Suppose you are a VC firm with $17 million and the opportunity to invest in 11 red-hot companies. Each of the startups has an 85% chance of returning 10 times the investment, but your LPs want a 95% chance that the $17 million they plowed into your fund will grow to $100 million. How do you invest the money to meet their goals while keeping as much cash as possible for follow-on funding?
Here’s New York University Prof. Dennis Shasha’s answer: “If each company’s probability of success is p, the probability of exactly k successes out of n companies is (n choose k) pk (1-p)(n-k) where (n choose k) is n!/((n-k)! k!) The probability of at least k successes is the probability of k successes + the probability of (k +1) successes + … + the probability of n successes.
“When n is 10 and k is 7, (n choose k) is 120, so the probability of exactly seven successes is 13%. The probability of exactly eight successes is 0.2758967, the probability of nine successes is 0.3474254, and the probability of 10 successes is 0.1968744. Adding these four probabilities, we find that the probability of at least seven successes is just over 95%.”
Don’t understand the math? Neither do we. All you need to know is that if you invest $1.43 million in each of 10 companies, the chance that at least seven will yield 10-fold returns is more than 95%. Assuming the 95% probability hits, the total return would be $100.1 million. Your LPs are happy and you have $2.7 million left over.