1 Context
Consider the Kelly Investment Criterion, which "is a formula used to determine the optimal size of a series of bets in order to maximize the logarithm of wealth". Put differently, the Kelly Criterion helps investors balance the trade off between maximizing their long-run expected value while also minimizing the chance that they ever go bust. Here is the formula for the "simple model" according to Wikipedia:
For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the Kelly bet is:
$$ f^{*} = \frac{bp - q}{b} = \frac{p(b + 1) - 1}{b}, $$
where:
- $f^{*}$ is the fraction of the current bankroll to wager, i.e. how much to bet;
- $b$ is the net odds received on the wager ("$b$ to $1$"); that is, you could win $b$ (on top of getting back your $\$1$ wagered) for a \$1 bet
- $p$ is the probability of winning;
- $q$ is the probability of losing, which is 1 − $p$.
The problem with the simple model is that it assumes you're investing 1 investment per period, in sequence. In the world of angel investing, you're making potentially many investments in parallel, with each investment paying off at some unknown date (i.e., one investment might pay out in 7 years, while another investment might pay out in 1 year). In addition, angel investment returns aren't necessarily binary (unlike in the simple model).
2 Question
How can the Angel Investor alter this model so that it accomodates investments with unknown payout periods (and such that the Angel Investor always has money left over for another investment, should an opportunity come along)? (It would also be nice to know how to make it accommodate investments that are non-binary, but we can assume binary outcomes for now).
3 Attempt
At the very least, we can try to accommodate the model for an unlimited number of investments (before any investment necessarily pays out). For example, if the simple model suggests that you ought to bet $x_1 \in [0,1]$ percent of your money in the first startup, and $x_2 \in [0,1]$ percent of your money in the second startup, then we could instead bet $\frac{1}{2^2} x_1$ of our money in the first startup, and $\frac{1}{3^2} x_2$ of our money in the second startup, and in general $\frac{1}{(n+1)^2} x_n$ of our money in the $n$th startup. In the long-run, this would ensure that we never bet more than
$$ \sum_{n = 1}^\infty \frac{1}{(n+1)^2} x_n \le \underbrace{\sum_{n = 1}^\infty \frac{1}{(n+1)^2} = \frac{1}{6}(\pi^2 - 6)}_{\text{fact from real analysis}} < 1 $$
percent of our money in any collection of startups, and would therefore allow us to make an unlimited number of investments without ever running out of money. But how would we know that this strategy is even close to Kelly optimal?