Venture Capital puzzles, solved step by step

Puzzles: 100
Traced to a firm: 7
Topics: 12
Hard: 30

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015You can commit Rs 10 crore to a startup now, or put in Rs 2 crore now and the other Rs 8 crore only if it hits a milestone, which it does 30% of the time. Once past the milestone, the company has a 40% chance of paying you Rs 100 crore for the full Rs 10 crore, and zero otherwise; if the milestone is missed, it pays nothing. What is the expected profit of each route, and what is the option to stop worth?Probability and expected valueHardSeed and early-stage VCDeep tech VC
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What is the option to stop after Rs 2 crore worth?
Show the worked solution
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.
The relationship
$EV_1 = 0.3 \times 0.4 \times 100 - 10 = 2 \qquad EV_2 = -2 + 0.3\,(0.4 \times 100 - 8) = 7.6$
0.3 chance the milestone is hit
0.4 chance of the Rs 100 crore payout once past the milestone
2, 8 the first and second tranches, Rs crore
What it says in wordsAverage every path's profit by its probability; the staged route only pays the Rs 8 crore on the paths where the milestone was hit.
Committing Rs 10 crore up front earns an expected Rs 2 crore, while paying Rs 2 crore first and Rs 8 crore only after the milestone earns Rs 7.6 crore, so the right to stop is worth Rs 5.6 crore, the Rs 8 crore saved in the 70% of cases that fail.
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.
Where candidates lose it
The common slip is to say both routes are the same because the total cheque and the payout are the same. The interviewer is testing whether you see that information arrives between the tranches, and that you can act on it.
The second loss is getting Rs 7.6 crore and not explaining it. The cleanest check is 0.7 x Rs 8 crore: the option is worth exactly the money you no longer risk in the failure cases.
What the interviewer asks next
The founder insists the second tranche is priced 50% higher. Is staging still better?
At what milestone probability does the option to stop become worthless?
Why do deep tech investors use milestone tranches more than consumer investors?
038A startup has 3 months of cash. Each month it either signs a contract worth one extra month of runway, with probability 0.4, or burns a month, with probability 0.6. What is the chance it reaches 6 months of cash, the level at which it can raise, before it reaches zero?Probability and expected valueHardSeed and early-stage VCIndia VC
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Starting halfway between zero and the raise, what is the chance of reaching 6 months first?
Show the worked solution
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.
With each month 60/40 against it, the company's chance of reaching six months before zero is 22.9% from a start of three, far below the 50% a fair walk would give from the same 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.
The relationship
$h(i) = \frac{1 - r^{i}}{1 - r^{N}}, \quad r = \frac{q}{p} = \frac{0.6}{0.4} = 1.5 \qquad h(3) = \frac{1 - 3.375}{1 - 11.39} = \frac{2.375}{10.39} \approx 0.229$
h(i) the chance of reaching the raise from i months of cash
p, q the chances of a good month, 0.4, and a bad one, 0.6
N the runway at which the company can raise, 6 months
What it says in wordsThe chance of reaching the top wall first depends on the start, the distance to each wall and how tilted each step is.
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.
Where candidates lose it
The common loss is answering 50% because the company starts in the middle, or 40% because that is the chance of a good month. Neither accounts for the walk ending at whichever wall comes first after many tilted steps.
The second loss is computing only the straight path, 0.4 cubed or 6.4%, and missing all the paths that wander down and back up. The recurrence, or the gambler's ruin formula, counts every path at once.
What the interviewer asks next
What is the chance of reaching 6 months if the odds of a good month rise to 0.5?
With the same 60/40 odds, from what starting runway does the company have a better than even chance of reaching 6 months?
If each contract added two months of runway instead of one, how would you set the problem up?
082You will meet 20 founders, one at a time, in random order, and can back only one. You must decide yes or no at the end of each meeting and cannot go back to anyone you passed. What rule maximises the chance that you back the single best founder, and what is that chance?Probability and expected valueHardSeed and early-stage VCMulti-stage VC
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Roughly how often does the best possible rule land the single best of the 20?
Show the worked solution
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.
The chance of backing the best of 20 founders rises from 5% if you back the first one to a peak of 38.4% if you let 7 pass, then falls slowly, to 22.2% if you let 15 pass, staying near the 1/e limit of 36.8% across a wide middle range.
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 relationship
$P(r) = \frac{r}{n}\sum_{i=r+1}^{n}\frac{1}{i-1} \qquad P(7) = \frac{7}{20}\left(\tfrac{1}{7} + \tfrac{1}{8} + \dots + \tfrac{1}{19}\right) \approx 0.384$
n founders in total, 20
r founders you let pass to set the bar
1/(i-1) weight for the best founder sitting at position i
What it says in wordsAverage, over every position the best founder could hold, the chance that nobody earlier stole the pick.
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.
Where candidates lose it
Most candidates either say 1 in 20, treating the choice as blind, or propose meeting everyone and then choosing, which the rules forbid. The interviewer is testing whether you see that watching without committing is itself a strategy.
The second loss is quoting 37% from memory without showing where it comes from. Give the r over i minus 1 argument in one sentence and the 7 for 20 falls out.
What the interviewer asks next
With 100 founders, how many do you let pass and what is your chance?
You are happy with either of the two best founders. Should you stop earlier or later?
A founder you passed is still available at the end half the time. How does the rule change?

0.3	chance the milestone is hit
0.4	chance of the Rs 100 crore payout once past the milestone
2, 8	the first and second tranches, Rs crore

h(i)	the chance of reaching the raise from i months of cash
p, q	the chances of a good month, 0.4, and a bad one, 0.6
N	the runway at which the company can raise, 6 months

n	founders in total, 20
r	founders you let pass to set the bar
1/(i-1)	weight for the best founder sitting at position i

Try it first

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

How do you work each route's expected profit?

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

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not 50% when the company starts halfway?

How do you solve it on a whiteboard?

What would change the company's odds most?

Where candidates lose it

What the interviewer asks next

Try it first

Why let good founders pass at all?

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

Where does the model stop describing real deal flow?

Where candidates lose it

What the interviewer asks next