Venture Capital puzzles, solved step by step

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.

The 3x in 3 years, at about 44.2% a year against 25.8%, a gap of about 18 points. IRR for a single cash in and out is the multiple's root over the years held. The cube root of 3 is about 1.44 because 1.44 cubed is 2.99. The seventh root of 5 is about 1.26 because 1.26 to the seventh is just over 5. Time sits under the multiple, so the smaller, quicker multiple earns the higher rate.

Why can a smaller multiple have the higher IRR?

Two runners: one covers 3 kilometres in 3 minutes, the other 5 kilometres in 7 minutes. The second went further, but the first was faster. A multiple says how far the money went; the IRR says how fast, so time sits in the denominator and a long hold drags the rate down. For a single cheque in and a single exit, the IRR is the yearly growth rate that turns 1 into the multiple over the holding period: the multiple's root over the years.

How do you take those roots without a calculator?

Guess and cube. 1.4 cubed is 2.744 and 1.5 cubed is 3.375, so the cube root of 3 is a little above 1.4; 1.44 cubed is 2.986, close enough, so 44%. For the seventh root of 5, use doublings: 5x is about 2.3 doublings, since 2 to the 2.3 is about 5, and 2.3 doublings in 7 years is one every 3 years, which the rule of 72 puts at about 24%. The rule of 72 runs a little low at rates this high, so nudge up: 1.26 to the seventh is just over 5, giving about 26%.

Does the higher IRR make the 3x the better investment?

Not on its own. The 3x deal wins on rate, but it hands money back after three years, and whether that is better depends on what the money can earn next. To match 5x over the same seven years, the 3x proceeds need to grow 5/3 times over the remaining four years, about 13.6% a year. If the fund or its LPs can redeploy above that, the quick 3x is better; if not, the 5x leaves more money in total. This is why LPs look at both IRR and multiple: a high IRR on a short hold can return fewer rupees.

About Rs 1,296 crore a year of packaged pet food spend, of which a brand winning 5% would take about Rs 65 crore. Take 2.5 crore households in the eight cities, 12% owning a dog or cat, 30% of those buying packaged food, and Rs 1,200 a month, or Rs 14,400 a year, per buying household. That is 9 lakh households and about Rs 1,296 crore. Pet ownership and the packaged share are the two numbers to defend.

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

Sizing a market is like estimating how many people in your building would join a weekend cricket league: count the flats, then the ones with a player, then the ones who would pay a fee. Each branch should be one filter you can name and defend separately, so a challenge to one number does not bring down the whole estimate. Here the branches are households, pet ownership, packaged-food use and spend. Every figure is an assumption to check against current survey data before it goes into a memo; the point in the room is the structure.

Which assumptions move the answer most?

Give each assumption a plausible range and move one at a time. Pet ownership from 8% to 16% and the packaged share from 20% to 40% each swing the answer from Rs 864 crore to Rs 1,728 crore, wider than either spend or household count. Spend from Rs 900 to Rs 1,500 a month moves it only to Rs 972 crore and Rs 1,620 crore, and the household count is the best known of the four. So the diligence effort goes on ownership and packaged share, and a founder's deck that is vague on those two is vague where it matters.

How do you get from the market to one brand's revenue?

The Rs 1,296 crore is spend across every brand and channel, including shops. A direct-to-consumer brand's potential is a share of that pool, and at 5% it would be about Rs 65 crore a year, which is the number to compare with the round size. State the limits: the packaged share is growing, so today's figure understates the pool in five years; and the estimate ignores smaller pets, treats and accessories, which a brand may sell alongside food.

They are almost equal: about 9.68p a year on the monthly plan against 9.6p on the annual plan. The monthly customer pays p in month one and is still paying with probability 0.96 to the k in month k plus one, so twelve months add to about 9.68p. The annual customer pays 12 x 0.8p, 9.6p, upfront. With revenue a near tie, the cash on day one and the lower churn of a committed customer decide it for annual.

How do you count revenue from a customer who might leave?

A gym that sells monthly memberships cannot count on twelve payments from every member who joins in January, because some stop coming in March. Expected revenue from a monthly customer is each month's price times the chance they are still a customer that month, added up. With 4% churn, the chance of still paying in month k plus one is 0.96 to the k: 1 in the first month, 0.96 in the second, and 0.61 by the twelfth. Twelve terms of a shrinking series add to about 9.68p, not 12p.

If revenue is a near tie, what decides it?

Everything the revenue sum leaves out. The annual plan collects 9.6p on day one, which funds growth without new equity, while the monthly plan collects its 9.68p over twelve months. Annual customers also cannot churn mid-year, and at renewal they tend to churn less than monthly customers, so year two favours annual too. The break-even is close: at monthly churn of 4.16% the two plans give the same first-year revenue, so at any churn above that the annual plan wins on revenue as well.

What could make the comparison misleading?

The sum assumes the same customer would choose either plan. In practice the customers who pick annual plans are already the more committed ones, so their lower churn is partly selection, not a result of the plan. A founder who shows that annual customers churn less is showing a correlation; the useful question is what happens to churn when the company pushes hesitant customers onto annual terms. The other limit is the discount itself: at 20% it is roughly break-even, but a deeper discount gives revenue away for cash that may be cheaper to raise elsewhere.

Rs 412.5 crore is left to invest, and the portfolio needs about 1.21x gross just to return the Rs 500 crore. Fees are 2% for five years, Rs 50 crore, plus 1.5% for five years, Rs 37.5 crore, a total of Rs 87.5 crore. LPs committed Rs 500 crore, so the Rs 412.5 crore that is invested must grow to Rs 500 crore: 500 over 412.5 is 1.21x before the fund earns a rupee of carry.

Why does a fund need more than 1x just to break even?

If a friend takes Rs 17.5 from every Rs 100 you give him to invest, the Rs 82.5 he does invest has to grow by a fifth just to give you your Rs 100 back. Management fees come out of commitments, so the portfolio is smaller than the money LPs put in, and it must earn the fees back before LPs see any profit. On a Rs 500 crore fund, ten years of fees at these rates take Rs 87.5 crore, 17.5% of the fund.

What multiple does the fund need to deliver a good result to LPs?

Run the same logic to a target. Say LPs want 2x net, Rs 1,000 crore, and the manager takes 20% carry on profit above the Rs 500 crore committed. The LPs' Rs 500 crore of profit is 80% of the total profit, so total profit must be Rs 625 crore and proceeds Rs 1,125 crore. That is 2.73x on the Rs 412.5 crore invested, so a 2x net fund is close to a 2.7x gross portfolio. The gap between gross and net is why LPs ask for both numbers and never compare one manager's gross with another's net.

Where does the clean answer go wrong in practice?

Funds often recycle: they reinvest early exit proceeds up to the amount of fees paid, so invested capital can approach the full Rs 500 crore and the break-even multiple falls towards 1x. The fee terms, the recycling allowance and the carry structure are set in each fund's limited partnership agreement, so confirm them there before using these figures for a real fund. Some funds also charge the later-period fee on invested capital rather than commitments, which lowers total fees.

36% if independent, 20% if perfectly correlated, and 30% if the chance both fail is 10%. In each case the chance of at least one failure is 20% plus 20% minus the chance both fail. Independent failures overlap 0.2 x 0.2 = 4%, giving 36%. Perfectly correlated failures overlap completely, 20%, giving 20%. With a 10% overlap the answer is 30%. Correlation makes any failure less likely but both failing far more likely.

Why can you not just add the two chances?

Think of two friends who each forget your birthday one year in five. If you add the chances you get two in five, but that counts the years when both forget twice, once for each friend. The chance of at least one event is the sum of the two chances minus the chance both happen, so the overlap decides the answer. For two independent events the overlap is the product, 0.2 x 0.2 = 4%, and the answer is 36%. The complement route checks it: neither fails with chance 0.8 x 0.8 = 64%.

What does correlation do to a venture portfolio?

It trades many small surprises for fewer, larger ones. As failures become more correlated, the chance of at least one failure falls from 36% to 20%, but the chance both fail rises from 4% to 20%, five times higher. The 10% case corresponds to a correlation of about 0.37 between the two failure events. A fund full of companies selling to the same customers, or depending on the same funding market, looks diversified by count but behaves like fewer, bigger bets.

Why would an LP care about this more than a single fund manager?

An LP holding several funds cares about how many bad outcomes can arrive at once. Correlated failures are what turn an ordinary bad year into a vintage where most companies struggle together, and an average-case model that treats failures as independent understates that risk. The limitation is that correlation is hard to measure for private companies with few data points; in practice investors judge it from shared exposures, such as the same sector, the same customers or the same reliance on new funding, rather than from a computed number.

About Rs 324 crore a year. Start with 1.2 crore stores. 15% sell more than Rs 20,000 a day, which leaves 18 lakh stores busy enough to need billing software. 30% of those will pay, which leaves 5.4 lakh paying stores. At Rs 500 a month, or Rs 6,000 a year, that is Rs 324 crore. The Rs 7,200 crore you get by charging every store is the total market, not the one this app can serve.

Why not multiply every kirana store by the price?

A tailor opening a shop in a colony of 2,000 homes does not have 2,000 customers. He has the homes that buy stitched clothes, at his prices, from a shop, rather than from a relative with a machine. A serviceable market counts only the buyers who feel the problem badly enough to pay for the fix, so every filter must describe a real reason a store would or would not buy. The Rs 20,000 daily sales cut is that reason here: a store taking in that much has enough bills, credit customers and stock lines that paper billing starts to hurt. Multiply all 1.2 crore stores by Rs 6,000 and you get Rs 7,200 crore, the total addressable marketRevenue if every possible buyer in the category bought the product at its price. A ceiling, rarely a plan., which describes the country rather than this app.

How do you say the answer so it survives the follow-up?

Say the chain as three multiplications and carry the unit at every step: stores, then stores, then rupees. The most common slip in this question is mixing a monthly price with an annual market, so say both numbers out loud: Rs 27 crore a month, Rs 324 crore a year. Then name the weakest link yourself. The 30% willingness to pay is the least grounded figure, and the answer moves one for one with it: at 20% the market is Rs 216 crore, at 40% it is Rs 432 crore.

Finish with what the number does and does not tell an investor. Rs 324 crore is a ceiling on this product's revenue, not a forecast; the app would win some share of the 5.4 lakh stores, and competitors are chasing the same ones. A partner hearing this will ask two things: what share is realistic in five years, and whether the stores can be sold a second product, such as working capital credit, that makes the market bigger than billing alone. The 1.2 crore store count is the question's premise; outside the room, confirm it against a published retail survey before building on it.

Up 50% in all, or about 14.5% a year. Turn each round into a multiplier: up 150% is x 2.5, down 60% is x 0.4, up 50% is x 1.5. The shortcut is that 2.5 x 0.4 is exactly 1, so the Series C fall wiped out the Series B gain, and the net is just Series D's x 1.5. The annual rate is the cube root of 1.5, which sits between 14% and 15%.

Why can the three percentages not simply be added?

A shopkeeper who marks a shirt up 150% and then runs a 60% off sale is not ahead by 90%. The markup took Rs 100 to Rs 250, and the sale took 60% of Rs 250, landing back at Rs 100. Each percentage is measured against whatever the value was just before it, so moves in a chain multiply rather than add. Adding gives +140%, a number that describes nothing in this company's history. Multiplying gives 2.5 x 0.4 x 1.5, which is 1.5: a valuation of 100 went to 250, back to 100, then to 150.

How do you find the annual rate without a calculator?

You need the number that, cubed, gives 1.5. Bracket it. 1.14 cubed is about 1.48 and 1.15 cubed is about 1.52, so the rate sits just below 14.5%, and saying 'about 14.5% a year' is the right precision for the room. The exact figure is 14.47%. Check it the other way: half of 50% is 25%, far too high, which is the arithmetic average and the error the question is fishing for.

One sentence of judgement helps. A company that fell 60% in one round and still ended up 50% ahead over three years has had a volatile path, and the steady 14.5% hides that volatility completely. Investors who entered at the Series B price are down 40% at Series D, which is why the entry round matters as much as the company's overall path.

Above about Rs 250 crore, and Series B sets that line. A non-participating holder converts when its share of the exit beats its preference, so each class's line is preference divided by stake: seed Rs 5 crore over 10% is Rs 50 crore, Series A Rs 20 crore over 20% is Rs 100 crore, Series B Rs 50 crore over 20% is Rs 250 crore. The class that paid the most per percentage point is the last to convert.

Why is the line for each class its preference divided by its stake?

Suppose a friend offers you a choice: your Rs 500 back, or a tenth of whatever the raffle raises. You take the tenth only once the pot is above Rs 5,000. A 1x non-participating preference is exactly that choice, so a class converts once its stake times the exit value beats its money back, and the crossover is preference divided by stake. Preference over stake is also the price the class paid for the whole company, which is why the class that bought at the highest valuation, here Series B at an implied Rs 250 crore, holds out longest.

Why does the question say 'roughly', and do the lower lines move?

The simple division assumes everyone else has converted. Below Rs 250 crore Series B keeps its Rs 50 crore, and that comes out of the pot before the others share it. When a senior class holds its preference, the pool left for everyone else shrinks, so the junior classes need a bigger exit before converting pays. Series A with B holding gets 20/80 of the exit less Rs 50 crore, which beats Rs 20 crore only above Rs 130 crore. Seed, with both A and B holding, needs Rs 100 crore. The top line does not move, because by the time Series B is deciding, everyone below it has already converted.

In the room, give Rs 250 crore and Series B first, then offer the refinement as a check. It shows you understand that preferences interact, which matters when you are the junior investor negotiating behind a large late round.

Rs 40 crore today, Rs 96 crore after the first round, Rs 64 crore after the second. Each round sells 20% of the company, so she keeps 80% of her stake: 40% becomes 32%, then 25.6%. Her rupees are that stake times the post-money value: 32% of Rs 300 crore is Rs 96 crore, and 25.6% of Rs 250 crore is Rs 64 crore. Dilution cut her percentage both times; only the down round cut her value.

Why does selling 20% leave her with 32%, not 20%?

Cut a pizza into ten slices and give four to a friend. If the host then takes a fifth of every plate for a late guest, the friend loses a fifth of her four slices, not two of them. A round that sells 20% of the company shrinks every existing holder by the same fifth, so her stake is multiplied by 0.8, not reduced by 20 points. 40% x 0.8 is 32%. The second round does it again: 32% x 0.8 is 25.6%.

If her percentage fell both times, why did her wealth rise the first time?

Because the first round was priced well above today's value. A Rs 300 crore post-money with Rs 60 crore raised means a pre-money of Rs 240 crore, so the company was repriced from Rs 100 crore to Rs 240 crore before the new money arrived. Dilution is a smaller slice; whether it costs you money depends entirely on the price the new investor pays for the slice. The second round sold 20% at Rs 250 crore post-money, which is Rs 50 crore raised on a pre-money of Rs 200 crore, below the Rs 300 crore she was last marked at. Her percentage fell by the same fifth, and this time her value fell too, from Rs 96 crore to Rs 64 crore.

One limit is worth stating. The Rs 64 crore is a paper value at the last round's price, and it assumes her shares are worth the same per share as the investors' preferred shares, which carry preferences hers do not. In a sale below the preferences, her real payout would be less.