Venture Capital puzzles, solved step by step

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.

The Series B follow-on, at 1.50x against 1.25x. Rs 10 crore at Rs 400 crore post buys 2.5%, worth Rs 50 crore in a Rs 2,000 crore exit; at a 30% chance that is Rs 15 crore expected. The seed buys 25%, halved to 12.5% by exit, worth Rs 250 crore; at 5% that is Rs 12.5 crore. The seed would need a 6% chance of the exit to draw level.

Why is the cheaper entry not automatically the better bet?

A lottery ticket costs very little and a fixed deposit costs a lot per rupee of payout, yet nobody thinks the ticket is the better deal just because it is cheap. What matters is the price multiplied by the chance of being paid. The expected multiple of a venture cheque is the chance of the exit times the stake you hold at exit times the exit value, divided by the cheque. The seed price is 10 times lower, so the stake starts 10 times bigger, but halving by dilution leaves it only 5 times bigger, and the odds are 6 times worse.

How do you compare them quickly in the room?

Compare the three ratios rather than the full sums. The seed stake at exit is 5 times the Series B stake, but its odds are 6 times lower, so its expected value is five sixths of the follow-on's. Five sixths of 1.5x is 1.25x. The same logic gives the break-even: the seed draws level only if its chance of the exit rises from 5% to 6%, or the Series B chance falls from 30% to 25%.

What does the expected multiple leave out?

Say the limits before the interviewer does. The seed's 1.25x comes with a 95% chance of losing everything, against 70% for the follow-on, so the two bets carry very different risk even before you compare averages. Time matters too: a Series B company is closer to exit, so the same multiple earns a higher annual rate. Against that, a seed fund's strategy depends on owning large stakes in a few outliers, and a partner may accept a lower expected multiple for the bigger stake. The arithmetic decides the comparison only once those preferences are stated.

Blended LTV to CAC is 1.33x; the marginal ratio on the last 1,000 customers is 0.86x. Total spend is Rs 50 lakh plus Rs 80 lakh plus Rs 1.4 crore, Rs 2.7 crore for 3,000 customers, an average of Rs 9,000, and Rs 12,000 over Rs 9,000 is 1.33. The last cohort costs Rs 14,000 for Rs 12,000 of contribution, so it destroys Rs 20 lakh.

Why can a healthy average hide a losing cohort?

Think of picking mangoes from a tree. The low branches take seconds, the middle ones need a ladder, and the top ones need a long pole and ten minutes each. The average minutes per mango still looks fine long after the top branches stopped being worth the effort. Customer acquisition works the same way: each extra customer from a saturating channel costs more than the last, so the average lags behind the cost of the customer you are adding now. The blended 1.33x looks acceptable. The marginal LTV to CACThe lifetime contribution of the next customer divided by the cost of acquiring that customer, as opposed to the average across all customers acquired so far. on the third cohort is 0.86x.

How much value does each cohort create or destroy?

Turn the ratios into rupees, because a ratio cannot be added up. The first cohort creates Rs 7,000 a customer, Rs 70 lakh. The second creates Rs 4,000 a customer, Rs 40 lakh. The third loses Rs 2,000 a customer, Rs 20 lakh. Stopping at 2,000 customers leaves Rs 110 lakh of value against Rs 90 lakh for all 3,000, so the third cohort makes the company smaller. At 2,000 customers the blend would also read better, 1.85x.

Why does a venture investor care about the difference?

A growth plan funded by a new round usually means spending more in the same channels. The return on that new money is set by the marginal ratio, not the blended one, so a company quoting 1.33x may be raising money to buy customers at 0.86x. Ask for acquisition cost by month or by spend band, not the lifetime average. Say the limitation as well: here every customer brings the same Rs 12,000, but a saturated channel often brings weaker customers too, so the real marginal ratio is likely worse than 0.86x.

The price implies a free cash flow margin of about -7%, and the line gives Rs 312.5 crore at the actual minus 30%. Rs 600 crore over Rs 50 crore is 12x ARR; divided by 0.25 that is a score of 48; less 55 points of growth leaves -7%. At minus 30% the score is 25, the multiple 6.25x and the EV Rs 312.5 crore, so the price assumes a 23-point margin gain worth Rs 287.5 crore.

How do you run a peer line backwards?

If you know that flats in a building sell for Rs 10,000 a square foot and one sold for Rs 1.2 crore, you can tell its size without measuring it: 1,200 square feet. A peer line works the same way. When the market prices software companies at a fixed multiple of their Rule of 40A score for software companies: revenue growth rate plus free cash flow margin, both in per cent. Forty or more is the conventional bar for a healthy balance of growth and cash generation. score, any observed price tells you the score the market is assuming. Rs 600 crore over Rs 50 crore is 12x ARR, and 12x over 0.25 is a score of 48.

What is the price paying for that the company does not yet earn?

Run the line forwards with the actual numbers. Growth of 55 plus a margin of minus 30 is a score of 25, worth 6.25x ARR, or Rs 312.5 crore. The gap of Rs 287.5 crore is what the market pays today for the company moving its margin from minus 30% to -7%, a 23-point improvement, without giving up growth. An investor buying at Rs 600 crore should ask how plausible that improvement is and how soon it must arrive.

How far can you trust the line?

A ten-company scatter fitted with one slope is a rough guide. The line explains the middle of the peer set well and the edges badly, so a company at either extreme of growth or margin will often sit far from it for reasons the score does not capture. The score also weights a point of growth the same as a point of margin, which markets do not always do. Treat the implied margin as a question to put to management rather than a fact about the company.

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.

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.

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.

Rs 5 crore, which is 10% of the round. Selling 25% for Rs 50 crore values the company at Rs 200 crore after the round. If you sit out, your 10% becomes 7.5%. To get back to 10% you need another 2.5% of a Rs 200 crore company, which costs Rs 5 crore. The general rule: to hold your stake, take the same share of the round as you own of the company.

Why does your stake fall to 7.5% if you sit out?

Think of a pizza cut into ten slices, one of them yours. If the owner adds enough new pizza that the old one becomes three quarters of the total, your slice is unchanged in size but now a smaller share of the whole. A new round does not take shares away from you; it adds shares for someone else, so every existing holder is scaled down by the same factor, here 0.75. Ten per cent times 0.75 is 7.5%, and the other holders' 90% becomes 67.5%.

Why is the answer exactly 10% of the round?

Do the sum, then see the pattern. The round sells 25% for Rs 50 crore, so the post-money is Rs 200 crore and 2.5% costs Rs 5 crore. Taking 10% of the round gives you 10% of the new shares, and you already hold 10% of the old ones, so you hold 10% of everything. This is what a pro rata rightA right, usually written into the investment terms, that lets an existing investor buy a share of a new round equal to its current ownership, so its stake is not diluted. gives an investor: the option to take its ownership share of each new round. It works the same at any round size.

When would a fund choose not to take its pro rata?

The arithmetic says what it costs; it does not say it is worth paying. Rs 5 crore at a Rs 200 crore post-money is a new investment decision at a higher price, and a seed fund with limited reserves may prefer to back a new company instead. The limitation of the clean answer is that it assumes the round stays at Rs 50 crore with your cheque inside it. If your money comes on top of a Rs 50 crore round from others, the round grows and you need slightly more to hold 10%.

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.