounding the 80% month down. 15.3 months must become 16, and fifteen gives 79.4%, which misses the target.
What the interviewer asks next
If the monthly chance falls by one point every month the company has been raising, how does the answer change?
What is the expected number of months to close, starting from day one?
How much extra runway, in months, buys the last 10 points of probability from 80% to 90%?
008You expect a company to exit at Rs 1,000 crore in six years and you need a 10x return on your cheque. Later rounds will dilute your stake by 40% before the exit. What is the highest post-money valuation you can pay today, and what ownership does a Rs 12 crore cheque need?Valuation riddlesCoreSeed and early-stage VCIndia VC
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What is the most you can pay, post-money?
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A post-money of Rs 60 crore, and a Rs 12 crore cheque needs 20% of the company. Rs 1,000 crore divided by 10 is Rs 100 crore, but later rounds leave you with only 60% of your stake, so the price falls to Rs 60 crore. Rs 12 crore buys 20% at that price, which dilutes to 12% by the exit and is worth Rs 120 crore, exactly 10x.
Why work backwards from the exit?
A shop owner who knows a sari will sell for Rs 10,000 and needs to double her money can pay the weaver at most Rs 5,000. She starts at the selling price and divides. A venture investor does the same: start at the exit value, divide by the multiple you need, and what is left is the most the company can be worth when you buy in. Here that is Rs 1,000 crore over 10, Rs 100 crore, before you account for what later rounds do to your stake. This is often called the venture capital method.
Where does the dilution go in the chain?
Later rounds do not change the exit value; they shrink the share of it you own. If you hold 20% today and keep 60% of it, you own 12% at the exit. Because you only collect 60% of the stake you buy, the company must be priced at 60% of the undiluted figure for the maths to work, Rs 60 crore post-money rather than Rs 100 crore. Rs 12 crore at Rs 60 crore post is 20%, and the pre-money is Rs 48 crore.
The relationship
$\text{Post} = \frac{V_{exit} \times (1-d)}{M} = \frac{1{,}000 \times 0.60}{10} = 60 \qquad \text{own} = \frac{12}{60} = 20\%$
V_exit the expected exit value, Rs crore
d dilution from later rounds, 40%
M the return multiple you need
own the stake the cheque must buy today
What it says in wordsThe price you can pay today is the exit value you will actually own a share of, divided by the multiple you need.
Rs 1,000 crore at exit divided by 10x and multiplied by the 60% you keep gives a Rs 60 crore post-money, where Rs 12 crore buys 20%, dilutes to 12% and returns Rs 120 crore; paying Rs 100 crore instead buys 12%, dilutes to 7.2% and returns only Rs 72 crore, 6x.
What do you add to show judgement?
Translate the multiple into a yearly rate and question the inputs. 10x over six years is about 47% a year, and the whole answer leans on a single exit value that is far from certain. If only one company in three reaches that exit, the honest calculation uses an expected exit, and the price you can pay falls by the same factor. Saying that sentence turns a formula into an investment view.
Where candidates lose it
The trap is stopping at Rs 100 crore. It forgets that the investor's 20% will not still be 20% at the exit, and a candidate who pays that price earns 6x, not 10x.
The second slip is applying the dilution backwards, dividing by 0.60 to get Rs 166.7 crore. Ask yourself whether dilution should make you willing to pay more or less; the answer is less.
What the interviewer asks next
If the company reaches the Rs 1,000 crore exit only one time in three, what post-money can you pay?
You negotiate pro rata rights and keep your stake by investing in later rounds. How does that change the price today?
Why might a seed investor accept a lower target multiple than 10x?
010Your investment committee meets in 45 minutes and you have no watch. You have two fuses and a lighter. Each fuse burns for exactly one hour, but unevenly, so half a fuse does not take half an hour. How do you measure exactly 45 minutes?Logic and brainteasersCoreSeed and early-stage VCMulti-stage VC
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What can you know for certain about a fuse lit at both ends?
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Light fuse A at both ends and fuse B at one end, at the same moment. Fuse A burns out after exactly 30 minutes, because two flames use up its hour twice as fast. At that instant fuse B has 30 minutes of burning left, so light its other end too. It goes out 15 minutes later, at exactly 45 minutes.
Why does lighting both ends give exactly 30 minutes?
Think of two people eating from opposite ends of a plate of biryani that one person would finish in an hour. One may eat faster on the rice and slower on the meat, but together they always clear the plate in half the time. A fuse holds a fixed amount of burning time, 60 minutes, and two flames consume it at twice the rate of one, so they meet after 30 minutes wherever along the fuse that happens to be. The unevenness only moves the meeting point.
The relationship
$t_1 + t_2 = 60,\; t_1 = t_2 \;\Rightarrow\; t = 30 \qquad \text{then } \tfrac{30}{2} = 15$
t_1 burn time of the stretch the left flame consumes
t_2 burn time of the stretch the right flame consumes
60 the whole fuse's burn time in minutes
What it says in wordsBoth flames burn for the same clock time and together use up all 60 minutes, so each burns for 30; a second flame on a 30-minute remainder halves it to 15.
Fuse A, lit at both ends, goes out at minute 30; at that moment fuse B, lit at one end, has 30 minutes of burning left, and lighting its second end makes it go out 15 minutes later, at exactly minute 45.
Why light fuse B at the start rather than later?
Fuse A is your clock for the first 30 minutes, and fuse B needs to spend those same 30 minutes burning so that exactly half its burn time is left. The trick is that fuse B stores time: after 30 minutes on one flame it holds exactly 30 minutes of burn, whatever its length, and a second flame halves that to 15. Lighting B only when A goes out would leave it with a full hour, and you could only measure 60 or 90 minutes from there.
What is the interviewer actually listening for?
Whether you separate what you know from what you do not. You do not know the burn rate along the fuse, so any plan that cuts or measures it by length is dead. You do know the total time, and the answer uses nothing else. That habit carries over to diligence: when a company's monthly numbers are noisy, lean on the totals you can verify rather than on a rate you are guessing.
Where candidates lose it
The first instinct is to fold or cut a fuse in half. The question rules that out by saying the burn is uneven, and candidates who reach for length show they did not hear the constraint.
The second loss is lighting fuse B only when fuse A goes out. Say the timing clearly: both fuses are lit at minute zero, A at both ends and B at one.
What the interviewer asks next
With one fuse only, which times can you measure?
Can you measure 15 minutes on its own with the same two fuses?
With three such fuses, can you measure 52.5 minutes?
011In your head, no calculator: a SaaS company has ARR of Rs 42 crore growing 65% a year, net burn of Rs 3.1 crore a month, net new ARR last quarter of Rs 6.2 crore and Rs 55 crore of cash. What are next year's ARR, the quarterly burn multiple and the runway?Mental maths and speed testsCoreVista Equity PartnersAustin · 2021
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What is the quarterly burn multiple?
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About Rs 69.3 crore of ARR, a burn multiple of 1.5 and about 18 months of runway. 42 x 1.65 is 42 plus 21 plus 6.3. A quarter's burn is 9.3, and 9.3 over 6.2 is 1.5 because both are multiples of 3.1. Cash of 55 over 3.1 a month is a shade under 55 over 3, so about 17.7 months.
How do you multiply by 1.65 without a calculator?
Break the awkward multiplier into pieces you already know. A shopkeeper adding 65% to a Rs 42 cost does not multiply by 0.65; he adds half, then a bit more. 65% is 50% plus 15%, so 42 x 1.65 is 42 plus 21 plus 6.3, which is Rs 69.3 crore. Fifteen percent is itself 10% plus half of that again: 4.2 plus 2.1. Every step is a halving or a shift of the decimal point, which is fast and hard to get wrong out loud.
Why does the burn multiple need care before any arithmetic?
Because the two inputs come in different periods. Burn is quoted per month and net new ARR per quarter. A burn multiple divides the cash burned by the new annual recurring revenue added over the same stretch of time, so three months of burn, Rs 9.3 crore, goes over the quarter's Rs 6.2 crore. Then notice that 9.3 is 3 x 3.1 and 6.2 is 2 x 3.1, so the ratio is exactly 3 over 2, or 1.5. Interviewers choose numbers like these on purpose; spotting the common factor is part of the test.
Splitting 1.65 into 1 plus a half plus 15% gives Rs 69.3 crore of ARR, matching periods turns the burn multiple into 9.3 over 6.2, which is 1.5, and rounding 3.1 down to 3 then shaving 3% gives about 17.7 months of runway.
The relationship
$\text{BM} = \frac{3 \times 3.1}{6.2} = 1.5 \qquad \text{Runway} = \frac{55}{3.1} \approx 17.7 \text{ months}$
3 x 3.1 net burn over one quarter, Rs crore
6.2 net new ARR added in the quarter, Rs crore
55 cash in the bank, Rs crore
What it says in wordsPut burn and new revenue on the same period before dividing, and divide cash by monthly burn for runway.
What do you say once the three numbers are out?
Check them against each other, because that is what a growth investor does next. Growing ARR by Rs 27.3 crore next year needs about Rs 6.8 crore of net new ARR a quarter, a little above the Rs 6.2 crore just achieved, so the 65% plan is plausible but not banked. At a 1.5 burn multiple that growth costs about Rs 41 crore of burn in the year, and with Rs 55 crore of cash and about 18 months of runway, the company will want to raise again within roughly a year, before the runway gets short. That one sentence turns arithmetic into a view on the company.
Where candidates lose it
The costly slip is dividing the monthly burn by the quarterly net new ARR and announcing a burn multiple of 0.5. It makes the company look three times more efficient than it is, and the interviewer chose mixed periods to see whether you notice.
The second loss is going silent while you calculate. Say the shortcut as you use it: half, then fifteen percent; three parts over two parts; fifty five over three, then shave a little.
What the interviewer asks next
If net burn rises 20% next year while ARR grows 65%, what happens to the burn multiple if net new ARR grows in step with ARR?
How much cash should the company raise to have 24 months of runway at the current burn?
Is a burn multiple of 1.5 good for a company of this size, and what would change your view?
Asked at Vista Equity Partners, Private Equity, Austin, 2021 (Wall Street Oasis): Mental math and tech/SaaS-specific sector insights
013A seed fund makes 25 equal bets. Each bet independently has a 4% chance of returning 50x and otherwise returns 0.5x. What is the chance the fund finds no 50x winner at all, and what is its expected multiple?Power law and portfolio mathsCoreSeed and early-stage VCMulti-stage VC
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What is the chance the fund ends with no 50x winner?
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About 36% of such funds find no winner, even though the expected multiple is 2.48x. Each bet misses with probability 0.96, so all 25 miss with probability 0.96 to the power 25, which is 0.360. Each bet is worth 0.04 x 50 plus 0.96 x 0.5 on average, 2.48x. The average hides a wide spread: a fund with no winner returns 0.5x.
Why is one winner not guaranteed when 25 times 4% is 100%?
A batsman who hits a six once every 25 balls on average can still face 25 balls and hit none. Twenty five times 4% gives the expected number of winners, one, not the chance of getting at least one. To find that chance, count the way the fund fails: every bet must miss, and since the bets are independent the misses multiply, 0.96 x 0.96 and so on, 25 times. That is 0.360, so about 36% of identical funds never see a 50x company.
The relationship
$P(\text{no winner}) = 0.96^{25} \approx 0.360 \qquad E[M] = 0.04 \times 50 + 0.96 \times 0.5 = 2.48$
0.96 the chance any one bet is not a 50x winner
25 the number of independent bets
E[M] the expected multiple of each bet, and so of the fund
What it says in wordsMultiply the miss chances for the no-winner case, and average the two outcomes for the expected multiple.
Across 25 independent bets at 4%, 36.0% of funds find no winner and return 0.5x, 37.5% find exactly one and return 2.48x, and 18.8% find two and return 4.46x, even though every fund has the same expected multiple of 2.48x.
What does the spread tell you about running a seed fund?
Each extra winner adds 49.5x on one twenty-fifth of the fund, about 1.98 turns of the whole fund. So the fund's result is set almost entirely by how many winners it happens to land: none gives 0.5x, one gives 2.48x and two give 4.46x. The most likely single outcome is exactly one winner, at 37.5%, but no winner at all is nearly as likely. This is why seed managers want more shots on goal: with 57 bets the chance of finding no winner falls below 10%.
Say the limitation too. Real outcomes are not two-point, and they are not independent: a funding drought hurts every company in a vintage at once, which fattens the no-winner tail rather than thinning it.
Where candidates lose it
The fast wrong answer is that a winner is certain because 25 times 4% is 100%. The interviewer wants to hear you separate an expected count from a probability.
The second loss is giving 2.48x and stopping. The point of the question is the gap between the average and the typical fund; say that more than a third of funds lose half their money while the average fund makes 2.48x.
What the interviewer asks next
How many bets would the fund need to cut the no-winner chance below 10%?
If the outcomes are correlated, does the no-winner chance go up or down?
What is the chance the fund returns at least 2x?
016A company has 1 crore shares outstanding and 20 lakh vested options with a strike price of Rs 50. The shares are worth Rs 200 each. How many shares does the treasury stock method count as fully diluted, and how many does counting every option as a share give?Dilution and ownership riddlesCoreGrowth equitySeries A to C VC
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How many new shares do the options add under the treasury method?
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The treasury method gives 1.15 crore shares; counting every option gives 1.20 crore. Exercising 20 lakh options at Rs 50 brings in Rs 10 crore, and at Rs 200 a share that cash buys back 5 lakh shares. So the options add a net 15 lakh, not 20 lakh. Counting all 20 lakh ignores the strike money the option holders pay.
Why does the strike price reduce the dilution?
Suppose five friends own a flat together and a sixth may join by paying Rs 50 lakh towards a share worth Rs 2 crore. The newcomer dilutes the others, but the Rs 50 lakh he brings in offsets a quarter of that. An option holder pays the strike to get a share, and that cash is worth something to the existing owners, so only the part of the share's value not covered by the strike is true dilution. Here each option pays Rs 50 for a Rs 200 share, so three quarters of each option is real dilution, and 20 lakh x 0.75 is 15 lakh.
How does the treasury method count it?
It pretends every in-the-money option is exercised and the company spends the strike cash buying its own shares at today's price. Rs 10 crore of strike cash at Rs 200 a share buys back 5 lakh shares, so the share count rises by 20 lakh and falls by 5 lakh, ending at 115 lakh, or 1.15 crore. Counting every option as a full share gives 1.20 crore, which overstates the dilution by the 5 lakh shares the strike money pays for.
The relationship
$\text{FD} = 100 + 20 - \frac{20 \times 50}{200} = 100 + 20 - 5 = 115 \text{ lakh}$
100 basic shares outstanding, lakh
20 vested options, lakh
50 strike price, Rs
200 share price, Rs
What it says in wordsAdd every option, then take off the shares the strike money could buy back at the current price.
Counting every option as a share gives 120 lakh shares, but the Rs 10 crore paid on exercise buys back 5 lakh at Rs 200, so the treasury method's fully diluted count is 115 lakh.
How do you check that the treasury count is right?
Price the company both ways. At Rs 200 a share, 115 lakh shares is an equity value of Rs 230 crore. A buyer paying Rs 200 for all 120 lakh shares would pay Rs 240 crore but collect Rs 10 crore of strike money back, a net Rs 230 crore. The two routes agree, which is the proof that the treasury method counts exactly what the options are worth, not more. Counting all options without the cash would value the company Rs 10 crore too high.
Where candidates lose it
The quick wrong answer is 1.2 crore shares, counting every option as a share. It forgets that option holders pay Rs 50 each, and the interviewer will ask where that Rs 10 crore went.
The other slip is ignoring the options entirely because they are not yet exercised. Vested in-the-money options are a claim on the company today; say they are counted and show the buyback.
What the interviewer asks next
If the share price falls to Rs 40, how many shares do the options add?
Should unvested options be counted in a fully diluted share count for an acquisition?
How does a convertible note enter a fully diluted count?
018A wedding-services startup asks you to size its market. Estimate the number of weddings in India each year, stating the population, age and marriage-rate assumptions you use.Market sizing and estimationCoreConsumer internet VCIndia VC
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Which number should the estimate be built on?
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About 1 crore weddings a year, within a range of roughly 85 lakh to 1.14 crore. Assume about 140 crore people spread over about 70 years of age, so about 2 crore per year of age, nudged up to 2.2 crore because India skews young. If 90% of them marry, 1.98 crore people marry each year, and two people per wedding gives about 99 lakh weddings.
Why start from one age cohort?
A school counting how many students graduate each year does not start from every student on its rolls; it counts one class, because each class graduates once. Weddings are a yearly flow, and each person typically marries once, so the yearly number of weddings is set by how many people reach marriage age in a year, not by the size of the population. If 140 crore people are spread across roughly 70 years of age, each single year of age holds about 2 crore people. India's population skews young, so the cohorts now reaching their twenties are larger than that average; 2.2 crore is a reasonable working figure.
Starting from about 140 crore people, one year of age holds roughly 2.2 crore, about 90% of them marry, and two people make each wedding, giving about 99 lakh weddings a year, while multiplying the whole population by the marriage share wrongly gives 63 crore.
How confident can you be, and how do you show it?
Give a range by moving the two softest assumptions. With a cohort between 2.0 and 2.4 crore and between 85% and 95% of people marrying, the answer runs from about 85 lakh to 1.14 crore weddings a year, so 1 crore is a sound central figure. The population input is an assumption to confirm against the latest census or official estimate. Remarriages and people who marry abroad move the answer by a few percent at most.
Cohort reaching marriage age 85% marry 90% marry 95% marry
2.0 crore 85.0 lakh 90.0 lakh 95.0 lakh
2.2 crore 93.5 lakh 99.0 lakh 104.5 lakh
2.4 crore 102.0 lakh 108.0 lakh 114.0 lakh
Weddings a year under each pair of assumptions, in lakh: the answer stays between 85 lakh and 1.14 crore.
What does an investor do with the number?
Narrow it to the startup's real market. A wedding-services company does not serve every wedding; it serves the urban weddings with a budget above some level, in cities where it operates. Say which cut you would apply next, for example the share of weddings in the top cities and above a spend threshold, and what you would multiply by to reach a rupee market size. That shows you know the count is the first step of a market sizing, not the answer.
Where candidates lose it
The common error is multiplying the whole population by the share who marry, which gives a lifetime stock of weddings rather than a yearly flow. The answer of tens of crores is absurd, and the interviewer will wait to see if you notice.
The second loss is forgetting that a wedding needs two people, which doubles the answer. Say the divide-by-two step out loud.
What the interviewer asks next
How would you turn the wedding count into a rupee market size for a photography startup?
What share of these weddings would you expect a city-focused startup to reach in five years?
How would you check the answer a second way, without the age cohort?
019A portfolio company says its revenue doubles every 9 months. What is that as an annual growth rate, and as a quarterly one?Growth and compoundingCoreSeed and early-stage VCConsumer internet VC
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What annual growth rate does doubling every 9 months imply?
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About 152% a year and 26% a quarter. A year holds 12/9, or 4/3, doubling periods, so revenue grows by 2 to the power 4/3, which is 2.52x. A quarter is a third of a doubling period, so it grows by the cube root of 2, 1.26x. Scaling the 100% in a straight line, 133% a year, understates the compounding.
Why is the answer not 100% times 12/9?
Picture a savings account that doubles every 9 months. In the three months after the first doubling, it grows on twice the original balance. Doubling compounds, so a fraction of a doubling period multiplies the amount by 2 raised to that fraction, not by 1 plus that fraction of 100%. A year is 4/3 of a doubling period, so revenue grows by 2 to the power 4/3. Since 2 to the power 1/3 is about 1.26, that is 2 x 1.26, about 2.52, or 152% growth.
The relationship
$g_{year} = 2^{12/9} - 1 \approx 152\% \qquad g_{quarter} = 2^{3/9} - 1 \approx 26\%$
12/9 doubling periods in a year
3/9 doubling periods in a quarter
2 the growth factor per doubling period
What it says in wordsRaise 2 to the number of doubling periods in the time window, then subtract one.
Doubling every 9 months reaches 2x at month 9 and 2.52x at month 12, so each year multiplies revenue by 2.52, 152% growth, and three years compound to 16x.
How do you get the cube root of 2 in your head?
Remember one anchor: 1.26 cubed is very close to 2, because 1.26 squared is about 1.59 and 1.59 x 1.26 is about 2.0. So a quarter, one third of a doubling period, grows by about 26%, and four quarters give 1.26 to the power 4, about 2.52, which checks the annual figure. The same anchor gives the monthly rate: 2 to the power 1/9 is about 1.080, roughly 8% a month.
What do you say about the founder's claim?
Translate it into something you can test. Doubling every 9 months means revenue 16 times larger in three years; ask what has to be true about the market and the team for that, and how long it has lasted. Founders often quote the doubling period from their best stretch. Ask for the doubling time over the last 24 months and over the last 6; if the recent one is slower, the growth is decaying, which is normal but changes the valuation.
Where candidates lose it
The trap is 133%, scaling 100% by 12/9 as if growth were a straight line. The interviewer is checking whether you compound fractional periods.
The quieter loss is getting the annual number right and then dividing it by four for the quarter, 38%. A quarter is a third of a doubling period, so it is the cube root of 2, about 26%.
What the interviewer asks next
What doubling time corresponds to 100% annual growth?
If growth slows so that the doubling time lengthens by 3 months every year, what is revenue after three years?
How would you check a founder's doubling claim from monthly revenue data?
020A company with no debt and no cash trades at 8x revenue and 40x earnings. Depreciation is 5% of revenue, there is no interest, and the tax rate is 25%. What are its net margin and its EBITDA margin?Valuation riddlesCoreGrowth equitySaaS-focused VC
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What is the net margin?
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The net margin is 20% and the EBITDA margin is about 31.7%. With no debt or cash, enterprise and equity value are the same, so 8x revenue equals 40x earnings and earnings are 8/40 of revenue. Grossing up 20% for 25% tax gives a pre-tax and EBIT margin of 26.7%, and adding back 5% of depreciation gives 31.7%.
How can two multiples give a margin?
If a flat sells for 25 times its yearly rent and 5 times the owner's yearly income from all sources, rent must be one fifth of that income, without knowing the price. When two multiples price the same value, dividing one by the other gives the ratio of their denominators. Here 8x revenue and 40x earnings price the same equity, because a company with no debt and no cash has enterprise value equal to equity value. So earnings over revenue is 8 over 40, a 20% net margin.
The relationship
$\frac{E}{R} = \frac{8}{40} = 20\% \qquad \text{EBITDA} = \frac{20\%}{1 - 0.25} + 5\% = 31.7\%$
E/R earnings over revenue, the net margin
8, 40 value over revenue and value over earnings
0.25 the tax rate
5% depreciation as a share of revenue
What it says in wordsDivide the revenue multiple by the earnings multiple for the net margin, then gross up for tax and add back depreciation.
On Rs 100 of revenue the company is worth Rs 800, so at 40x its earnings are Rs 20; grossing that up for 25% tax gives EBIT of Rs 26.7, and adding back Rs 5 of depreciation gives EBITDA of Rs 31.7, a 31.7% margin.
How do you climb from net profit back to EBITDA?
Undo each deduction in reverse order. Tax is 25% of pre-tax profit, so net profit is 75% of it, and pre-tax profit is 20% divided by 0.75, about 26.7% of revenue. With no interest, pre-tax profit equals EBIT. Adding back depreciation of 5% gives EBITDA of 31.7% of revenue. A common slip is grossing up by 1.25 instead of dividing by 0.75, which gives 25% instead of 26.7%.
Does the answer make sense for a company priced like this?
Check with a third multiple. The EV/EBITDA is 8 divided by 0.317, about 25.3x. A 20% net margin with a 40x earnings multiple describes a profitable business that the market expects to keep growing quickly, which is the profile of a mature software company rather than an early startup. Say the assumption that made the trick work, no debt and no cash; with net debt, EV and equity value differ and you need the balance sheet to link them.
Where candidates lose it
Candidates reach for a share price or revenue figure that the question never gave. The insight is that the two multiples share a numerator, so their ratio is the margin; say that first.
The second loss is the gross-up. Net profit is 75% of pre-tax profit, so divide by 0.75; multiplying by 1.25 understates EBIT and EBITDA.
What the interviewer asks next
If the company had net debt equal to 1x revenue, what would the net margin be?
What EV/EBITDA multiple does this company trade at?
If the P/E fell to 30x with the revenue multiple unchanged, what does that imply?

900	what the fund actually returned
540	what the best company returned
360/29	one more company earning the average of the other 29
300	the fund's committed capital

0.9	the chance of not closing in any one month
n	months of raising before the cash runs out

V_exit	the expected exit value, Rs crore
d	dilution from later rounds, 40%
M	the return multiple you need
own	the stake the cheque must buy today

t_1	burn time of the stretch the left flame consumes
t_2	burn time of the stretch the right flame consumes
60	the whole fuse's burn time in minutes

Try it first

Why does one company out of thirty move the whole fund?

How does that compare with the cost of the failures?

What do you add after the number?

Where candidates lose it

What the interviewer asks next

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Why not just add 10% a month?

How do you find the month it must start?

What would you say to a founder about this?

Where candidates lose it

What the interviewer asks next

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Why work backwards from the exit?

Where does the dilution go in the chain?

What do you add to show judgement?

Where candidates lose it

What the interviewer asks next

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Why does lighting both ends give exactly 30 minutes?

Why light fuse B at the start rather than later?

What is the interviewer actually listening for?

Where candidates lose it

What the interviewer asks next

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How do you multiply by 1.65 without a calculator?

Why does the burn multiple need care before any arithmetic?

What do you say once the three numbers are out?

Where candidates lose it

What the interviewer asks next

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Why is one winner not guaranteed when 25 times 4% is 100%?

What does the spread tell you about running a seed fund?

Where candidates lose it

What the interviewer asks next

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Why does the strike price reduce the dilution?

How does the treasury method count it?

How do you check that the treasury count is right?

Where candidates lose it

What the interviewer asks next

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Why start from one age cohort?

How confident can you be, and how do you show it?

What does an investor do with the number?

Where candidates lose it

What the interviewer asks next

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Why is the answer not 100% times 12/9?

How do you get the cube root of 2 in your head?

What do you say about the founder's claim?

Where candidates lose it

What the interviewer asks next

Try it first

How can two multiples give a margin?

How do you climb from net profit back to EBITDA?

Does the answer make sense for a company priced like this?

Where candidates lose it

What the interviewer asks next