The Kelly criterion in allocation

Mean-variance theory tells you the mix of assets; the Kelly criterion addresses a different question: given an edge, how much should you bet to grow wealth fastest over the long run?

The idea

Devised by John Kelly in 1956 for information theory and adopted by gamblers and investors, Kelly finds the bet size that maximizes the long-run growth rate of capital — the geometric, compounding return, not the single-period average.

For a simple bet with probability p of winning b-to-1 odds:

Kelly fraction = p - (1 - p) / b

A concrete example

Probability of winning p = 0.6
Win pays even money, so b = 1
    Kelly fraction = 0.6 - 0.4 / 1 = 0.20

So you should stake 20 percent of your capital on this favorable bet — and re-stake 20 percent of your new total each time, scaling with wealth.

In a portfolio setting

Generalized to investing, the Kelly fraction is roughly the expected excess return divided by the variance of returns:

Kelly fraction (approx.) = excess return / variance

This is closely related to maximizing the Sharpe ratio, and it formalizes a deep truth: betting too small leaves growth on the table, but betting too large is catastrophic — because compounding punishes losses asymmetrically. Lose 50 percent and you need a 100 percent gain just to recover.

Why nobody bets full Kelly

Full Kelly is brutally volatile — it can produce 50 percent drawdowns that are mathematically optimal but emotionally and practically intolerable. Worse, Kelly assumes you know your edge precisely; overestimate it and full Kelly overbets into ruin. For these reasons practitioners use fractional Kelly (often half or a quarter), trading a little long-run growth for a large reduction in volatility and a margin of safety against estimation error.