Venture Capital puzzles, solved step by step

On these numbers, no: waiting is worth Rs 79 crore on average against a certain Rs 80 crore. Weight each outcome by its chance: half of 110 is 55, three tenths of 80 is 24, and the no-deal branch adds nothing. Waiting loses a crore of expected value and adds a one in five chance of having no round at all, which for a company that needs the money can be far worse than zero.

How do you compare a sure thing with a gamble?

A shop offers you Rs 800 for your old phone today, or you can try an online sale that might fetch Rs 1,100, might fetch Rs 800, or might not sell at all. Put a number on the gamble by weighting each outcome by its chance and adding, then compare that expected value with the sure offer. Here waiting gives 0.5 x 110 + 0.3 x 80 + 0.2 x 0, which is Rs 79 crore of pre-money on average, against Rs 80 crore she can sign today.

Why is the expected value not the whole answer?

Because the outcomes are not equally painful. A one crore gap in expected value understates the case against waiting, because the 20% branch is not a lower valuation but no round at all. A company with six months of cash that misses a round may have to cut staff or raise on much worse terms later. Even if waiting were worth slightly more on average, a founder who cannot survive the bad branch should still sign. Turn it round to show the margin: holding the no-deal chance at 20%, waiting only beats signing if the chance of Rs 110 crore rises above 53.3%; holding the 50% upside, the no-deal chance must fall below 18.75%.

What would you ask before trusting the probabilities?

Where the 50% comes from. Founders tend to overrate the chance of a better offer because the meetings that went well are the ones they remember. Ask how many investors are past a partner meeting, and whether the current lead would return at the same price after being turned down, which this puzzle quietly assumes. A pre-money figure is also not the founder's wealth: the stake she keeps depends on how much she raises as well.

One seed bet has a 4% chance, and it takes 57 independent bets to have a 90% chance of at least one such exit. Multiply the stage rates: 0.4 x 0.5 x 0.5 x 0.4 = 0.04. For a portfolio, find the chance that every bet misses: 0.96 to the power n. It first falls below 10% at n = 57, where the chance of at least one exit is 90.2%. Correlated bets would need more.

How do the stage rates turn into one chance?

A cricket team must win the quarter final, the semi final and the final; its chance of the trophy is the product of the three, not the average. Each financing stage is a gate the company must pass, so the chance of reaching the end is the product of the conversion rates. 40% to Series A, then half, then half, then 40% to the large exit: 0.4 x 0.5 x 0.5 x 0.4 is 0.04. Out of 100 seed companies, 4 get there.

Why can you not just add the chances across bets?

Adding 4% twenty five times gives 100%, which would mean 25 bets make a hit certain. For at least one success, work with the opposite: the chance that every bet misses, which multiplies. Each bet misses with probability 0.96, so n independent bets all miss with probability 0.96 to the power n. Twenty five bets all miss 36% of the time, so the chance of at least one hit is only 64%. To push the miss chance under 10%, take logs: n is at least ln 0.1 over ln 0.96, which is 56.4, so 57 bets. At 56 bets the chance is still 89.8%.

What does this tell you about how seed funds are built?

It explains why seed portfolios hold dozens of companies rather than a handful. With a 4% hit rate, a portfolio of 25 companies misses entirely about a third of the time, however good the picks look one by one. Then say the limit: the bets are not independent. Companies from the same year share a funding climate and an exit window, so a weak market lowers every stage rate at once. Correlation makes the true number of bets needed higher than 57, and the 4% itself is an assumption to test against a fund's own record.

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 30.6% a year, not 36%; and 10% a year needs monthly churn of about 0.87%. Each month keeps 97% of the customers still there, so a year keeps 0.97 to the power 12, which is 69.4%. Going the other way, 90% retention a year means a monthly retention of 0.9 to the power one twelfth, about 99.13%, so monthly churn of 0.87%. This counts one starting group of customers and ignores new sign-ups.

Why is it not simply twelve times 3%?

A water tank that loses 3% of what is in it every hour loses less each hour, because there is less water to lose. Monthly churn is a share of the customers still there, so it compounds on a shrinking base: the yearly survival rate is the monthly survival rate raised to the twelfth power. After six months 0.97 to the sixth leaves 83.3%, and after twelve 69.4%. Twelve times 3% would be right only if each month lost 3% of the original group, which nobody measures.

How do you get the monthly rate for 10% a year without a calculator?

Dividing 10% by 12 gives 0.83%, and it is close because small rates compound gently. The exact figure is slightly higher, 0.87%, because the monthly losses come out of a shrinking base and so each month must lose a touch more to reach 10% in total. Say the division as a first estimate, then the correction. For large rates the gap grows: 3% a month gives 30.6% a year, against 36% by multiplication.

Why does an investor care which convention a company used?

Because companies quote whichever looks better. A business that reports 3% monthly churn and 30.6% annual churn is internally consistent; one that reports 3% monthly and 25% annual is either measuring different customers or counting new sign-ups against losses. Ask what base the churn is measured on and whether it counts logos or revenue, since revenue churn can be much lower when the customers who leave are the small ones.

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.