Venture Capital puzzles, solved step by step

Without recycling the fund invests Rs 82.5 crore and needs 3.03x; with full recycling it invests Rs 97.5 crore and needs about 2.72x. Investors want Rs 250 crore back. Without recycling, Rs 82.5 crore must produce all of it. With recycling, Rs 97.5 crore must produce Rs 265 crore, because Rs 15 crore of proceeds was put back to work rather than paid out. Carry is ignored throughout.

Why can a fund not invest all of its commitments?

A family sets aside Rs 1,00,000 for a wedding and the planner's fee is Rs 17,500 of it: only Rs 82,500 buys food and flowers. Management fees are paid out of the same commitments that fund the investments, so a fund that never recycles invests less than its headline size. Rs 17.5 crore could be 2% a year for five years and 1.5% for five more; schedules vary, but the arithmetic is the same. To turn Rs 100 crore into Rs 250 crore for investors, Rs 82.5 crore has to earn 250 / 82.5 = 3.03x.

How does recycling change the multiple, and why is it not 250 over 97.5?

RecyclingReinvesting proceeds from early exits into new companies instead of distributing them, usually capped at about the amount of fees paid. lets the fund put Rs 15 crore of early exit money back into new companies, so invested capital reaches Rs 97.5 crore. But that Rs 15 crore was proceeds the portfolio had already earned and did not pay out, so the portfolio must produce Rs 250 crore for investors plus the Rs 15 crore it recycled. Gross proceeds of 265 on cost of 97.5 is 2.72x: still well below 3.03x, but not the 2.56x a quick division gives.

Say the limit too. Recycling delays money investors would have received early, so it can lift the multiple while lowering the IRR, and it works only if the fund finds good companies for the recycled rupees. It is a tool for getting fees back to work, not free return.

About 26%; the exact figure is 25.7%. Two anchors bracket it: 2x in 3 years is 26.0% and 3x in 5 years is 24.6%. For a sharper figure, take ln 2.5, about 0.92, divide by 4 to get 0.23, and add half its square: about 25.6%. Check by squaring twice: 1.26 squared is 1.59, and 1.59 squared is about 2.5.

Which anchors should you carry into the room?

You judge a distance on a road by the milestones, not by pacing it out. Carry a few multiple and years pairs and interpolate between them rather than calculating from scratch. 2x in 3 years is 26.0%, 3x in 5 years is 24.6%, 2x in 4 years is 18.9% and 3x in 4 years is 31.6%. The target, 2.5x in 4 years, lies between the last two and is bracketed by the first two.

How do you land on 26% out loud?

Route one: interpolate on a log scale. 2.5 sits about 55% of the way from 2 to 3 in log terms, so the IRR sits about 55% of the way from 18.9% to 31.6%, near 26%. Route two uses the log directly. The continuous growth rate is the log of the multiple divided by the years, and adding half its square converts it to an annual rate: 0.23 plus 0.026 is about 25.6%.

Then check. Squaring twice is the fastest test of a four year rate: 1.26 squared is 1.59, and 1.59 squared is 2.52, close enough to 2.5. The check matters more than the method, because it catches a slip in either route in five seconds.

Split the envelopes into three groups of three and weigh two of the groups against each other. The heavier pan holds the cheque; if they balance, it is in the group left aside. Then weigh two envelopes from that group: the heavier one is it, and a balance means the third. Each weighing has three outcomes, so two weighings tell apart 3 x 3 = 9 envelopes.

Why is a balance scale worth three answers, not two?

A traffic signal carries three messages, not two, because amber counts. A balance scale is the same: left heavy, right heavy, or level. The balanced outcome is information too, so the best weighing splits the suspects into three equal groups, two on the scale and one off it. Most people instinctively halve, as they would with a yes or no question, and halving throws away the third answer.

How do you prove two weighings are enough and one is not?

Count end points. With k weighings and three outcomes each, the scale can produce at most 3 to the power k different results, and each envelope needs its own result. One weighing gives three results, too few for nine envelopes. Two give nine, exactly enough, and the tree in the figure shows a plan that uses all nine. Three weighings would handle 27 envelopes the same way.

What does the interviewer want to hear beyond the procedure?

Say the counting argument, not just the steps. It shows you can tell when a plan is optimal rather than merely working. The habit carries over to diligence: a good question is one whose every answer, including the dull one, changes what you do next. A reference call where only a glowing answer would move you is a weighing that wastes the balanced outcome. Then state the assumption the puzzle rests on: you know the odd envelope is heavier, not merely different.

2.25x your money, about 31% a year over three years. ARR grew 3 x 2 x 1.5, nine times. The multiple fell from 40x to 10x, a quarter of where it started. Value is ARR times the multiple, so it grew 9 x 0.25 = 2.25 times. The company did nine times better as a business, and the price of that business fell by three quarters along the way.

Why do revenue growth and the multiple multiply?

A shopkeeper who sells three times as many mangoes at half the price takes in one and a half times the money, not three. Valuation is ARR times the multiple, so the change in value is the growth in ARR times the change in the multiple. Entry: ARR of 1 at 40x is a value of 40. Exit: ARR of 9 at 10x is 90. Ninety over forty is 2.25x. Compound the growth rates first: 3 x 2 x 1.5 is 9, not 3 + 2 + 1.5.

Is 2.25x in three years a good outcome?

It is about 31% a year, respectable, but look at where it came from. Nine times the business produced barely two and a quarter times the money, because the entry price assumed growth that the exit price no longer rewarded. The break-even exit multiple is 40 / 9, about 4.4x ARR: below that, a company that grew nine times would still lose money. That is the risk of paying a high multiple, and it is why growth investors ask what multiple they need at exit before they ask how fast the company will grow.

What did the puzzle leave out on purpose?

Dilution: any round in those three years shrinks the stake, so the real multiple is lower. Preferences: in a weak exit a preferred holder may be paid ahead of common, which can lift the investor's multiple above the common one. The path of the multiple: the figure draws an even decline, but only the two ends are given, and the multiple of money depends only on them. State these and the answer becomes an investor's view rather than an arithmetic result.

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.

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.

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.

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.