Venture Capital puzzles, solved step by step

About 61%, and for an 80% chance it must start raising after month 2. The chance of failing all nine months is 0.9 to the power 9, about 39%, so it closes about 61% of the time. An 80% chance needs 0.9 to the power n below 20%, which first happens at 16 months (81.5%). Eighteen months of cash less 16 means raising from month 2.

Why not just add 10% a month?

Think of a friend who picks up the phone one time in ten. Call nine times and you will not reach them 90% of the time, because every call after the one they answer is wasted. With repeated independent tries, count the chance of failing every time and subtract it from one; adding the chances double counts the months after a close. Adding 10% a month would give 100% at ten months and more than 100% after that, which is the tell that the method is wrong.

How do you find the month it must start?

Set 0.9 to the power n below 0.2 and solve. Taking logs, n is at least ln 0.2 over ln 0.9, about 15.3, so round up to 16 whole months. Fifteen months gives 79.4%, just short of 80%, and sixteen gives 81.5%. With 18 months of cash, sixteen months of raising means starting after month 2. Doubling the time spent raising only lifts the odds from 61% to about 81%, because each extra month adds 10% of a shrinking remainder.

What would you say to a founder about this?

The model is simple and its lesson is not: fundraising odds climb slowly, so a company that waits until half its runway is gone has already given up a large share of its chances. The limitation is worth naming. Real months are not independent; a round that has not closed in six months usually gets harder, not equally likely, so the true curve is flatter still.

Committing now earns an expected Rs 2 crore; staging earns Rs 7.6 crore, so the option to stop is worth Rs 5.6 crore. Committing pays Rs 100 crore only 12% of the time, worth Rs 12 crore, against Rs 10 crore invested. Staging risks Rs 2 crore first and adds Rs 8 crore only after the milestone, so in the 70% of cases that fail it saves Rs 8 crore: 0.7 x 8 is Rs 5.6 crore.

Why is waiting worth anything if the payout is the same?

Think of booking a wedding venue with a small refundable deposit rather than paying in full a year out. If the engagement is called off, you lose the deposit, not the whole fee. Staging a cheque does not change what you win; it changes what you lose in the worlds where things go wrong, because you stop paying once you learn the milestone was missed. Here the milestone fails 70% of the time, and each of those times the staged route has spent Rs 2 crore instead of Rs 10 crore.

How do you work each route's expected profit?

Committing now: the company pays Rs 100 crore only if it passes both hurdles, 0.3 x 0.4, which is 12% of the time. That is worth Rs 12 crore against Rs 10 crore paid, an expected profit of Rs 2 crore. Staging: pay Rs 2 crore for sure, then in the 30% of worlds that hit the milestone, pay Rs 8 crore for a 40% shot at Rs 100 crore, which is worth Rs 32 crore at that point. Minus 2, plus 0.3 x 32, gives Rs 7.6 crore.

What would a founder say about this, and what is the limit?

A founder will rarely give you the same price for both tranches, because staging moves risk onto the company. The Rs 5.6 crore is the most it is worth paying for the right to stage, in a higher second-tranche price or a smaller stake. Staging also has costs the model ignores: a company that has to hit a milestone to get its money may be run for the milestone rather than for the business, and the uncertainty can scare off other investors.

About 45%, below a coin flip. Exactly two pilots convert in three ways: pilots 1 and 2 only, 0.6 x 0.5 x 0.7 = 0.21; pilots 1 and 3 only, 0.6 x 0.5 x 0.3 = 0.09; pilots 2 and 3 only, 0.4 x 0.5 x 0.3 = 0.06. All three convert with probability 0.09. Adding the four gives 0.45.

How do you make sure you count every way to survive?

Think of a cricket team that needs two of its three openers to score fifty. You would list who scores and who does not, because each opener's failure matters to the case where the other two succeed. List the outcomes that satisfy the condition, write each as a product of the chance each pilot converts or does not, and add them. At least two out of three means exactly two, which can happen three ways, or all three.

Why do the quick shortcuts fail?

Multiplying the two best pilots, 0.6 x 0.5, gives 30% and forgets that pilots 1 and 3, or 2 and 3, also keep the company alive. Adding the three pair products, 0.30 + 0.18 + 0.15, gives 63% and makes the opposite error. Each pair product already includes the case where the third pilot converts too, so the all-three outcome is counted three times and must be taken off twice: 0.63 minus 2 x 0.09 is 0.45. That correction gives the same answer as the list, which is a useful check in the room.

What would you tell the investment committee?

That three pilots which each sound promising still leave the company more likely to fail than survive. The answer also depends heavily on independence: if the pilots share a buyer type or a product gap, they tend to succeed or fail together, and the real survival chance could be noticeably higher or lower. Ask what the pilots have in common before trusting the 45%.

About 23.7% after five years and about 5.6% after ten. Each year the company survives with probability 0.75, and the years are independent, so the chances multiply: 0.75 to the fifth is 0.237. Ten years is that number squared, 0.237 x 0.237, about 0.056. Half the companies are gone before the end of year three.

Why do you multiply survival rather than add up the deaths?

Think of a batch of 100 phones, each with a one in four chance of breaking in every year you own it. In year one about 25 break and 75 are left. In year two the 25% applies to those 75, not to the original 100, so about 19 break and 56 are left. A yearly death rate only acts on the companies still alive, so survival shrinks by the same factor each year instead of falling by the same amount. Adding 25% five times gives 125%, and a probability above 100% is the sign you are using the wrong operation.

How do you get 0.75 to the fifth in your head?

Build it from squares. 0.75 squared is 0.5625, call it 0.56. Squared again, 0.56 x 0.56 is about 0.316, which is year four. One more 0.75 takes 0.316 to 0.237 for year five. Year ten is year five squared, so once you have 0.237 the second answer is one step: 0.237 x 0.237 is about 0.056. Saying the squaring route out loud shows the interviewer a method rather than a memorised number.

What does a flat death rate mean for a seed portfolio?

Two more numbers fall out of the same 25%. The average company lives four years, one over the yearly death rate, and half are gone within about 2.4 years, where 0.75 to the n crosses one half. A seed fund of 30 companies with this death rate expects only about 1.7 of them alive at year ten, which is why seed funds are sized for most companies failing. Say the limitation too: real death rates are not flat. They are highest in the first two years and fall for companies that find a market, so a flat 25% overstates late deaths and understates early ones.

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.

36% if independent, 20% if perfectly correlated, and 30% if the chance both fail is 10%. In each case the chance of at least one failure is 20% plus 20% minus the chance both fail. Independent failures overlap 0.2 x 0.2 = 4%, giving 36%. Perfectly correlated failures overlap completely, 20%, giving 20%. With a 10% overlap the answer is 30%. Correlation makes any failure less likely but both failing far more likely.

Why can you not just add the two chances?

Think of two friends who each forget your birthday one year in five. If you add the chances you get two in five, but that counts the years when both forget twice, once for each friend. The chance of at least one event is the sum of the two chances minus the chance both happen, so the overlap decides the answer. For two independent events the overlap is the product, 0.2 x 0.2 = 4%, and the answer is 36%. The complement route checks it: neither fails with chance 0.8 x 0.8 = 64%.

What does correlation do to a venture portfolio?

It trades many small surprises for fewer, larger ones. As failures become more correlated, the chance of at least one failure falls from 36% to 20%, but the chance both fail rises from 4% to 20%, five times higher. The 10% case corresponds to a correlation of about 0.37 between the two failure events. A fund full of companies selling to the same customers, or depending on the same funding market, looks diversified by count but behaves like fewer, bigger bets.

Why would an LP care about this more than a single fund manager?

An LP holding several funds cares about how many bad outcomes can arrive at once. Correlated failures are what turn an ordinary bad year into a vintage where most companies struggle together, and an average-case model that treats failures as independent understates that risk. The limitation is that correlation is hard to measure for private companies with few data points; in practice investors judge it from shared exposures, such as the same sector, the same customers or the same reliance on new funding, rather than from a computed number.

Rs 8 crore of expected profit against Rs 6.5 crore, so the choice is worth Rs 1.5 crore. A good round turns Rs 5 crore into Rs 25 crore, a Rs 20 crore profit; a bad one leaves Rs 2.5 crore, a Rs 2.5 crore loss. Forced to invest, you expect 0.4 x 20 minus 0.6 x 2.5, or Rs 6.5 crore. With the right you pass on bad rounds, so you expect 0.4 x 20, or Rs 8 crore. The Rs 1.5 crore gap is the price of the bad branch you no longer have to take.

Why is a right worth more than an obligation to do the same thing?

A season ticket that lets you skip any match you like is worth more than one that forces you to sit through the washed-out ones, even though the matches are the same. A right is worth exactly the losses it lets you refuse, so its value is the expected loss on the branches you would walk away from. Here the bad branch costs Rs 2.5 crore 60% of the time, an expected Rs 1.5 crore, and the right lets you skip all of it. The good branch is the same Rs 20 crore profit either way, so it adds nothing to the difference.

Where does this simple answer overstate the right in real life?

The question lets the round's pricing tell you for certain whether it is good. In practice the signal is noisy, so a pro rata right is worth less than Rs 1.5 crore here: you will sometimes pass on a good round and sometimes take a bad one. If your read were no better than a coin, the choice would be worth nothing, because you could not tell the branches apart. Two more limits: a hot round may leave you no allocation even with the right, and a fund must hold reserves to exercise it, and that cash earns nothing while it waits.

This is why seed funds fight for pro rata rights and why the best later investors try to cut them back. The value sits with whoever gets to look before deciding, and it grows with how far apart the good and bad outcomes are. Say that last point in the room: a wider spread between good and bad rounds makes the right worth more, exactly as a wider spread makes any option worth more.

About 31%, not 80%. Picture 1,000 companies. 100 are good and your screen passes 80 of them. 900 are bad and it passes 20% of those, 180 companies. So 260 pass, and only 80 of them are good: 80 divided by 260 is 30.8%. The screen raises the odds from 10% to about 31%, which is useful, but most companies that pass are still bad because bad ones are so common.

Why is the answer so far below 80%?

A smoke alarm that rings for every real fire and for one in five burnt toasts will mostly ring for toast, because toast is far more common than fire. When the thing you are looking for is rare, even a small false alarm rate applied to the large crowd of negatives produces more false passes than true ones. Here 20% of 900 bad companies is 180, against 80% of 100 good companies, 80. The screen's 80% describes how it treats good companies; the question asks what a pass tells you, which also depends on how many bad companies get in.

What would actually improve the screen?

Run the alternatives. Halving the false pass rate to 10% lifts the answer to 80 out of 170, about 47%, while raising the hit rate from 80% to 95% only takes it to 95 out of 275, about 35%. When good companies are rare, cutting false passes matters more than catching every good one. A second, independent check helps the same way: if a passing company goes through another screen with the same rates, the base rate is now 31%, and a second pass takes it to about 64%.

The limit is that the two checks must be independent; a second partner who looks at the same deck with the same biases is not a second screen. And the 10% base rate is an assumption about your deal flow; a fund with better sourcing starts from a higher base and every pass means more.

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.

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.