Venture Capital puzzles, solved step by step

About 68 covers a day, a little over one turn of its 60 seats. Each cover leaves Rs 390 once food is paid for. Fixed costs are Rs 8 lakh a month, so the restaurant needs 8,00,000 / 390 = 2,051 covers a month, or 68.4 a day over 30 days. That is 1.14 turns. Open 26 days a month and the bar rises to about 79 a day.

What does each cover actually contribute?

A tea stall that sells a cup for Rs 15 and spends Rs 5 on milk, leaves and sugar has Rs 10 a cup to pay its rent and wages. Break-even is fixed costs divided by contribution per unit, the price less the cost that rises with each unit, never fixed costs divided by the full price. Here a Rs 600 bill with 35% food cost leaves Rs 390 a cover. The rent, the staff and the other Rs 1 lakh do not change with covers, so they add to Rs 8 lakh a month that the Rs 390s must clear.

Is one turn a day an easy bar or a hard one?

It depends on the meals the restaurant serves. A place open for lunch and dinner gets two chances to fill the room; at 1.14 turns it needs each seat filled a little over once across both services. The sensitivity matters more than the point estimate: if the average bill falls to Rs 500, the bar rises to 82 covers a day, and if it closes four days a month it rises to about 79. At 90 covers a day it earns about Rs 2.5 lakh a month before tax, which shows how much of each extra cover falls to profit once fixed costs are covered.

What else would an investor ask about?

This puzzle leaves out costs that scale with sales besides food: delivery platform commissions, card fees and electricity on busy nights. Each one lowers the Rs 390 and raises the break-even. Then there is the opening cost, the fit-out and deposit, which break-even ignores but payback does not. A growth investor asks the same three questions of a chain: what each outlet contributes per cover, how many covers a mature outlet does, and how long a new one takes to get there.

14.9% without the credit line and 18.9% with it, and the multiple is 2.0x either way. IRR measures time as well as money. Without the line, Rs 100 doubles over five years: 2 to the power one fifth, less 1. With it, the investors' Rs 100 is out for four years, and 2 to the power one quarter, less 1, is 18.9%. Nobody made an extra rupee; with 8% interest, investors actually make less.

Why does paying later raise the IRR?

Lend a friend Rs 100 and get Rs 200 back: whether that is a good deal depends on whether it took five years or four. IRR is the yearly rate that turns the money paid in into the money paid out, so shortening the time the money is out raises the IRR even when the rupees are identical. Without a credit line, the investors' Rs 100 grows to Rs 200 over five years, 14.9% a year. With one, the investors pay at year 1 and still receive Rs 200 at year 5: four years, 18.9% a year.

What happens once you count the interest?

The bank is not free. At 8% for one year, the fund calls Rs 108 at year 1 to repay Rs 100 plus interest. The investors now pay more for the same Rs 200, so the multiple falls to 1.85x, yet the IRR is still 16.7%, higher than the 14.9% without the line. Less money, better-looking IRR. That gap is why investors in a fund ask for the multiple and the IRR side by side, and increasingly ask for the IRR with the credit line stripped out.

Why would a fund use a credit line at all?

There are honest reasons: it lets the fund close a deal quickly without waiting ten business days for a capital call, and it smooths many small calls into a few large ones, which investors find easier to manage. The concern is the IRR effect. A fund that ranks high on IRR partly because of the line may have made less money for its investors than one that ranks lower. Say both reasons, then say which number you would compare funds on: the multiple, and the IRR measured from the date the investment was made.

About 30 months. By the rule of 72, 8% a month doubles ARR roughly every 9 months: Rs 20 crore at month 9, 40 at month 18, 80 at month 27. Ten times is about 3.3 doublings, so about 30 months. The exact figure is ln 10 over ln 1.08, which is 29.9, so ARR first passes Rs 100 crore in month 30. Few companies hold 8% a month that long.

How do you get to 30 in your head?

Money in a deposit that doubles every nine years has gone up eight times after twenty seven. Turn the growth rate into a doubling time with the rule of 72, then count doublings: ten times is a little over three doublings, because 2 to the power 3.3 is about 10. 72 divided by 8 is 9 months per doubling, and 3.3 x 9 is about 30 months. The exact doubling time at 8% is 9.01 months, so the rule is very close here.

Check the answer: at month 29 ARR is Rs 93.2 crore and at month 30 it is Rs 100.6 crore. The crossing falls just before the end of month 30, so say about 30 months, or month 30 if the interviewer wants a whole number.

Is 8% a month believable for two and a half years?

8% a month compounds to about 152% a year, two and a half times ARR every twelve months. Young companies from a small base sometimes do that for a while; holding it from Rs 10 crore to Rs 100 crore is rare. Monthly growth rates usually decay as the base grows, so a plan built on a constant 8% overstates how soon the company reaches scale. Saying that limit turns a compounding exercise into a judgement about the plan in front of you.

With two rolls, keep a 4, 5 or 6 and the game is worth 4.25; with three, keep only a 5 or 6 on the first roll and it is worth about 4.67. Work backwards. The last roll is worth 3.5 on average, so on the roll before it keep anything above 3.5. That makes two rolls worth 4.25, which becomes the bar for the first of three rolls: only a 5 or 6 beats it.

Why work backwards from the last roll?

Deciding whether to take a flat today depends on what the next viewing is likely to offer, and that depends on whether there is another after it. The value of rolling again is the value of the game that remains, so solve the last stage first, where there is no choice, and carry its value back as the bar for the stage before. With one roll left she takes whatever comes, worth 3.5 on average. That 3.5 is what she gives up by keeping an earlier roll.

How do you compute each value?

With two rolls: keep 4, 5 or 6 on the first, each with chance one in six, and roll again on 1, 2 or 3, which is worth 3.5. Each stage's value is the average, over the six faces, of the better of keeping that face and rolling on. That gives (4 + 5 + 6) / 6 + (3/6) x 3.5 = 2.5 + 1.75 = 4.25. With three rolls the bar is 4.25: keep 5 or 6, worth (5 + 6) / 6, and roll on otherwise, worth (4/6) x 4.25. Together that is 4.67.

What does this say about shopping a real term sheet?

Two things. The bar for accepting rises with the number of chances left, so an early offer that is merely average should be declined if the process has room. But each extra roll adds less: the second roll adds 0.75, the third only about 0.42, and in real fundraising each roll costs weeks and the dice are not fair, since an offer turned down rarely comes back and a long process can scare off the next investor. The model is a way to think about the bar, not a reason to keep shopping.

25 lakh shares under full ratchet and about 11.1 lakh under broad-based weighted average. Full ratchet resets the conversion price to the new Rs 40, so the Rs 10 crore converts into 10 crore / 40 = 25 lakh shares. Weighted average resets it to 100 x (100 + 8) / (100 + 20) = Rs 90, so the investor gets 10 crore / 90 = 11.1 lakh. Either way, the extra shares dilute the other holders.

What does anti-dilution protection actually change?

A shop that promises to refund the difference if the price drops within a month is protecting its customer against a later, cheaper sale. Anti-dilution keeps the investor's rupees the same and lowers the price at which they convert into common shares, so the investor ends up with more shares. The investor put in Rs 10 crore at Rs 100. If the conversion price falls to P, it converts into 10 crore / P shares. The whole question is how far P falls, and the two formulas answer differently.

How does each formula set the new price?

Full ratchet is blunt: the conversion price becomes the new round's price, Rs 40, however few shares were sold there. Broad-based weighted average moves the price only in proportion to how much cheap stock was issued relative to the whole company. The Rs 8 crore raised would have bought 8 lakh shares at the old Rs 100; it actually bought 20 lakh. Against a fully diluted base of 1 crore shares, the price moves by 108 over 120, to Rs 90.

Who pays for the extra shares?

Everyone without the protection, mostly the founders and employees. With no protection the investor holds 8.3% after the round; under weighted average 9.2%; under full ratchet 18.5%. Full ratchet hands the investor 15 extra lakh shares against about 1.1 lakh under weighted average, and the other holders' 90 lakh shares fall from 75.0% to 66.7% of the company. That is why broad-based weighted average is the common market term and full ratchet a sign that a company had little negotiating power. New investors in the down round also dislike a ratchet, because it dilutes them too, and often ask for it to be waived as a condition of investing.

150 decks. Reduce everything to one unit first: 4 associates working 3 hours is 12 associate-hours, and 60 decks over 12 is 5 decks per associate-hour. Six associates for five hours is 30 associate-hours, so 30 x 5 = 150. As a check, scale the original: 60 x 6/4 x 5/3 is also 150. This assumes everyone works at the same steady rate.

Why reduce to a rate per associate-hour?

If 2 cooks make 40 rotis in an hour, one cook makes 20 an hour, and any kitchen size or shift length follows from that one number. Work problems become simple once you find the output of one worker in one unit of time, because the total is then that rate times workers times hours. Here 60 decks came from 4 x 3 = 12 associate-hours, so the rate is 5 decks per associate-hour. Everything else is multiplication.

How do you avoid the slip under time pressure?

Ask which way each change pushes the answer before you multiply. More associates and more hours both raise output, so both ratios must be greater than one: 6/4 and 5/3, never 4/6 or 3/5. The fast wrong answer, 90, applies the 6/4 and stops. On a timed test the danger is not hard arithmetic but a missed step, so a five-second check that the answer moved in the right direction by roughly the right amount is worth it: one and a half times the people for two thirds more time should give about two and a half times the output, and 60 x 2.5 is 150.

Where would this break down in a real deal team?

The model assumes every associate screens at the same steady pace and that output scales with hours. Real screening slows late in a long day, and a bigger team spends time coordinating and avoiding duplicate work. Saying so is unnecessary on the test itself, but it is the right instinct if the same question comes up in an interview: the arithmetic answer is the ceiling, and the real number is usually lower.

Post-money Rs 150 crore, pre-money Rs 120 crore, and existing holders keep 70%. The cheque sets the price: Rs 30 crore for 20% means the whole company after the round is 30 / 0.2 = Rs 150 crore, so the pre-money is Rs 120 crore. The pool is 10% of 150, Rs 15 crore, created inside the pre-money. Existing holders keep 70%, Rs 105 crore, which is their real price.

How does a cheque and a percentage set the valuation?

If Rs 30 buys a fifth of a pizza, the whole pizza costs Rs 150. Post-money is the cheque divided by the share it buys, and pre-money is post-money less the cheque. Rs 30 crore for 20% gives Rs 150 crore post-money and Rs 120 crore pre-money. The founder's cap on the raise works here as a cap on the valuation too: at 20%, raising only Rs 24 crore would set the post-money at Rs 120 crore.

Where does the pool come from, and why does it matter?

The lead wants the pool created before its money arrives, so new shares for the pool are carved out of the pre-money. A pool inside the pre-money dilutes only the existing holders, so their shares are really valued at the pre-money less the pool: 120 - 15 = Rs 105 crore. The headline pre-money of Rs 120 crore is true for the company and flattering for the founders, which is why experienced founders negotiate the pool size as hard as the valuation.

One assumption to state: there is no existing unallocated pool. If there were, it would count towards the 10%, the new carve-out would be smaller, and the existing holders would keep more.

About 12% for 20 bets and 59% for 5 bets. In both strategies a single hit returns at least the whole fund: 20x on 5% is 1x, and 20x on 20% is 4x. So a strategy loses money only when every bet misses. Twenty bets all miss with chance 0.9 to the twentieth, 12.2%; five bets all miss with chance 0.9 to the fifth, 59.0%. Expected value is 2x either way.

When does each strategy lose money?

Buying one lottery ticket in each of twenty draws and buying four tickets in each of five draws can cost the same and win the same on average, yet the second leaves you empty-handed far more often. Find the outcome that loses money first: here one hit already pays back the whole fund in both strategies, so losing means getting no hits at all. With 20 bets each worth 5% of the fund, a hit returns 20 x 5% = 1x the fund, exactly the money back. With 5 bets of 20%, a hit returns 4x. Either way, zero hits is the only losing outcome.

If the expected value is the same, why does it matter?

Because a fund's investors live through one draw, not the average of many. Concentration leaves the expected multiple at 2x but stretches the outcomes: B loses money 59% of the time, yet also returns 4x or more 41% of the time, against 13% for A. A returns at least 3x 32% of the time and rarely does spectacularly. Neither is better in the abstract; the choice depends on how much the fund's investors can bear a blank.

What does the model leave out?

It assumes the bets are independent and identical. In practice a concentrated fund argues that it can pick better and support each company more, raising p; a diversified one argues that nobody can pick reliably at seed. Correlation also matters: if all twenty companies depend on the same funding climate, the 12% understates how often A has a blank decade. Say the result, then say which assumption you would test first.

Applied to every share, the offer implies Rs 720 crore, but it prices common shares, not the company. The Rs 1,200 crore figure was a price for preferred shares, which are paid back first in a sale and carry protections. Common is paid last, is illiquid and usually needs company consent to sell. A buyer discounts for all of that. Neither Rs 1,200 crore nor Rs 720 crore is the company's value.

Why do the two prices differ for the same company?

A first-class and a general ticket on the same train cost different amounts, but the fare gap says nothing about the train's speed. Preferred and common are different securities in the same company: preferred is paid back before common in a sale, so a rupee of preferred is worth more than a rupee of common whenever a weak exit is possible. Multiplying the preferred price by every share gives the headline Rs 1,200 crore; multiplying the common offer, 60% of that price, by every share gives Rs 720 crore. Both are prices of a security scaled up, not a measure of what the business is worth.

Where does the discount come from?

Take an illustration: preferred holders own 40% and are owed Rs 400 crore first in any sale. In a Rs 400 crore sale they take everything and common gets nothing; only above Rs 1,000 crore, where 40% of the sale beats Rs 400 crore, do the two classes share pro rata. Common holds the first loss in the bad outcomes. On top of that come illiquidity, the company's right to block or match a sale, and the lack of information rights. Each adds to the discount, and none says the company is worth less than at the last round.

When would a secondary price be a real markdown signal?

When the discount is to the same security, or much larger than the structure explains. A buyer offering a deep discount for preferred shares from the latest round is a stronger signal, because the security is identical to the one that set the headline. Ask which class is being sold, how big the preference stack is relative to the likely exit range, and how much stock is on offer; a small, forced sale tells you about the seller's need for cash as much as the company.

1.25 on the headline numbers, and 0.75 once the one-off services are removed. The magic number is the quarter's revenue increase, annualised, divided by the previous quarter's sales and marketing spend. Headline: Rs 5 crore x 4 = Rs 20 crore over Rs 16 crore is 1.25. But only Rs 3 crore will recur; Rs 12 crore over Rs 16 crore is 0.75. Two crore of services flattered the efficiency by two thirds.

What is the magic number measuring?

A coaching centre that spends Rs 1.6 lakh on advertising one month and signs students paying an extra Rs 50,000 a month in fees has bought Rs 6 lakh a year of fees for Rs 1.6 lakh. The magic number is the annualised increase in recurring revenue divided by the sales and marketing spend that produced it, usually the previous quarter's. It is a quick read on how efficiently a company turns sales spend into new revenue. Annualise because the spend buys a full year of revenue from each new customer, not just one quarter.

Why does the one-off revenue matter so much?

Because the measure is built on a small difference. A Rs 2 crore swing in a Rs 5 crore increase is 40% of the numerator, so a one-off item moves the magic number far more than it moves revenue. Read as recurring revenue, a magic number of 1 means a year of the new revenue repays the quarter's sales spend; 1.25 suggests about 9.6 months, but the true 0.75 means about 16 months, and that is before the cost of serving the customers. Some investors multiply by gross margin for that reason: at a 75% margin the recurring figure becomes 0.56.

Ask three things before trusting the number: is the revenue recurring, is the spend fully counted, including sales salaries and commissions, and is one quarter representative. A single strong quarter can follow a deal that slipped from the quarter before, so investors look at a rolling average across several quarters.