Position sizing with the Kelly criterion (and why most traders use a fraction of it)

Finding a positive expected value is half the problem. The other half is deciding how much to bet. Bet too little and an edge grows slowly. Bet too much and a normal losing streak can wipe you out, even with a real edge.

The Kelly criterion, published by John L. Kelly Jr. at Bell Labs in 1956, answers the sizing question precisely, under some strong assumptions.

#The idea

Kelly chooses the fraction of your bankroll to bet that maximizes the long-run growth rate of the bankroll, equivalently the expected logarithm of wealth. It balances two forces:

• Bigger bets grow the bankroll faster when you win.
• Bigger bets shrink it more when you lose, and losses compound. Losing 50% requires a 100% gain to recover.

Above the Kelly fraction, adding risk actually lowers long-run growth. At twice the Kelly fraction, expected growth falls to roughly zero.

#The formula for a binary contract

For a general bet that pays net odds b to 1 and wins with probability p:

f* = (b × p − (1 − p)) / b

A YES contract bought at price c that pays $1 has net odds b = (1 − c) / c. Substituting and simplifying gives a compact formula:

f* = (p − c) / (1 − c)

where f* is the fraction of your bankroll to spend buying contracts. The numerator is your edge per contract; the denominator is how much each contract can win.

#Worked example

You estimate an event's probability at 60%. The YES ask is 50¢, and for simplicity there are no fees.

f* = (0.60 − 0.50) / (1 − 0.50) = 0.10 / 0.50 = 0.20

Full Kelly says to spend 20% of your bankroll. With a $10,000 bankroll, that is $2,000, or 4,000 contracts at 50¢.

Now change one input. If your probability is 55% instead:

f* = (0.55 − 0.50) / 0.50 = 0.10

A 5-point drop in your estimate halves the bet. Kelly is very sensitive to p, which is exactly the number you know least well.

#Including fees

Treat the fee as part of the price. With price c and fee f per contract, use c' = c + f:

f* = (p − c') / (1 − c')

In a spreadsheet:

B2  Probability          0.60
C2  Ask                  0.50
D2  Fee                  0.01
E2  Effective cost       =C2 + D2
F2  Kelly fraction       =MAX(0, (B2 - E2) / (1 - E2))
G2  Half Kelly           =F2 / 2
H2  Bankroll             10000
I2  Contracts (½ Kelly)  =FLOOR(G2 * H2 / E2, 1)

The MAX(0, …) matters: when the edge is negative, Kelly's answer is "don't bet," not "bet a negative amount" on this side.

#Why full Kelly is too aggressive in practice

Kelly is optimal if you know the true probability and outcomes are independent. Neither is true in real trading.

• Estimation error. Your p comes from a model or a backtest and carries error. Because overbetting hurts growth more than underbetting by the same amount, uncertainty about p pushes the right bet size below Kelly.
• Drawdowns. Even with perfect knowledge, full Kelly routinely produces deep drawdowns. A full-Kelly bettor has a 50% chance of seeing their bankroll halve at some point along the way, and few people can stick with a strategy through that.
• Correlated bets. Kelly for one bet assumes it is the only bet. Ten positions on correlated events, such as several games on the same weekend or several outcomes of the same economic release, are closer to one large bet.

#Fractional Kelly

The common practical answer is to bet a fixed fraction of the Kelly amount: half Kelly or quarter Kelly.

Half Kelly keeps roughly three quarters of the growth rate while halving the volatility, and it is much more forgiving of an overestimated edge. That trade is why most professionals who use Kelly at all use a fraction of it.

#Kelly with many positions

When you hold several positions at once, compute each one's Kelly fraction, then scale the whole set down so that:

• total capital at risk stays within a limit you set (for example 25% of the bankroll), and
• correlated positions are treated as one combined exposure.

Exact multi-asset Kelly requires optimization, but the scaled-down approach captures most of the benefit with far less modeling.

#Frequently asked questions

#What is the Kelly criterion?

The Kelly criterion is a formula for the fraction of capital to stake on a bet with a known edge that maximizes the long-run growth rate of wealth. It was introduced by John L. Kelly Jr. in 1956 and is widely used in betting and investing, usually in a reduced, fractional form.

#What is the Kelly formula for a prediction market contract?

For a contract that pays $1, bought at price c, with your probability p that it pays out, the Kelly fraction is (p − c) divided by (1 − c). Include fees by adding them to c. If the result is zero or negative, do not take the position.

#Why do traders use half Kelly?

Because the true probability is never known exactly and full Kelly produces large drawdowns. Half Kelly gives up about a quarter of the theoretical growth rate in exchange for half the volatility and much more protection against an overestimated edge.

#Does the Kelly criterion work for stock trading?

The same principle applies, but stock returns are continuous rather than binary. A common continuous approximation is a fraction equal to expected excess return divided by variance of returns. The estimation-error problem is even larger for stocks, so fractional sizing is standard.