Venture Capital puzzles, solved step by step

Faster than 33.3% a year; at exactly one third her stake's value stands still. Each round leaves her 75% of her previous ownership, so her stake's value changes by 0.75 times one plus the valuation growth. That equals 1 when growth is 1 over 0.75 minus 1, a third. At 20% growth her stake loses 10% of its value every year, even while the company grows.

Why is the break-even growth a third and not a quarter?

A shop sells an item at a 25% discount and wants its revenue back to where it was: it needs a third more units, not a quarter more, because 75 x 1.333 is 100. After giving up a quarter, she holds three quarters, and three quarters of anything must grow by a third to get back to the whole. This is the same asymmetry as recovering from a loss: the recovery is measured on a smaller base, so it needs a bigger percentage.

What happens at 20% and at 50% growth?

At 20%, each year multiplies her stake's value by 0.75 x 1.2 = 0.9. After five rounds the company is worth 2.5 times as much, and her stake is worth only 59% of what it was. At 50%, each year multiplies it by 1.125, and five rounds take it to 1.80 times. Same dilution, opposite outcomes, decided entirely by whether growth clears one third.

Why would an investor ask this?

Because it is the test every round must pass for existing holders. With a quarter sold, post-money growth of a third is the same as the new round's pre-money equalling the last round's post-money. A round only makes existing holders richer if the new pre-money beats the old post-money, which is what an up round means. Say that link and the puzzle becomes a point about pricing, not just arithmetic.

Let the first 7 founders go, then back the first one who is better than all of them; you back the single best about 38% of the time. The first 7 set the bar. You win when the best founder comes after them and nobody earlier than the best beats the bar first. Across all cut-offs, 7 gives the highest chance, 38.4%, against 5% for picking at random.

Why let good founders pass at all?

Renting a flat in a tight market works the same way: the first few viewings teach you what a good flat looks like, and you commit to the next one that beats them. The early meetings are the price of learning where the bar sits, and the rule trades a small chance that the best founder is among them for a much better read on everyone after. Commit too early and you have no bar; wait too long and the best has probably already gone.

How do you work out the chance for a given cut-off?

Say you let r founders pass. Suppose the best founder is at position i, after the first r. You back that founder only if nobody between r and i beats the bar first, which happens exactly when the best of the first i minus 1 founders sits inside the first r, a chance of r over i minus 1. Each position is equally likely to hold the best, one in 20, so add up across positions.

The curve is flat near the top: letting 6 pass gives 37.9% and 8 gives 38.2%. The general rule is to let about n/e go, 37% of the field, and win about 37% of the time; for 20 founders that is 7.4, which rounds to 7.

Where does the model stop describing real deal flow?

It assumes you only value the single best, can only rank founders against each other, see them in random order and never get a second chance. A real investor is happy with a top three founder, has an absolute sense of quality from past deals, and can sometimes return to a founder a week later. Each of those argues for committing earlier. Say that limit after the number; it shows you know what the model is for.

The cap bites at an exit of Rs 220 crore and converting takes over above Rs 300 crore; in between, the investor's payout is stuck at Rs 60 crore. Below Rs 20 crore the investor takes everything. Above it, the investor takes Rs 20 crore plus 20% of the rest until the total reaches 3x, Rs 60 crore. Only when 20% of the whole exit exceeds Rs 60 crore does converting pay more.

What does a capped participating preference mean in plain words?

Picture a relative who lends to your shop on the terms: my money back first, then a fifth of whatever is left, but I will never take more than three times what I put in; if a fifth of the whole shop is ever worth more, I will take that instead. The investor first takes back Rs 20 crore, then shares 20% of the remainder, until the total reaches the cap of 3x, Rs 60 crore. Above the cap, the investor can give up the preference and convert to plain common shares. Here the cap covers the total return, preference included, which is the usual reading; say so before you calculate.

Where exactly do the two kinks sit?

Set each piece equal to the cap. Participation reaches Rs 60 crore when 20 plus 20% of the exit less 20 equals 60, which is an exit of Rs 220 crore; converting beats Rs 60 crore when 20% of the exit exceeds it, above Rs 300 crore. At a Rs 100 crore exit the investor receives Rs 36 crore against Rs 20 crore as plain common; at Rs 400 crore it receives Rs 80 crore, exactly its 20%.

Why does the flat band matter at the negotiating table?

Inside the band the investor gains nothing from a better price, while every extra rupee goes to the common holders. Between Rs 220 crore and Rs 300 crore the investor is indifferent to the price, so a quick, certain sale at the bottom of the band can suit it more than a long push for the top. A founder who knows where the band sits knows when interests diverge, and can plan the board conversation before a buyer appears.

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.

Offer B leaves the founders 66.2% against 58.3% under Offer A, and its effective pre-money is Rs 69.5 crore against Rs 67.0 crore. A pool created inside the pre-money is paid for by the existing holders. Offer A's 20% pool is Rs 23 crore of a Rs 115 crore post-money, so the founders' shares are really valued at Rs 67 crore. Offer B's 10% pool costs only Rs 10.5 crore.

Why does a pool inside the pre-money lower the real price?

A flat sold for Rs 90 lakh, where the seller must also leave Rs 23 lakh of new furniture for the buyer, really sold for Rs 67 lakh. A pool created inside the pre-money is paid for entirely by the existing holders, so the effective pre-money is the headline pre-money minus the pool's value. Offer A: post-money is 90 + 25 = Rs 115 crore, the pool is 20% of that, Rs 23 crore, and the founders' shares are worth 90 - 23 = Rs 67 crore. Offer B: post-money Rs 105 crore, pool Rs 10.5 crore, effective pre-money Rs 69.5 crore.

What share do the founders keep under each offer?

Take the investor's share and the pool off the whole. The founders keep one minus the investor's share minus the pool: 58.3% under Offer A and 66.2% under Offer B. The investor gets a smaller share under A, 21.7% against 23.8%, which is why the headline looks better; the extra 10 points of pool more than cancel it. This assumes the founders own everything before the round and there is no existing pool.

What would you push back on?

The pool size, not the headline. A pool should be sized to the hires planned before the next round, not to a round number. If the founder can show that 10% covers the hiring plan, Offer A's pool falls to Rs 11.5 crore, its effective pre-money rises to Rs 78.5 crore and the founders keep 68.3%, better than either offer as written.

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.

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.