The Kelly Criterion and Position Sizing Beyond the 1% Rule

f* = p - q/b

That is the entire criterion. p is the probability of winning, q = 1 - p is the probability of losing, b is the amount won per unit risked when the bet works, and f* is the fraction of capital that maximizes the long run growth rate of the account. Position sizing is the one corner of trading where an exactly correct answer exists, and this is it.

The answer is also, in almost every real case, far too large to use, and working out why takes about fifteen minutes of arithmetic that changes how a trading account is run more than any signal does. The 1 percent rule and R multiples are the prerequisite if expectancy is unfamiliar.

Two bets, the same answer

A coin that pays even money and lands your way 55 percent of the time:

f* = 0.55 - 0.45/1 = 0.10

A setup that wins 40 percent of the time and pays two units on a win against one on a loss:

f* = 0.40 - 0.60/2 = 0.40 - 0.30 = 0.10

Ten percent of capital per bet. Two distributions with nothing in common except their expected value. If that number feels reckless, hold onto the feeling. The arithmetic agrees with you. The rest of this guide is the explanation.

What the fraction is actually maximizing

Kelly maximizes the expected logarithm of wealth, which is the same thing as the long run compound growth rate. For the even money case, growth per bet is

G(f) = p x ln(1 + bf) + q x ln(1 - f)

Push the 55 percent coin through it at three sizes.

Fraction bet	Calculation	Growth per bet
5 percent, half Kelly	0.55 x ln(1.05) + 0.45 x ln(0.95)	0.375 percent
10 percent, full Kelly	0.55 x ln(1.10) + 0.45 x ln(0.90)	0.501 percent
20 percent, double Kelly	0.55 x ln(1.20) + 0.45 x ln(0.80)	about 0.000 percent

The middle row by hand, because it is the one worth trusting: 0.55 x 0.09531 = 0.052421, 0.45 x (-0.10536) = -0.047412, and the sum is 0.005009, or 0.501 percent per bet.

Three results fall out of that table. Half Kelly delivers 0.375 / 0.501 = 75 percent of the maximum growth rate. Double Kelly delivers nothing whatsoever, despite every individual bet carrying a positive expected value. And past double Kelly the growth rate turns negative, so an account with a genuine edge grinds steadily toward zero while its owner keeps making good bets.

The reason is compounding asymmetry. A 50 percent loss needs a 100 percent gain to undo, which the drawdown recovery calculator makes uncomfortably concrete, and bigger bets buy a higher arithmetic average at the cost of a lower geometric one, with only the geometric one sitting in your account.

Half Kelly is the best trade available anywhere

The growth function is a downward parabola, so near the peak it is nearly flat, while volatility of the equity curve scales roughly linearly with the fraction bet, which means you can halve the bet and slide a long way across the horizontal axis and barely at all down the vertical one.

Three quarters of the growth for half the risk. Very little in markets is priced that generously, which is why practitioners who take this arithmetic seriously almost never bet the full fraction.

Your inputs are worse than you think

The mathematical argument for fractional sizing is good. The statistical argument is the one that should settle it.

Suppose you believe your win rate is 55 percent and size at the resulting 10 percent, when the true rate is 52 percent and the correct fraction was 0.52 - 0.48/1 = 0.04. You are betting two and a half times Kelly, and the table above already told you what that does.

Now look at how the 55 percent was estimated. Say 60 wins in 110 trades. The standard error on a proportion from 110 observations is sqrt(0.55 x 0.45 / 110) = 0.047, so a two standard error band runs from roughly 46 percent to 64 percent, and at the bottom of that band the Kelly fraction is negative, meaning the strategy has no edge at all and should not be traded. Your point estimate and “stop trading immediately” are inside the same confidence interval.

That is the honest position of nearly every discretionary trader with a year of records, and it is the reason quarter or third Kelly is less a preference than an admission, so run the Kelly criterion calculator at the top and the bottom of your own confidence band and size for the bottom.

The continuous form, and why volatility targeting won

For an asset traded continuously, the optimal fraction is

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

expected excess return over variance. An asset with a 6 percent expected excess return and 16 percent annualized volatility gives 0.06 / 0.16^2 = 0.06 / 0.0256 = 2.34, which is 234 percent of capital.

That number is useful precisely because it is ridiculous. It shows how violently the answer depends on the expected return input, which is the one input nobody can forecast: halve the expected return to 3 percent and the prescription halves to 117 percent of capital, still absurd, on one assumption changed by three percentage points.

This is where the textbook stops being useful. Volatility, unlike expected return, is persistent and forecastable from recent data, so practitioners fix the numerator and scale on the denominator:

weight = target volatility / realized volatility

Target 10 percent annualized against realized volatility of 20 percent and you hold 0.5 units. When realized volatility falls to 12 percent you hold 0.10 / 0.12 = 0.83 units, so risk contribution stays roughly constant across regimes and the input is one you can actually measure. Most systematic managers size this way, including those running the factor portfolios on the Pro Desk, and you can measure your own inputs with the Sharpe ratio calculator and the tools in the portfolio risk guide.

From a fraction to a share count

The formula gives a fraction of capital at risk. A trade ticket needs a number of shares. The stop distance is the bridge.

Take a $200,000 account, a computed Kelly fraction of 10 percent, and a decision to trade at one third Kelly. The risk budget is 200,000 x 0.10 x 0.333 = $6,660. The stock trades at $84. The level that invalidates the idea sits at $79.20, so you risk $4.80 per share. The position is 6,660 / 4.80 = 1,387 shares, a notional of about $116,500, which is 58 percent of the account.

Read that twice. A cautious sounding risk fraction produced an enormous notional position. The stop was close. Tight stops let Kelly buy a great deal of exposure, and they also make the position acutely sensitive to the execution problems in market microstructure, where a gap through a thin book turns your $4.80 of risk per share into $9.

Where Kelly breaks

Correlation breaks it first. The single bet formula assumes the bet is the only thing happening. Six positions that are all long beta are one position sized six times, and the multi asset form of the criterion needs the full covariance matrix to say anything sensible. The practical substitute is portfolio heat: cap total risk across open positions, cap any correlated cluster below that, and count four technology positions risking 1 percent each as a 4 percent single risk when their pairwise correlation is 0.8. The diversification arithmetic shows why the naive count misleads so badly.

Fat left tails break it second. A short volatility strategy might show a 90 percent win rate and a 0.3 win to loss ratio over a sample containing no crash, giving f* = 0.90 - 0.10/0.3 = 0.567. The formula cheerfully recommends 57 percent of capital on a strategy whose defining event is missing from the data, which is the February 2018 lesson in one line of arithmetic.

Unstable edges break it third. Kelly assumes the same bet repeats forever at the same odds, but market edges decay, and a size computed from last year’s statistics is a claim about the next one that nothing in the data supports.

Size at a quarter to a half of the computed fraction in the Kelly criterion calculator, cap it with a fixed percentage rule on top, and then read pairs trading and statistical arbitrage, where the sizing question and the hedge ratio turn out to be the same question.