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:

$$ f^{*} = \frac{bp - q}{b} = \frac{p(b + 1) - 1}{b}, $$


  • $f^{*}$ 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?


2 Answers 2


Unfortunately, the solution isn't simple in that you can pick up a piece of paper or pencil, but with software it isn't actually as bad as it is about to sound.

To begin with, note that the Kelly Criterion is precisely equivalent to assuming logarithmic utility and maximizing the utility of wealth. This is valuable in two ways. First, it allows you to model your actual problem rather than impose a solution on a problem that resembles your real problem. Second, it allows you to fully integrate constraints into the model assumptions and not merely returns.

This would begin as with any basic economic model with $$\max_{\boldsymbol{\alpha}}[\mathcal{U}(\tilde{w})],$$ where $\boldsymbol\alpha$ is a vector of allocations to be chosen from. It would not end up looking like the Capital Asset Pricing Model or the APT though.

Instead, you would model cash flows. You would formally model the probability of bankruptcy, the probability of the firm being merged into another firm for cash, the probability of the firm being bought for stock, the probability of the firm going public and exiting in an IPO. For reasons that are too long and extensive to discuss here, there is no Frequentist solution to this problem, only Bayesian solutions.

Assuming you lack access to industry data, you could condition what you believe the various probabilities are and reduce risk by creating low likelihood, planned likelihood, and high likelihood probabilities. You need to model cash reinvestment rates if the money is not thrown off.

In the event of bankruptcy, you need to model cash recoveries from the bankruptcy trustee. In the event of a merger, you should model the discounts for lack of marketability or liquidity. Ashok Abbott provides good estimators in his chapter of the Valuation Handbook. The ISBN is 9780470385791 and I have used this to lecture from in graduate courses and as course reading.

If you have access to market data, then your Bayesian likelihood function for each cash flow from debt and preferred stock should be modeled as a Bernoulli trial as either it will be paid or it will not be paid. The principal of both would be modeled using the normal distribution for the expected recoverable principal. The likelihood function for dividends should be $$\frac{\sigma}{\sigma^2+(\delta-f(x))^2},$$ if the dividends are not expected to remain constant, but should grow with time. In this representation, $\delta$ is a specific dividend and $f$ is the function of data that allows you to estimate its amount. If you believe there will be constant dividends, declining dividends or dividends that grow linearly with time, then you would use the normal distribution.

You do need to be careful with distributions of the general form $$\frac{\sigma}{\sigma^2+(x-\mu)^2}.$$ They have no mean or variance and you cannot use short cuts to do the estimations. You have to do it as a full Bayesian estimate.

On the positive side, these highly separated parts can be constructed as a very large, very high-dimension, Bayesian decision theory problem for a whole portfolio. For that you should read Parmigiani's book on Decision Theory as a primer. It covers both Frequentist and Bayesian decision theory, but will give you enough of a background to think about how to construct this as an abstraction and once you can abstract it, then you can write an algorithm.

For more information on the dividend process see

Harris, D.E. (2017) The Distribution of Returns. Journal of Mathematical Finance, 7, 769-804

For more information on decision theory see

Parmigiani, G. and Inoue, L. (2009) Decision Theory: Principles and Approaches. Wiley Series in Probability and Statistics, Chichester, 155-171.

  • $\begingroup$ By sheer coincidence, I am reading another text on Decision Theory by James Berger. Does this alternative text look comparable to the one you linked to (in the sense that it should prepare me to be able to answer questions like this)? $\endgroup$
    – George
    Commented May 6, 2018 at 20:30
  • $\begingroup$ Also, are you aware of pre-made financial model templates or other software that can help compute Kelly bets under all of these constraints? $\endgroup$
    – George
    Commented May 6, 2018 at 20:31
  • $\begingroup$ I have read both and Berger's text is a more formal work. Parmigiani's is a bit more approachable, depending entirely on your math and academic background. $\endgroup$ Commented May 6, 2018 at 20:31
  • $\begingroup$ I am not aware of any that are valid. Part of the issue is that you will model your specific problem and your specific problem isn't really going to match others. $\endgroup$ Commented May 6, 2018 at 20:32
  • $\begingroup$ I will point out that other than being time-consuming, this won't end up working out to be that bad, particularly if you "cheat" and use approximations such as best case, worst case and planned case. As far as I know, all the separate parts are out in the literature. $\endgroup$ Commented May 6, 2018 at 20:33

No real answer here, but rather some thoughts:

  1. Kelly can deal with any distribution, not necessarily binary outcomes. See examples here.
  2. The more severe issue IMO is the fact that investment opportunity arrival times are unknown. Knowing this sequence in advance would substantially improve the gains. There is a method in computer science called "Competitive Analysis", which helps analyzing the performance of online algorithms (namely, algorithms that get their input in pieces) and compare it to their offline (algorithms that know the entire input in advance) counterparts.

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