Venture Capital puzzles, solved step by step

About Rs 212.5 per Rs 100 committed, a multiple of 2.125x. Ten years at 1% takes Rs 10 in fees, so Rs 90 reaches the underlying funds. At 2.5x that becomes Rs 225. The profit over the Rs 100 committed is Rs 125, and 10% carry on it is Rs 12.5. The LP keeps Rs 212.5, so the extra layer turns 2.5x into about 2.1x.

Why does the fee cost more than its 10% headline?

Think of buying vegetables through a cousin who takes Rs 10 out of every Rs 100 you give him before he reaches the market. The vendor's prices may be excellent, but you only ever buy Rs 90 worth. Fees come out before the money is invested, so the underlying 2.5x is earned on Rs 90, not on Rs 100. Rs 90 at 2.5x is Rs 225, which is 2.25x of what the LP committed. The 1% a year looks small, but over ten years it is a tenth of the commitment that never works.

Which layer costs the LP more, the fees or the carry?

Split the 0.375x gap between 2.5x and 2.125x. Fees cost 0.25x and carry costs 0.125x, so the fixed fee does twice the damage of the profit share at this return. That ranking flips at higher returns, because carry grows with profit while the fee does not. At 4x underlying, the same structure would cost 0.4x in fees and about 0.26x in carry. The assumption here is that carry is charged on profit over the full commitment, with fees returned first and no hurdle; a structure that charges carry from the first rupee of profit costs more.

What does an LP get for the extra layer?

A fund of funds sells access and diversification: places in funds that are hard to get into, spread across managers and years, with someone else doing the selection and monitoring. The LP should compare the net-of-everything multiple with what it could earn going direct, not compare the fund of funds' fees with zero. The limitation is that this one number hides timing. The fund of funds' fees start on day one, while the underlying funds return money late, so the gap in annual rate terms is wider than the gap in multiples suggests.

About 23%. This is a random walk between two walls, zero and six months, starting at three. With r the ratio of down to up probabilities, 0.6 over 0.4 or 1.5, the chance of hitting the top first from rung i is (1 minus r to the i) over (1 minus r to the 6). From three that is 2.375 over 10.39, or 22.9%, against 50% if the odds were even.

Why is the answer not 50% when the company starts halfway?

Picture someone walking along a narrow wall in the wind, three steps from either end, and the wind pushes them back towards the start a little more often than forward. Each single step is only slightly unfair, but the walk ends at whichever end comes first, and over many steps the small push decides it. A 60/40 tilt per month sounds mild, yet over the many months the walk can last it compounds into odds of roughly three to one against reaching the raise. Only a perfectly fair walk gives 50% from the midpoint.

How do you solve it on a whiteboard?

Let h(i) be the chance of reaching 6 from rung i. One month from now the company is at i plus 1 with chance 0.4 or i minus 1 with chance 0.6, so h(i) is 0.4 h(i + 1) plus 0.6 h(i - 1), with h(0) = 0 and h(6) = 1. The solution of that recurrence is the gambler’s ruinThe classic problem of a player betting one unit at a time until reaching a target or losing everything. It gives the chance of hitting either wall of a random walk first. formula, (1 - r to the i) over (1 - r to the N), with r = q over p = 1.5. Then it is arithmetic: 1.5 cubed is 3.375 and 1.5 to the sixth is 3.375 squared, about 11.39, so h(3) = 2.375 / 10.39, about 0.229.

What would change the company's odds most?

The formula shows two levers. Moving the bar closer helps a lot: if the company could raise at 4 months instead of 6, the chance from 3 rises to 58.5%. Cutting the tilt helps even more, because the damage comes from compounding a bad ratio over many steps, which is why investors push early teams to cut burn rather than hope for a run of contracts. The limit of the model is that months are not independent coin flips: contracts cluster, and a founder can change the step size by cutting costs. It is a way to see the shape of the risk, not a forecast.

About Rs 8.33 crore, roughly 5% of the sale, although they own half the company. The investors compare their Rs 150 crore preference with 40% of Rs 160 crore, Rs 64 crore, and take the preference. That leaves Rs 10 crore for common shareholders. Founders hold 50 of the 60 points of common, so they get five sixths of Rs 10 crore, and employees get Rs 1.67 crore.

Why do founders who own half the company get so little?

If you own half a house with a large loan against it and sell it for slightly more than the loan, the bank is paid first and your half of the small remainder is all you get. A liquidation preference works like that loan: investors are paid their Rs 150 crore before common shareholders see anything, so in a sale close to the preference, ownership percentages barely matter. This is the preference overhangThe total of liquidation preferences ahead of common shareholders. Until a sale exceeds it by a wide margin, common holders receive little, whatever their ownership. founders worry about after raising a lot of money at high valuations.

At what sale price does ownership start to matter again?

Investors convert when 40% of the sale beats Rs 150 crore, which happens above Rs 375 crore. Between Rs 150 crore and Rs 375 crore every extra rupee goes to common, so founders get five sixths of each rupee; above Rs 375 crore everyone shares by ownership and founders get half. At exactly Rs 375 crore the two rules give the same answer: founders take five sixths of Rs 225 crore, Rs 187.5 crore, which is half the sale. That is the price at which the founders' 50% stake finally means 50% of the money.

What does this mean for an investor sitting on the board?

A founder team facing Rs 8 crore from a Rs 160 crore sale has little reason to work for it, and a board that needs the founders to close the deal has a problem. That is why sales near the preference overhang often include a management carve-out, a slice of proceeds set aside for the team ahead of or alongside the preferences. The limits of the clean answer: real stacks have several series with different seniority, and any carve-out, transaction costs or debt come off the top first, which leaves common with even less.

About 5.2 years, not 3.4. Paying 50x when the exit multiple is 10x means ARR must grow 5x before you get your money back. At 60% a year that takes 3.4 years. But each round leaves you 85% of your previous stake, so your share of value grows by 1.6 x 0.85, or 36% a year. Solving 1.36 to the power n equal to 5 gives about 5.2 years, and that is only break-even.

What does paying 50x when peers trade at 10x actually commit you to?

Buying a mango sapling at the price of a grown tree is not a mistake if the sapling grows; it is a bet on how fast. When you pay five times the multiple a mature business earns, the company's revenue has to grow fivefold just for you to break even, because the multiple will fall back to the mature level by the time you sell. Growth first pays back the premium; only growth beyond that makes money. At 60% a year, 1.6 to the power n equals 5 gives 3.4 years.

Why does dilution stretch the answer by almost two years?

Think of a family shop that takes in a new partner every year, each time giving away 15% of everyone's share. The shop can double and your slice can still barely move. Your stake's value grows at the company's growth rate times what you keep after each round, so 60% growth with 15% dilution is 1.6 x 0.85, or 36% a year for you, not 60%. The bar the company must clear keeps rising: 5x becomes 5 divided by 0.85 after one year, then again after two, which is the red curve in the figure.

Say the limits before the interviewer does. The model holds 60% growth flat for five years, which few companies manage; growth that fades makes the wait longer. It treats each round's new cash as spent on the growth already assumed, rather than sitting on the balance sheet adding to value. And break-even after 5.2 years is a 0% return on money that a venture fund needs to multiply several times, so the honest conclusion is that the entry price has already spent most of the upside.

Ten tests, run at once. Number the statements and write each number in binary, ten digits long. Test pool 1 holds every statement whose first digit is 1, pool 2 every statement whose second digit is 1, and so on. A week later the ten results, read as positive equals 1, spell out the forged statement's number. Ten is the minimum because nine tests give only 512 possible result patterns, fewer than the 1,000 statements.

Why can you not just halve the pile and test again?

Halving is the right instinct when you can see each answer before asking the next question, like guessing a number with higher or lower clues. Here every result takes the full week, so all the testing has to be designed up front. When questions must be asked in parallel, each statement needs its own unique pattern of answers, and the puzzle becomes how many yes or no answers it takes to give 1,000 statements a distinct pattern each. Ten answers give 2 to the 10, or 1,024 patterns; that is enough.

How do you prove ten is the fewest?

Count the outcomes. Each test has two results, so n tests have at most 2 to the n distinct result patterns, and you need at least one pattern per possible forgery. Nine tests give 512 patterns for 1,000 suspects, so at least two statements would share a pattern and you could not tell them apart. Ten give 1,024, which is why 10 is both achievable and the minimum. One small detail: if you number the statements 1 to 1,000, the all-negative pattern is never used, which is fine because one statement is definitely forged.

Say where the trick breaks. It relies on exactly one forgery; with two forged statements, the positives are the union of two binary numbers and no longer name either one, so you would need more tests and a different design. It also assumes the test is perfectly sensitive in a pool of 500 statements. In real diligence a forger rarely fakes only one document, so the useful habit is the reasoning, counting outcomes before designing the checks.

The 12% deal made Rs 21 crore against Rs 1.5 crore, fourteen times as much. The quick flip shows a 32% IRR because 1.15x in six months annualises to 1.15 squared, about 1.32. But LPs spend rupees, not rates. If the Rs 11.5 crore then sat idle for the remaining nine and a half years, the deal is 1.15x over ten years, an IRR of about 1.4%. The ten-year deal's 3.1x is about 12% a year on the whole Rs 10 crore for the whole time.

How can a 32% IRR make less money than a 12% IRR?

A cab that charges a high rate per minute for a two-minute ride earns less than a modest-rate cab hired for the whole day. IRR is a rate per year while the money is out, so a short deal can post a huge IRR on a tiny rupee profit, and the rate says nothing about how long the capital earned it. On Rs 10 crore the flip made Rs 1.5 crore. The ten-year deal made Rs 21 crore, with a lower rate sustained over twenty times as long.

Why does it matter what the cash does after the exit?

The 32% assumes the Rs 11.5 crore can be put straight back to work at a similar rate. For an LP whose money was committed for ten years, cash returned early earns only what the LP can do with it next, and if that is nothing, the deal's true rate over the commitment is about 1.4%. This is why LPs read IRR beside the money multiple, and why a fund boasting a top-quartile IRR on a 1.3x fund gets hard questions. Some funds also time capital calls with credit lines to shorten the period the money counts as out, which flatters IRR without adding a rupee.

The fair limit: early cash is worth something if the LP really can reinvest it, and the reinvestment rate decides by how much. In the room, say both numbers for every deal, multiple and IRR, and say which one an LP banks.

Roughly 35,000 to 50,000 trucks a day, call it 40,000. From the goods: about 3 crore people, each accounting for some 10 kg of goods leaving the region daily, gives 3 lakh tonnes; at 12 tonnes a truck that is 25,000 loaded trucks, or about 35,714 once empty departures are added. From the trucks: about 4 lakh trucks serving NCR, each leaving once every 8 days, gives 50,000. Two routes within 1.4x of each other make the range defensible.

Why build the number twice?

A carpenter measures twice before cutting, not because the first measurement was careless but because a second, separate measurement catches the slip the first one hid. An estimate built from one chain of assumptions can be precise and still out by ten times; a second route built from different assumptions is the only check you can run in the room. Here the first route starts from what has to move, goods, and the second from what moves it, trucks. They share no inputs, so their agreement means something.

Walk route A slowly. Assume about 3 crore people in NCR. Outbound freight is the manufactured goods from the industrial belts plus wholesale trade re-sent to the rest of north India; assume 10 kg per resident a day, so 3 lakh tonnes. A loaded truck averages perhaps 12 tonnes across small and large vehicles, giving 25,000 loaded departures. Trucks that came in loaded often leave empty; if 30% of departures are empty, total departures are 25,000 divided by 0.7, about 35,714. Route B: assume 4 lakh intercity trucks, from NCR and outside, serve the region, and an average round trip takes 8 days, so 50,000 leave each day.

How would you check it outside the room?

The best check is a third, measured source: GST e-way bills, which are generated for goods moved above a value threshold and are published at state level, and toll plaza counts of goods vehicles on the main exits from NCR. Each has its own bias; e-way bills miss low-value loads and toll counts mix through traffic with NCR departures, so use them to test the range rather than replace it. A morning spent with brokers at a transport hub would test the 30% empty share and the 8-day trip, the two inputs most likely to be wrong.

Then say what the number is for. Trucks leaving a day is not the market; the marketplace earns on loads it matches. At 40,000 departures a day, if a third are booked through brokers and the platform takes a fee per load, the next step is the fee times the brokered loads. The truck count is the base everything else multiplies, which is why it deserves two routes.

About 12.6x, against 32x at a steady 100%. Multiply the yearly factors one at a time: 2 x 1.8 is 3.6, x 1.64 is 5.9, x 1.512 is 8.9, x 1.41 is about 12.6. Steady doubling gives 2 to the fifth, or 32. A growth rate that fades by a fifth each year leaves you with less than 40% of the steady outcome, which is why the fade assumption moves a valuation more than the opening growth rate.

How do you compute this quickly without losing the thread?

Turn each year's growth into a factor and keep a running product, like a cricket scorer adding each over to the total rather than recomputing the innings. Revenue after five years is the product of the five yearly factors, so a fading rate must be multiplied year by year; there is no single rate you can raise to the fifth power. The running total goes 2, 3.6, 5.9, 8.9 and 12.6. Rounding each step to one decimal keeps it mental and still lands within a few per cent.

Which matters more, the starting growth rate or how fast it fades?

Run a few cases. A company that starts at 150% but keeps only 60% of its growth each year ends at about 11.6x, below the 100% company that keeps 80%, and well below a 100% company that keeps 90%, which reaches about 19.7x. The opening rate is the number in the pitch deck; the fade is the number that decides the outcome. Diligence time is better spent on why growth should persist, such as retention and new markets, than on last year's growth figure.

The limit: a constant fade is a convenient shape, not a law. Real growth can stall, then re-accelerate on a new product, and a smooth curve will miss both. Use it to show the sensitivity, then test the specific reasons this company's growth would or would not hold.

Both reach Rs 337.5 crore, but only A keeps growing when new sales stop. Each year A's ARR becomes 130% from existing customers plus 20% from new ones, and B's becomes 90% plus 60%: both 1.5x. Three years of 1.5x takes Rs 100 crore to Rs 337.5 crore. Stop new sales and A still grows 30% a year, to about Rs 741 crore in three more years, while B shrinks 10% a year, to about Rs 246 crore.

How can a company losing revenue from its existing customers grow as fast as one expanding them?

Two water tanks can both rise by 50 litres an hour: one has a strong tap and a small leak, the other a small tap and no leak at all, plus a booster on the old water. ARR growth is net retention plus new-customer ARR, so the same headline growth can come from customers who spend more each year or from a sales team refilling a leaking base. A gets 30 points from existing customers and 20 from new ones; B loses 10 points from existing customers and replaces them with 60 points of new sales. Both add 50%.

Why would an investor pay more for A at the same ARR?

Because A's growth does not depend on the sales team hitting target every quarter. High net retention is growth the company has already earned; new-logo growth must be bought again every year, usually at a high sales and marketing cost. B has to replace 10% of its base before it grows at all, and as B gets bigger that hole gets bigger in rupees: Rs 10 crore in year one, Rs 22.5 crore in year three. If its market saturates or a recession slows buying, B shrinks; A keeps compounding from the customers it already has.

State the assumptions. The model applies the same retention to new customers from their first year, which flatters B if new customers churn faster than old ones, as they often do. It also treats retention as constant; very high NRR tends to fall as customers reach full deployment. A real diligence would read retention by customer cohort, not a single blended number.

About 13.3%. Every fund's estimate is right on average, but the deal goes to the fund with the highest estimate, and the highest of five errors is not average. For errors spread evenly from minus 20% to plus 20%, the highest of n sits on average (n - 1)/(n + 1) of the way to the top: with five bidders that is 4/6 of 20%, or 13.3%. Winning is itself evidence that you guessed high.

Why does winning tell you that you overestimated?

Ask five friends to guess the weight of a goat at a fair and give the goat to whoever guesses highest; the winner will nearly always have guessed too much. When the prize goes to the highest estimate, the winner is selected for being too optimistic, so even unbiased bidders overpay on average. This is the winner's curseThe tendency for the winner of an auction with uncertain value to have overestimated that value, because winning selects the most optimistic estimate.. No one bid foolishly here; each estimate was fair. The overpayment comes entirely from the selection.

What should a disciplined bidder do about it?

Bid below your estimate, and more so when there are more bidders or more uncertainty. The curse grows with the number of bidders and with the width of the errors: two bidders overpay by 6.7% on average, five by 13.3%, ten by 16.4%. So a fund in a crowded growth round should shade its bid by roughly the expected overestimate, about 12% of its estimate here, and a fund that keeps winning contested deals should ask whether it is better informed or just more optimistic. A simulation of 200,000 auctions in the source file gives 13.3%, matching the formula.

The limit: the model assumes every fund's estimate is equally noisy and that all bid their raw estimate. Better-informed funds face a smaller curse, and in practice everyone shades, so real overpayment is smaller than the raw figure. The direction of the effect is what to carry into the room.