The Kelly criterion: how much to bet for maximum growth

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The Kelly criterion answers a deceptively simple question: what fraction of your wealth should you put on a bet or a risky investment so that your wealth grows as fast as possible over the long run? The answer John L. Kelly gave in 1956 maximizes geometric growth, but it often stakes far more than investors can stomach. That is why many practitioners use only a fraction of it.

The idea: growth, not expected value

A bet with a positive expected value tempts you to stake as much as possible. But if you always bet your entire wealth, the first loss wipes you out, even if every single bet makes money on average. When you bet repeatedly, what matters is not the average of the outcomes but their product: your wealth after many rounds is starting capital times (1 + return₁) times (1 + return₂), and so on. Long-run growth is therefore driven by the expected value of the logarithm, not of the return.

Kelly developed the idea as an engineer at Bell Labs while working on information transmission: Kelly (1956) showed that the highest exponential growth rate of the capital of a gambler with inside information equals the transmission rate of the information channel. Edward Thorp and others later did most to bring it to the stock market (Thorp 2006).

The formula for a simple bet

Two features stand out. First, the growth curve is flat around f*: half the Kelly stake still delivers 1.50%, about 75% of the maximum growth, with far less volatility. Second, the curve drops steeply beyond f*: at double the Kelly stake, growth is roughly zero or negative. Betting too much is more dangerous than betting too little.

Kelly for stocks and portfolios

For investments with many possible outcomes, there is a continuous-time approximation. It follows from the same objective, the expected logarithm of wealth, and matches the optimal stock allocation that Merton (1969) derived for an investor with logarithmic utility:

Plugging f* into the growth formula reveals an elegant link to the Sharpe ratio: the maximum achievable additional growth is half the squared Sharpe ratio of the investment. In the example, the Sharpe ratio is 0.05 / 0.18 = 0.28, and 0.28² / 2 gives the 3.86% from above. An investment with twice the Sharpe ratio therefore allows four times the additional growth, provided you lever it up accordingly.

With several investments, the formula generalizes to f* = Σ⁻¹(μ − r): the Kelly portfolio is the tangency portfolio of mean-variance theory, levered to a particular level of risk. It therefore inherits all the problems of optimizing with estimated returns.

Advanced: drawdowns and fractional Kelly

The full Kelly stake maximizes growth, but at the price of very deep interim losses. In the continuous approximation (ignoring interest), you can calculate the probability that your wealth will at some point fall to a fraction x of its current value. If you bet the fraction c of the Kelly stake, the following holds:

With the full Kelly stake, there is a 50% chance that your wealth will be cut in half at some point. For most investors, that is intolerable. Half the stake lowers this probability to one in eight but costs only a quarter of the growth. MacLean et al. (2010) sum up the good and bad properties: the highest long-run growth, but large swings and a long wait before the advantage is secure.

A second reason for fractional Kelly: the inputs are uncertain. If you overestimate the excess return, you automatically end up at a multiple of the true Kelly stake, and thus on the steep part of the curve. Seen this way, half or quarter Kelly is insurance against estimation error.

What the research says

The criterion has been debated for decades. Its advocates stress that over the long run a Kelly strategy almost surely ends up with more wealth than any essentially different strategy. Samuelson (1971) sharply disagreed: it does not follow that maximizing the geometric mean is optimal over a finite horizon. Whether it is depends on your risk preferences. Only with logarithmic utility does the optimum coincide with the Kelly stake.

Thorp (2006) describes how he applied it to blackjack, sports betting and the stock market and, from experience, recommends betting fractions of the Kelly stake. Merton (1969) shows in continuous time that the optimal stock allocation of an investor with constant relative risk aversion γ equals (μ − r) / (γ σ²). Kelly corresponds to γ = 1, half Kelly to γ = 2.

Limitations and common mistakes

• The inputs are unknown: with a coin toss you know p; with stocks you do not. The expected excess return can be estimated from past data only very imprecisely, and errors have a quadratic effect on the Kelly stake.
• Leverage is expensive and limited: a Kelly fraction of 154% assumes you can borrow at the risk-free rate. Higher borrowing costs, margin requirements and forced selling after losses change the math.
• Fat tails: the continuous-time formula assumes normally distributed returns. Rare, very large losses (skewness and kurtosis) make the full stake even more dangerous.
• Horizon and goals: Kelly optimizes for an infinitely long horizon. If you need a specific amount in ten years or plan to make withdrawals, your goal is something other than maximum growth.
• Not a license to concentrate: a high f* for a single stock almost always rests on overly optimistic assumptions about its return.

What Kelly means for individual investors

The key lesson is not the exact formula but the shape of the growth curve: more risk brings more growth only up to a point, and less growth beyond it. It is the same effect as volatility drag: a portfolio with very large swings grows more slowly than its average return suggests. For most investors, a sensible stock allocation lies far below the full Kelly fraction, because they would not sit through the deep interim losses.

A step-by-step approach for your own portfolio

1. Estimate your portfolio's excess return over money market rates conservatively, for example from long time series or valuation models, not from the last five good years.
2. Estimate the volatility, for example from monthly returns over ten years or more.
3. Calculate f* = excess return / volatility² and use at most half of it.
4. Check the result against your loss tolerance: how far can your portfolio fall before you would sell? The smaller of the two numbers wins.

For a broad stock portfolio, this often lands at allocations around or above 100%; your loss tolerance is then the real limit. Above all, the Kelly criterion explains why, beyond a certain point, taking more risk no longer pays.

How Portfolio Lab calculates

Portfolio Lab does not calculate a Kelly fraction. What you can use instead: the Monte Carlo simulation shows the range of wealth paths for a portfolio and the growth rate (CAGR) of each path. Use it to compare portfolios with different stock allocations. With your own assumptions, it works parametrically: the return you enter is the arithmetic mean, and typical growth comes in about half the variance below it. On the Tactical page, the “Target volatility” model scales a portfolio to a fixed level of risk, with an exposure of up to 200%. Details in the Monte Carlo methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.