Legal Tech Investing for Angel Investors Part 4: From Paradox to Kelly

In the comments on the St. Petersburg Paradox piece, I promised a proper treatment of the Kelly criterion.

Unfinished business from the previous post

The St. Petersburg paradox was never really a puzzle about expected value. The math there is straightforward, even if the answer is strange: a coin-flip game with infinite expected value that no rational person would pay more than a modest sum to play. The important takeaway, however, is that maximizing expected value, taken as a governing principle, produces answers no sane actor would act on once a bet is large enough, or repeated enough, or unbounded enough. That still leaves the point about the optimal size of the bet open.

From whether to how much

As lawyers, we love a rigorous answer. In this case, that answer was developed (almost coincidentally) by John Kelly, a Bell Labs scientist. His formula for optimal bet sizing has since become foundational in both information theory and quantitative finance. Kelly recognized that instead of asking which single bet maximizes expected value this round, we should ask which bet size maximizes the rate of compound growth across many rounds. Slightly oversimplifying, St. Petersburg asked what a bet is worth. Kelly asks how much of your capital to stake on it.

The formula, and what it actually says

I would not dare reproduce the actual math here, but here is the simplified version of the formula (with f* standing for the optimal bet size):

f* = edge / odds

The layperson's translation: bet a fraction of your capital proportional to how strong your edge is, and inversely proportional to how much you stand to lose if you are wrong. If you have no edge, bet nothing — no amount of odds makes a coin flip worth staking capital on. As your edge grows, bet more, but never your entire capital (I can almost hear someone sighing with relief “finally, I knew this would link to diversification somehow”), and never more than the formula prescribes. This last part is the real counter-intuitive part – when someone is faced with a “sure thing,” the instinct is to bet as much as possible. However, betting past f* is not a more aggressive version of the optimal strategy. Math shows that this strategy is sub-optimal. Past the Kelly fraction, additional size actually lowers your long-run growth rate (i.e., the formula has a peak).

What this means for how a check actually gets sized

In the investing world, Kelly matters enormously, but – interestingly enough – is often disregarded in standard practice. Most funds size positions by working backward from an ownership target (e.g., 15 to 20 percent at Series A) and back into the check size (and the relevant valuation). It is a heuristic used instead of math. It is certainly not irrational, but an ownership target is a statement about desired position in the cap table, and does not take edge into account.

Kelly is the question a fund is implicitly answering every time it sizes a check. Not "what ownership do we want," but "how much of our capital does our actual edge in this company justify." The gap between the two answers is where a good deal of unexamined risk tends to live.

To be clear, I don't mean to suggest that any fund — including GrayHair — can or should operate purely on Kelly principles. In practice, being overly dogmatic with Kelly can exclude an investor from a great round (e.g., if the amount calculated using Kelly is just short of the minimum check in the round), or, conversely, cause it to overcommit and strain liquidity, creating problems down the road. But directionally, ignoring Kelly's logic entirely (which is what happens when investors size check size purely off ownership targets with no reference to edge or conviction) is likely detrimental to performance over time. And before someone asks: “I write the same check into every opportunity that I like,” while often mistaken for discipline, is equally deficient from the math standpoint.

Limitations of Kelly (almost a legal disclaimer) – Gambler’s Ruin

Kelly assumes you know your edge with reasonable precision, which is obviously not always the case (although you can and should assess the strength of your (relative) edge at least qualitatively). It also assumes you are playing a repeatable game, where "long-run growth" is a coherent thing to optimize for in the first place. What it does not tell you is what happens if a bad stretch of variance takes you below the capital threshold you need just to keep playing. The problem here is not sizing, but rather survival, and the optimization is not for growth, but for the avoidance of “ruin”. I will run about that in due course (it is relevant for every investor, but it is probably more of a consideration for funds).

P.S. – the real legal disclaimers

Of course, all of my posts on investing are for informational purposes only, and should not be interpreted as investment recommendations.