Venture Capital puzzles, solved step by step

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.

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.

A proposes A Rs 98 crore, B nothing, C Rs 1 crore, D nothing and E Rs 1 crore. A needs three of five votes, its own and two more. If A were removed, B's winning plan would give C and E nothing, so Rs 1 crore each buys their votes. Working backwards from two partners up gives every step of the chain.

Where do you start a problem like this?

At the end, where the answer is obvious. A chess player thinks about the final position and works back to the move in front of her. Backward induction solves the smallest game first, then uses its answer as each partner's fallback in the next larger game. With only D and E left, D proposes Rs 100 crore for itself; its own vote is one of two, which is half, so it passes. E gets nothing, and everyone further up the table knows it.

How does each proposer buy votes as the table grows?

With three left, C needs two votes. E gets nothing if C is removed, so Rs 1 crore buys E: C 99, D 0, E 1. With four left, B needs two votes and D is the partner who would get nothing in C's plan, so Rs 1 crore buys D: B 99, C 0, D 1, E 0. Each proposer buys exactly the votes it needs from the partners whose fallback is lowest, paying one crore more than that fallback. With five left, A needs three votes; C and E get nothing in B's plan, so A pays them Rs 1 crore each and keeps Rs 98 crore.

What assumption is doing the work, and what does the puzzle say about real funds?

The tie rule matters. If an indifferent partner voted yes, A could offer C, D and E nothing and keep all Rs 100 crore, because they get nothing anyway. Stating how indifferent players vote is the step that separates a full answer from a lucky one. The real-world point is that self-interest plus a voting rule rewards whoever controls the agenda, which is why carry splits at real firms are set in writing, with vesting, rather than left to a vote among partners.

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.

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.

16 months. The customer pays Rs 10,000 a month, but at a 75% gross margin only Rs 7,500 of that is left after the cost of serving them. That gross profit is what repays the Rs 1.2 lakh spent to win the customer, and 1,20,000 divided by 7,500 is 16 months. Dividing by revenue instead gives 12 months, which flatters the business by a third.

Why does payback use gross profit and not revenue?

A tea stall that spends Rs 1,200 on a signboard has not earned it back when it has sold Rs 1,200 of tea, because the milk, sugar and gas cost money too. CAC is repaid only by what the customer leaves behind after the cost of serving them, which is gross profit, not revenue. Here the cost of hosting, support and payment processing takes a quarter of each month's Rs 10,000, leaving Rs 7,500 to pay back the Rs 1.2 lakh.

What does 16 months tell an investor?

It says how long the company's cash is tied up in each new customer. A company that grows fast with a 16-month payback consumes cash for well over a year on every customer it adds, so faster growth means a bigger cash need, not a smaller one. Many SaaS investors treat payback under about 12 to 18 months as healthy for mid-market customers, though the benchmark moves with customer size and churn. If this customer stays four years, it leaves Rs 3.6 lakh of gross profit, three times its acquisition cost.

Name the limitation as well. The simple payback assumes the customer never leaves. If 2% of customers churn each month, the average customer takes about 19 months to repay its CAC, because some leave before they have paid back.

Committing now earns an expected Rs 2 crore; staging earns Rs 7.6 crore, so the option to stop is worth Rs 5.6 crore. Committing pays Rs 100 crore only 12% of the time, worth Rs 12 crore, against Rs 10 crore invested. Staging risks Rs 2 crore first and adds Rs 8 crore only after the milestone, so in the 70% of cases that fail it saves Rs 8 crore: 0.7 x 8 is Rs 5.6 crore.

Why is waiting worth anything if the payout is the same?

Think of booking a wedding venue with a small refundable deposit rather than paying in full a year out. If the engagement is called off, you lose the deposit, not the whole fee. Staging a cheque does not change what you win; it changes what you lose in the worlds where things go wrong, because you stop paying once you learn the milestone was missed. Here the milestone fails 70% of the time, and each of those times the staged route has spent Rs 2 crore instead of Rs 10 crore.

How do you work each route's expected profit?

Committing now: the company pays Rs 100 crore only if it passes both hurdles, 0.3 x 0.4, which is 12% of the time. That is worth Rs 12 crore against Rs 10 crore paid, an expected profit of Rs 2 crore. Staging: pay Rs 2 crore for sure, then in the 30% of worlds that hit the milestone, pay Rs 8 crore for a 40% shot at Rs 100 crore, which is worth Rs 32 crore at that point. Minus 2, plus 0.3 x 32, gives Rs 7.6 crore.

What would a founder say about this, and what is the limit?

A founder will rarely give you the same price for both tranches, because staging moves risk onto the company. The Rs 5.6 crore is the most it is worth paying for the right to stage, in a higher second-tranche price or a smaller stake. Staging also has costs the model ignores: a company that has to hit a milestone to get its money may be run for the milestone rather than for the business, and the uncertainty can scare off other investors.

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.

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.

Carry is Rs 120 crore, and the fund is 3.0x gross and 2.6x net. The profit is Rs 900 crore less the Rs 300 crore of capital returned, Rs 600 crore. The GP takes 20% of that, Rs 120 crore. LPs receive Rs 780 crore on Rs 300 crore, which is 2.6x, against 3.0x for the portfolio before carry.

What is the carry charged on?

Think of a tutor paid a fifth of whatever a student's marks improve by, not a fifth of the total marks. Carried interest is a share of the profit, so the LPs' own capital comes back to them first and only the gain above it is split. The fund returned Rs 900 crore on Rs 300 crore, so the gain is Rs 600 crore and the GP's 20% is Rs 120 crore. A hurdleA minimum return LPs must receive before the GP earns any carry. This fund has none. would delay the carry but, once caught up, would not change its size at this level of return.

Why is net 2.6x and not 2.4x?

Taking 20% off 3.0x would charge carry on the capital as well as the profit. The gap between gross and net is 20% of the profit multiple, 0.2 x 2.0, which is 0.4 turns, so 3.0x gross becomes 2.6x net. The same logic gives a quick rule: at any gross multiple M with 20% carry and no fees, net is 1 plus 0.8 x (M minus 1). A 2.0x gross fund nets 1.8x; a 5.0x fund nets 4.2x.

What would an LP add?

That fees push net lower still. A typical venture fund also charges an annual management fee on committed capital, often around 2% for the investment period, which reduces both the money invested and the net multiple; confirm the actual terms in the fund's agreement. An LP compares managers on net multiples and net IRR, because that is the only number that reaches its own balance sheet. A GP quoting 3.0x is quoting the portfolio, not the investor's outcome.

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.

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.

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.

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.

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.

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.

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.