Kelly Criterion: The Formula for Optimal Position Sizing

Key Takeaways

• Kelly criterion gives the fraction of capital that maximises long-run geometric growth: betting more than Kelly guarantees eventual ruin, betting less sacrifices compounding.
• For a simple binary bet with 55% win rate and 1.5:1 payoff, full Kelly suggests staking 25% of capital per trade, far too aggressive for most portfolios with uncertain parameters.
• Half Kelly is the standard practitioner adjustment: it keeps roughly 75% of theoretical growth while cutting portfolio volatility roughly in half.
• The multi-asset Kelly portfolio requires the covariance matrix and reduces to the mean-variance optimal portfolio, so single-bet Kelly applied to correlated positions dramatically overbets.

What It Is

The Kelly criterion solves a specific problem: given a bet with known probability and known payoff, what fraction of wealth should you stake to maximise the long-run geometric growth rate of capital?

Betting less than Kelly sacrifices growth. Betting more increases growth in the short run but guarantees eventual ruin as the number of bets grows large. Kelly is the unique fraction that maximises the expected logarithm of terminal wealth.

The Intuition

Imagine a favourable coin: 60 percent chance of winning, even money on each flip. If you bet 100 percent of your bankroll every flip, the first loss wipes you out and the edge is worthless. If you bet 1 percent, you barely grow. Somewhere in between there is a single fraction that compounds fastest over many flips. Kelly tells you what that fraction is.

The deeper insight is that compounding is multiplicative. A 50 percent loss needs a 100 percent gain to recover. Maximising the expected log of wealth, rather than the expected wealth itself, accounts for this asymmetry automatically. Expected-wealth maximisers take bets so aggressive they go broke on the way to a fortune that never arrives.

How It Works

For a simple binary bet where you win b units on a win and lose 1 unit on a loss:

f* = (b * p - q) / b

Where f* is the Kelly fraction of current wealth, p is the probability of winning, q = 1 - p is the probability of losing, and b is the net odds received on the win (so 2-to-1 odds give b = 2).

For trading with asymmetric payoffs, an equivalent form is useful. If you win W and lose L on each unit staked:

f* = p / L - q / W

And for a continuous investment decision with excess return mu - r and variance sigma^2, the Kelly fraction on a single asset is approximately:

f* = (mu - r) / sigma^2

Each version says the same thing: bet more when the edge is larger, less when the payoff ratio or variance is worse.

Fractional Kelly is the practitioner adjustment. Most traders use half Kelly or quarter Kelly, meaning they stake 0.5 * f* or 0.25 * f*. The reason is not timidity. It is a direct response to parameter uncertainty: p, mu, and sigma are estimated, not known, and Kelly is very sensitive to estimation error on the upside. Half Kelly keeps roughly three-quarters of the theoretical growth rate while cutting volatility in half, a widely cited trade-off.

Worked Example

You have a trading strategy with a 55 percent hit rate. Average winner is 1.5 R, average loser is 1 R, where R is the unit risked.

Using the asymmetric form:

p = 0.55, q = 0.45
W = 1.5, L = 1.0

f* = p / L - q / W
   = 0.55 / 1.0 - 0.45 / 1.5
   = 0.55 - 0.30
   = 0.25

Full Kelly says stake 25 percent of capital per trade. That is extreme. A half-Kelly user stakes 12.5 percent, a quarter-Kelly user stakes 6.25 percent. In practice, running half or quarter Kelly is standard for real strategies with uncertain parameters.

Now stress the inputs. Suppose the true hit rate is 52 percent instead of 55. Recomputing gives f* = 0.52 - 0.48 / 1.5 = 0.20. A five-percentage-point error on p moves the Kelly size by a fifth. This is why fractional Kelly exists.

Frequently Asked Questions

Q: What is the Kelly criterion in simple terms? Kelly criterion tells you what fraction of your capital to stake on each bet to grow it as fast as possible over the long run. Bet the Kelly fraction and your wealth compounds at the maximum possible rate. Bet more and you will eventually go broke, even with a genuine edge.

Q: How does the Kelly criterion affect investment decisions? It turns a qualitative edge into a precise position size. Traders use it to scale position sizes to their actual win rate and payoff ratio, rather than using an arbitrary 2% or 5% rule. The formula penalises bets with wide dispersion of outcomes relative to their expected value.

Q: What is a real-world example of the Kelly criterion? A strategy with a 55% win rate and 1.5:1 average win-to-loss ratio produces a Kelly fraction of 25%. Full Kelly says risk 25% per trade. Half Kelly users take 12.5%, quarter Kelly users take 6.25%. In practice, very few systematic traders use more than 10% per position because parameter uncertainty makes full Kelly dangerous.

Q: How can investors use fractional Kelly safely? Use half Kelly as a default and re-estimate the win rate and payoff ratio regularly. A 5-percentage-point downward revision in win rate typically reduces the Kelly fraction by 20–25%, so treating the estimates as fixed substantially overbets when the true edge is smaller than measured.

Q: How is the Kelly criterion different from fixed-percentage position sizing? Fixed-percentage sizing (e.g., always risk 2%) is simple but ignores the actual edge and payoff ratio of each bet. Kelly sizing adjusts the stake to the measured advantage: bigger stakes when the edge is large and the odds are favorable, smaller stakes when the edge is thin. Fixed sizing can simultaneously overbet weak signals and underbet strong ones.

Common Mistakes

1. Treating estimated parameters as known. Kelly assumes you know p, W, and L. You do not. You have sample estimates with standard errors. The safe Kelly fraction under realistic estimation error is roughly half of the point estimate, which is the main rationale for fractional Kelly, not risk aversion in the classical sense.

2. Applying single-bet Kelly to correlated positions. The simple formula assumes independent bets. A portfolio of ten long equity positions is not ten independent bets; it is mostly one bet on the market. Summing per-ticker Kelly fractions overbets dramatically. The multi-asset Kelly requires the covariance matrix and reduces to f = Sigma^-1 * (mu - r) under standard assumptions, which is the same form as the mean-variance optimal portfolio.

3. Ignoring the infinite-horizon assumption. Kelly maximises asymptotic growth. If you might be forced to withdraw capital at a bad time, or your strategy has a finite shelf life, the Kelly fraction is too aggressive. Samuelson's 1979 critique turned on exactly this point: log-utility is not the right utility for every investor, and the long-run argument does not rescue an investor who cannot survive the interim path.

4. Treating fractional Kelly as arbitrary. Half Kelly is often described as a gut-feel safety margin. It is better understood as a Bayesian shrinkage toward zero given uncertainty in the inputs. The academic literature on parameter uncertainty arrives at fractional Kelly from first principles, not from squeamishness.

5. Forgetting how large full Kelly can recommend. With a strong edge and a decent payoff ratio, full Kelly easily suggests staking 30 to 50 percent of capital on a single bet. The drawdowns that go with that are eye-watering: full Kelly can experience 50 percent drawdowns with non-trivial probability even when the edge is real. Very few investors have the stomach to ride that out, and forced de-risking at the wrong moment destroys the advantage.

Sources

1. Kelly, J.L. Jr. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal, 35(4), 917-926. https://archive.org/details/bstj35-4-917
2. Thorp, E.O. "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." https://www.eecs.harvard.edu/cs286r/courses/fall12/papers/Thorpe_KellyCriterion2007.pdf
3. MacLean, L.C., Thorp, E.O., Ziemba, W.T. "Good and Bad Properties of the Kelly Criterion." https://www.stat.berkeley.edu/~aldous/157/Papers/Good_Bad_Kelly.pdf
4. Samuelson, P.A. (1979). "Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long." Journal of Banking and Finance, 3, 305-307. https://ideas.repec.org/h/wsi/wschap/9789814293501_0034.html
5. Ziemba, W.T. "A Response to Professor Paul A. Samuelson's Objections to Kelly Capital Growth." LSE Research Online. https://eprints.lse.ac.uk/119002/1/dp_52.pdf

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.