Venture Capital puzzles, solved step by step

Contribution is Rs 90 crore, a 75% margin on revenue, and the company is valued at 8.3x revenue and 11.1x contribution. A 12% take rate on Rs 1,000 crore of GMV is Rs 120 crore of revenue. Payment costs of Rs 20 crore and refunds of Rs 10 crore leave Rs 90 crore. The 1x GMV headline is the same price as 8.3x revenue.

Why is GMV not the company's revenue?

A property broker who helps sell a Rs 1 crore flat does not earn Rs 1 crore; she earns her commission. GMVGross merchandise value: the total value of goods or services sold through a marketplace, before the platform takes its cut. is the value flowing through the platform, and only the take rate, here 12%, is the platform's revenue. On Rs 1,000 crore of GMV that is Rs 120 crore. The other Rs 880 crore belongs to sellers, so a multiple of GMV prices something the company never keeps.

What is left after the costs that scale with every order?

Payment processing costs 2% of GMV, Rs 20 crore, and refunds the platform absorbs cost 1%, Rs 10 crore. Both are charged on GMV but paid out of revenue, so a 3% cost on GMV eats a quarter of a 12% take rate. Contribution is Rs 90 crore: 75% of revenue, but only 9% of GMV. This is the money available to pay for marketing, salaries and offices, and eventually profit.

What would you say about the valuation?

That the headline multiple flatters the price, and the real question is what the contribution can grow into. Paying 11.1x contribution is reasonable only if contribution grows fast or the take rate can rise without losing sellers. Two marketplaces at 1x GMV can be wildly different prices: one with a 25% take rate is valued at 4x revenue, one with a 5% take rate at 20x. Always convert GMV multiples into revenue and contribution before comparing companies.

About 45%, below a coin flip. Exactly two pilots convert in three ways: pilots 1 and 2 only, 0.6 x 0.5 x 0.7 = 0.21; pilots 1 and 3 only, 0.6 x 0.5 x 0.3 = 0.09; pilots 2 and 3 only, 0.4 x 0.5 x 0.3 = 0.06. All three convert with probability 0.09. Adding the four gives 0.45.

How do you make sure you count every way to survive?

Think of a cricket team that needs two of its three openers to score fifty. You would list who scores and who does not, because each opener's failure matters to the case where the other two succeed. List the outcomes that satisfy the condition, write each as a product of the chance each pilot converts or does not, and add them. At least two out of three means exactly two, which can happen three ways, or all three.

Why do the quick shortcuts fail?

Multiplying the two best pilots, 0.6 x 0.5, gives 30% and forgets that pilots 1 and 3, or 2 and 3, also keep the company alive. Adding the three pair products, 0.30 + 0.18 + 0.15, gives 63% and makes the opposite error. Each pair product already includes the case where the third pilot converts too, so the all-three outcome is counted three times and must be taken off twice: 0.63 minus 2 x 0.09 is 0.45. That correction gives the same answer as the list, which is a useful check in the room.

What would you tell the investment committee?

That three pilots which each sound promising still leave the company more likely to fail than survive. The answer also depends heavily on independence: if the pilots share a buyer type or a product gap, they tend to succeed or fail together, and the real survival chance could be noticeably higher or lower. Ask what the pilots have in common before trusting the 45%.

The price is Rs 434.78 a share and the investor gets 2,87,500 new shares. The Rs 50 crore pre-money is divided across 11.5 lakh shares, the 10 lakh existing plus the 1.5 lakh pool, so each is worth Rs 434.78. Rs 12.5 crore buys 2,87,500 shares, 20% of the 14,37,500 after the round. With the pool left out of the pre-money the price would be Rs 500.

Why does the pool change the price but not the headline valuation?

Picture four friends agreeing that a flat is worth Rs 1 crore, and then agreeing, inside that same Rs 1 crore, to set aside a fifth room for a flatmate who has not arrived yet. The value of the flat has not moved, but each friend's share of it is smaller. A pool created inside the pre-money is paid for entirely by the existing holders, because the fixed pre-money value is spread across more shares before the new money comes in. This is the option pool shuffleSetting the size of the employee option pool inside the pre-money valuation, so the dilution from the pool falls on existing shareholders rather than on the new investor., and the price falls from Rs 500 to Rs 434.78 even though the term sheet still says Rs 50 crore.

What is the founders' real pre-money, and what does each party own?

Value the shares the founders actually hold. 10 lakh shares at Rs 434.78 is Rs 43.48 crore, so the founders have in effect accepted a pre-money of Rs 43.48 crore, not Rs 50 crore. The Rs 6.52 crore gap is the pool, valued at the round price. A 2.5 lakh pool on the same terms would cut their real pre-money to Rs 40.00 crore, which is why founders negotiate the pool size as hard as the headline.

Then give the ownership table, because the interviewer will ask for it. After the round there are 14,37,500 shares: founders hold 69.6%, the pool 10.4% and the investor 20.0%. The investor's 20% is fixed by Rs 12.5 crore over a Rs 62.5 crore post-money; the pool only decides who inside the other 80% gives it up. Had the same pool been created after the round, the investor would share the dilution: founders 71.4%, investor 17.9%, pool 10.7%.

Outside investors should cut the chance the company is good from 50% to 12.5%. Take 200 seed companies: 100 good and 100 bad. The fund backs 90 good and 30 bad ones, and passes on 10 good and 70 bad. A pass therefore comes from a good company 10 times out of 80. An insider investing, by contrast, lifts the chance to 75%.

Why does one fund's decision carry so much information?

Imagine a restaurant owner who eats at her own restaurant nearly every day, and then one week stops. Regulars notice, because she knows the kitchen better than anyone. An existing investor has a board seat, the monthly numbers and a view of the team, so its choice to follow on or not is a signal from the best-informed party in the room. This is the signalling riskThe damage to a startup's fundraising when a well-informed existing investor declines to invest again, because outsiders read the decision as bad news. founders weigh when they take seed money from a large multi-stage fund.

How do you turn the percentages into the answer without a formula?

Use counts, not probabilities; they are harder to get wrong under pressure. Start with 200 companies because it makes every branch a whole number. Of 200, the fund passes on 10 good and 70 bad companies, so a pass is compatible with a good company only 10 times in 80. The same tree answers the mirror question for free: of the 120 companies it backs, 90 are good, so an insider investing lifts the chance from 50% to 75%.

When is the insider's pass weaker evidence than this?

Say the limit before the interviewer does. The numbers assume the fund passes only for reasons of quality. A fund that has run out of reserves, or whose cheque size no longer fits the round, passes for reasons unrelated to the company, and the fall from 50% should then be much smaller. That is why a good outside investor asks why the insider passed before reading the pass as a verdict, and why founders prefer insiders who state their follow-on policy in advance.

190 one-to-ones in all, and 150 of those pairs never meet if talk stays within tables. Each of 20 founders pairs with 19 others, 380 counted from both sides, so 190 pairs. A table of five holds 5 x 4 / 2 = 10 pairs, and four tables hold 40. That leaves 190 minus 40, or 150 pairs, about 79% of all possible meetings, unmet.

Why do you divide by two?

Think of handshakes at a wedding. If you asked every guest how many hands they shook and added up the answers, each handshake would be counted twice, once by each person in it. A pair is one meeting seen from two sides, so the number of pairs among n people is n times (n minus 1), halved. For twenty founders that is 20 x 19 / 2, which is 190.

How many pairs does the seating leave out, and why is it more than three quarters?

Count what the tables allow, then subtract. A table of five holds 10 pairs, so four tables hold 40. Splitting a group into four tables does not cut the meetings to a quarter; it cuts them to about a fifth, because pairs grow with the square of group size. 40 out of 190 is 21%, so 150 pairs never meet. Check it from one founder's seat: each founder meets 4 people and misses 15, and 20 x 15 / 2 is also 150.

Why would a venture interviewer ask about pairs?

Pair counting sits under network effects. A marketplace with n users has about n squared over two possible connections, which is why tripling users multiplies the possible connections roughly ninefold. It is also why platform teams at funds run founder dinners in rotating rounds: moving people between tables is the cheapest way to raise the share of pairs that meet. Say the limit as well: a possible connection is not a valuable one, and most of the 190 pairs would have little to discuss.

About 1.97 and about 0.48. For the first, the rule of 72 says 7% doubles money in about 10.3 years, so ten years gives just under 2. For the second, ln 0.93 is about minus 0.0725, ten years of it is minus 0.725, and e to that is about 0.48. They are not reciprocals because 0.93 is not 1 over 1.07: undoing a 7% rise takes only a 6.5% fall.

How do you get each number without a calculator?

Start with the one you know. The rule of 72A shortcut for compounding: money growing at r per cent a year doubles in roughly 72 divided by r years. says money growing at r% doubles in about 72 over r years, so at 7% it doubles in about 10.3 years and ten years leaves you just short of 2. For the fall, work in natural logs, which turn compounding into adding. ln(1 minus x) is about minus x minus half of x squared, so ln 0.93 is about minus 0.07 minus 0.00245, which is minus 0.0725. Ten years gives minus 0.725. e to minus 0.693 is exactly one half, and minus 0.725 is a little further down, so the answer is a shade under a half: about 0.48.

Why are the two answers not mirror images?

Everyday version first: a shirt marked up 7% and then marked down 7% ends below its starting price, because the markdown is taken on the higher price. The x squared term in the log has the same sign whichever way the rate goes, so it drags both paths down: the up path gains a little less than 7% a year in log terms and the down path loses a little more. The true mirror of a 7% rise is a 6.5% fall. Put the two paths together and 1.07 x 0.93 is 0.9951 a year, so ten years of each leaves 0.952: a round trip that still loses about 5%.

Then say why a growth investor cares. Swings cost compound growth even when the average yearly change is zero. A company whose revenue alternates between up 7% and down 7% averages zero change but ends smaller, losing roughly half the square of the swing every year. The limit is that this is a small effect at 7%; it becomes large at the 30% and 50% swings early-stage revenue can show.

Rs 1,440 crore after five years, a compound rate of about 135% a year. The yearly multipliers chain: 3 x 3 x 2 x 2 x 2 is 72, and Rs 20 crore times 72 is Rs 1,440 crore. The annual rate is the fifth root of 72, about 2.35, so revenue grows about 135% a year on average, even though no single year actually grew at that rate.

Why do the multipliers multiply rather than add?

Think of a sapling that triples its height in its first year and triples again in its second. A one metre sapling is three metres after a year and nine after two, not six, because the second tripling acts on the three metres it has already grown. Each year's multiplier applies to the revenue the company already has, so five multipliers combine by multiplication into a single 72x. Adding them gives 12, which is the wrong operation and the most common slip.

Walk the path aloud so the interviewer can follow: Rs 20 crore, then 60, 180, 360, 720 and 1,440. Saying each year also gives you the intermediate figures that a follow-up question usually asks for, such as when the company crosses Rs 500 crore, which here happens during year four.

How do you find the annual rate in your head?

Bracket the fifth root first. 2 to the fifth is 32 and 3 to the fifth is 243, so the answer is between 2 and 3, and much closer to 2. Try 2.5: 2.5 squared is 6.25, to the fourth about 39, to the fifth about 98, too high. The fifth root of 72 is about 2.35, so the compound rate is about 135% a year. If you know your logs, ln 72 is ln 8 plus ln 9, about 4.28; a fifth of that is 0.855, and e to 0.855 is about 2.35.

What does the average rate hide?

The 135% is a summary, not a description. The path is front-loaded: after one year the company is at Rs 60 crore against Rs 47 crore on the steady path, and the gap persists until the two meet at year five. That matters to an investor because the early years carry most of the growth, and a company that misses its first tripling cannot recover the 72x by doubling later. Say the limitation too: this path is a widely quoted benchmark for the best software companies after they find their market, not a forecast, and very few companies sustain it.

About Rs 144 crore a year. Three lakh dentists at two per clinic is 1,50,000 clinics. Forty per cent of those, the city clinics likely to buy software, is 60,000. At Rs 24,000 a year each, that is Rs 144 crore. The figure most open to challenge is dentists per clinic: at three per clinic the market is Rs 96 crore, at one and a half it is Rs 192 crore.

Why start by turning dentists into clinics?

If you were sizing the market for kitchen ovens you would count kitchens, not cooks, because a house with four cooks still buys one oven. A market is counted in the unit that pays, and practice-management software is bought once per clinic, so the first step converts people into paying units. Three lakh dentists at two per clinic gives 1,50,000 clinics. Skip the step and you double the answer before you have made a single judgement.

Which assumption would a partner push on first?

Rank the steps by how wrong they could be. The dentist count comes from a register and is unlikely to be far off. Dentists per clinic is the step to defend, because many dentists practise alone while others work across two or three clinics, and the answer moves in proportion to it. At three per clinic the market is Rs 96 crore; at one and a half it is Rs 192 crore. The price is next: Rs 24,000 a year is Rs 2,000 a month, which a solo clinic may resist and a multi-chair chain may pay several times over.

What exactly does Rs 144 crore measure?

Be precise about the label. The 40% filter removes clinics unlikely to buy, so this is closer to a serviceable market than to the total, and it assumes every likely buyer pays full price to someone. It is the size of the prize across all vendors, not the revenue one startup can expect, and a fund will ask what share is winnable and how quickly. State the limitation plainly: every input here is an assumption to verify, the dentist count against the current professional register in particular, before the number goes into an investment memo.

With B senior, B takes Rs 40 crore and A Rs 5 crore; pari passu, A takes Rs 9 crore and B Rs 36 crore. The preferences add to Rs 50 crore against a Rs 45 crore sale, so someone is short. A senior B is paid in full first and A gets what is left. Pari passu, each recovers 90% of its preference. Common shareholders get nothing either way, and seniority moves Rs 4 crore from A to B.

What do senior and pari passu mean when the money runs short?

Two friends lent Rs 10,000 and Rs 40,000 to a third, who can now repay only Rs 45,000. If the bigger lender was promised first claim, he takes his Rs 40,000 and the other gets Rs 5,000. If they agreed to stand together, they each take 90 paise in the rupee. Seniority decides the order of payment; pari passuLatin for equal step. Claims that rank pari passu are paid at the same time, sharing any shortfall in proportion to the amounts owed. means paying everyone at once in proportion to what each is owed. Both rules give the same answer whenever the sale covers every preference; they only differ in a shortfall like this one.

Would either series rather convert to common?

Check it, because non-participating preferred can always choose to convert. When the total preference exceeds the sale price, converting cannot help either series, because a converted holder still stands behind the other series' preference and shares only what is left. If A converted under B seniority, it would share the Rs 5 crore left with the other common holders, getting less than Rs 5 crore. If B converted under pari passu, it would share the Rs 35 crore left after A's Rs 10 crore, less than its Rs 36 crore. So the stacks above are the answer, whatever the stakes are.

Why does the later investor usually ask for seniority?

The later investor writes the bigger cheque at the higher price, and a down-side sale hurts it most. Seniority protects the Rs 40 crore cheque at the expense of the Rs 10 crore one, so it moves Series A's recovery from 90% to 50%. That is why earlier investors negotiate hard for pari passu when a new round comes in. The limitation of this example is that it is a single sale at a single price; at any exit above Rs 50 crore the ranking stops mattering for the preferences and only the conversion decisions remain.

There are two equilibria: both fund, worth Rs 26 crore each, and both refuse, worth Rs 9 crore each. If the other funds, your best reply is to fund, Rs 26 crore against Rs 9 crore. If the other refuses, your best reply is to refuse, Rs 9 crore against minus Rs 1 crore. Both funding is better for both, but it is only stable if each trusts the other, so bridges fail on coordination rather than on the economics.

How do you find the equilibria in a two-by-two game?

Two friends agree to meet for a film, and each will only buy a ticket if the other turns up; alone, the ticket is wasted. Both going is best, but both staying home is also a stable outcome, because neither wants to be the one sitting alone. An equilibrium is a pair of choices where each player's choice is the best reply to the other's, so neither gains by switching alone. Work it from each player's seat: fix the other's move, compare your two payoffs, and mark your best reply. Cells where both players' best replies meet are the equilibria.

Where do the payoffs come from?

Each payoff is the value of a 30% stake less the cash put in. Both fund: 30% of Rs 120 crore is Rs 36 crore, less Rs 10 crore, Rs 26 crore. One funds: 30% of Rs 30 crore is Rs 9 crore, so the funder ends at minus Rs 1 crore and the refuser at Rs 9 crore. Neither funds: Rs 9 crore each. Funding together creates Rs 34 crore of value for the two of them, but neither will move first if it fears being left as the lone funder. This is the structure economists call a stag huntA coordination game with two stable outcomes: a better one that needs both players to cooperate, and a safer one each can reach alone..

How do investors get to the good equilibrium in practice?

Remove the fear of funding alone. Bridges are usually written so that no money moves unless the full Rs 20 crore is committed, which deletes the minus Rs 1 crore outcome and leaves funding as the better choice whatever the other does. A lead investor committing first, publicly, does the same job. The limitation of the model is that it fixes both stakes at 30%; in practice the bridge would buy new shares, which raises the reward to funding and makes coordination easier, and a refusing investor may also face penalties such as losing its preferences.