Position Sizing and the Kelly Criterion: Fixed Fractional, Full Kelly, Half Kelly and Why Overbetting Ruins

Position sizing is the rule that decides how much of your account to risk on each trade. The Kelly criterion is the particular fraction that maximizes the long-run growth rate of capital: for a bet that wins b units per unit risked with probability p, it is f* = p - (1 - p) / b of current capital. Full Kelly is rarely used by professionals because it produces very deep drawdowns, and because the edge it depends on is estimated with error; overestimate the edge and Kelly bets become too large, and betting about twice the true Kelly fraction leaves you with roughly zero growth.

In short: for a signal that wins 40% of the time at 2:1, Kelly says risk 10% of the account per trade. In our simulation of 500 such trades, full Kelly had a median maximum drawdown of 89%, and 46% of paths fell to half their starting value at some point. Half Kelly kept 76% of the growth rate, cut the median drawdown to 61%, and cut the chance of ever halving to 12%, close to Edward Thorp’s closed-form result of 1 in 8. Because a retail track record of a few hundred trades cannot pin down the win rate within several percentage points, the honest Kelly fraction is a range, and the safe end of it is far below the headline number.

This article explains a sizing method and its mathematics. It is not a recommendation to trade or to use any particular risk level.

A common misconception: “if the edge is real, bet bigger”

Signal sellers rarely talk about position sizing, and when they do it is usually one of two extremes: a fixed “risk 1% to 2% per trade” rule with no explanation, or the suggestion that a high win rate justifies large lots. Both skip the central fact that sizing determines whether an edge turns into growth. A positive average return per trade is necessary but not sufficient. Size the trades too large and a winning strategy can still shrink the account; this is a mathematical result, not a matter of discipline.

Fixed fractional sizing

The common professional baseline is fixed fractional sizing: risk a constant fraction f of current equity on every trade, where “risk” means the loss if the stop is hit. Translating that into lots needs the stop distance and the pip value (see pips, lots and signal math):

lots = (account x f) / (stop in pips x pip value per lot)

With a $10,000 account, f = 1%, and a 25-pip stop on EUR/USD, where one standard lot is worth $10 per pip: lots = $100 / (25 x $10) = 0.4 lots.

Fixed fractional sizing has two useful properties. Position sizes shrink automatically after losses and grow after gains, so the account can in principle never reach zero from a sequence of stopped-out trades. And every trade is measured in the same unit, R (one R is the amount risked), which is what makes statistical testing possible (see statistical significance of a track record).

The question fixed fractional sizing does not answer is which f to choose. That is the question Kelly answered.

Kelly (1956): the gambler with a private wire

John L. Kelly Jr., working at Bell Labs, published “A New Interpretation of Information Rate” in the Bell System Technical Journal (vol. 35, no. 4, July 1956, pp. 917-926). His setup is a gambler who receives the results of games over a noisy communication channel before the bookmakers adjust the odds. Each received symbol is correct with probability q and wrong with probability p, and the bets pay even money.

Kelly first notes that betting everything on each signal maximizes the expected value of capital, but that the gambler “would be broke with probability one if he continued indefinitely.” He then supposes the gambler bets a fixed fraction l of capital each time, and defines the exponential growth rate

G = q log(1 + l) + p log(1 - l)

which holds “with probability one” in the long run. Maximizing G gives l = q - p, and the maximum growth rate equals Shannon’s information rate of the channel. His argument for this choice is not that it is optimal for every gambler, but that in a game that never ends, the bettor using this fraction “will, with probability one, eventually get ahead and stay ahead” of one using any other fixed fraction.

The important shift is from maximizing expected wealth to maximizing the expected logarithm of wealth, which is the same as maximizing the median long-run growth rate.

Thorp: from blackjack to markets

Edward O. Thorp, who was introduced to Kelly’s paper by Claude Shannon at MIT in 1960, applied it to blackjack card counting and later to sports betting and securities. His survey, “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market” (originally a 1997 conference paper; reprinted in the Handbook of Asset and Liability Management, vol. 1, edited by Zenios and Ziemba, North-Holland, 2006), gives the formulas traders use.

For a bet paying b to 1 with win probability p and loss probability q = 1 - p:

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

For an even-money coin with p = 0.53, Thorp’s Example 2.1 gives f* = 0.53 - 0.47 = 0.06, or 6% of current capital per bet.

For a continuous-price investment with excess return m and volatility s (and a zero risk-free rate), Thorp’s equations (7.6) give f* = m / s^2, and the growth rate at a fraction c of Kelly relative to the maximum is

g(c x f*) / g(f*) = c x (2 - c)

So half Kelly (c = 0.5) earns 0.5 x 1.5 = 75% of the maximum growth rate, and double Kelly (c = 2) earns zero. Growth falls off symmetrically on either side of f*, but risk does not: volatility of the growth path scales with c, so overbetting adds risk while reducing return.

Thorp also derives the probability that capital ever falls to a fraction x of its starting value, again in the continuous approximation:

Prob(capital ever falls to x times the start) = x^(2/c - 1)

For full Kelly (c = 1) that is simply x: a 50% chance of ever losing half, a 30% chance of ever falling to 30%. For half Kelly it is x^3: the chance of ever losing half drops to 1/8. Thorp adds that the probability of doubling before halving is 2/3 at full Kelly and 8/9 at half Kelly, and observes that most people “strongly prefer the increased safety and psychological comfort of ‘half Kelly’ (or some nearby value), in exchange for giving up 1/4 of their growth rate.”

Fraction of Kelly	Ever fall to 70% of start	Ever fall to 50%	Ever fall to 30%	Ever fall to 10%
Full (c = 1)	70.0%	50.0%	30.0%	10.0%
Half (c = 0.5)	34.3%	12.5%	2.7%	0.1%
Quarter (c = 0.25)	8.2%	0.8%	0.02%	about 0%

These are infinite-horizon probabilities from Thorp’s formula, computed in Python.

MacLean, Thorp and Ziemba: the good and bad properties

Leonard MacLean, Thorp and William Ziemba summarize the case for and against Kelly in “Good and bad properties of the Kelly criterion” (2010; reprinted in their edited volume The Kelly Capital Growth Investment Criterion, World Scientific, 2011). On the good side: maximizing the expected log of wealth asymptotically maximizes the growth rate, asymptotically minimizes the expected time to reach a distant goal, and never risks ruin.

On the bad side, the abstract is blunt: “The main disadvantage of the Kelly criterion is that its suggested wagers may be very large. Hence, the Kelly criterion can be very risky in the short term.” Their examples include an illustration from Ziemba and Hausch in which 700 bets, each with a 14% advantage, can turn $1,000 into $18 under full Kelly, while half Kelly “does not help much” in that worst case, where $1,000 can still fall to $145.

Two of their observations matter most for traders:

Overbetting is strictly worse. “It never pays to bet more than the Kelly strategy because then risk increases (lower security) and growth decreases.” In continuous time, betting exactly double Kelly produces a growth rate equal to the risk-free rate.
Errors in the mean dominate. Citing Chopra and Ziemba (1993), they note that errors in means, variances and covariances were about 20:2:1 in importance, measured by cash-equivalent loss: errors in the mean matter roughly ten times as much as errors in the variance. Their conclusion: “Given the extreme sensitivity of E log calculations to errors in mean estimates, these estimates must be accurate and to be on the safe side, the size of the wagers should be reduced.”

Worked example 1: Kelly for a 40% win rate at 2:1

Take a signal that wins +2R 40% of the time and loses -1R 60% of the time. Expectancy is 0.4 x 2 - 0.6 = 0.20R per trade, a strong edge by any retail standard. Kelly: f* = (2 x 0.4 - 0.6) / 2 = 0.10, or 10% of the account at risk per trade.

Growth per trade at each fraction, g(f) = 0.4 x ln(1 + 2f) + 0.6 x ln(1 - f):

Fraction of Kelly	Risk per trade	Log growth per trade	Share of maximum growth	Typical growth over 500 trades
0.10	1.0%	0.00189	19.5%	2.6x
0.25	2.5%	0.00433	44.5%	8.7x
0.50	5.0%	0.00735	75.7%	39x
0.75	7.5%	0.00913	94.0%	96x
1.00	10.0%	0.00971	100.0%	129x
1.50	15.0%	0.00743	76.5%	41x
2.00	20.0%	0.00070	7.2%	1.4x
2.50	25.0%	-0.01042	negative	0.01x

The growth rate hits zero at 20.4% risk, 2.04 times Kelly, matching Thorp’s continuous result almost exactly. “Typical” here means the median outcome, exp(500 x g).

Note how flat the curve is near the top: 0.75 Kelly gets 94% of the growth for three quarters of the risk.

Worked example 2: simulating the paths

Growth rates describe the long run. Traders live through the paths. We simulated 20,000 independent sequences of 500 trades of the same signal at five sizing levels:

Sizing	Risk per trade	Median ending equity	5th percentile ending equity	Chance of ending below start	Median max drawdown	Chance of a 50%+ drawdown	Chance of ever falling to half the start
Quarter Kelly	2.5%	8.7x	2.29x	0.3%	35.6%	8.7%	0.7%
Half Kelly	5.0%	39.4x	2.82x	0.9%	61.1%	84.7%	12.0%
Full Kelly	10.0%	128.5x	0.73x	6.5%	88.8%	100.0%	46.4%
1.5x Kelly	15.0%	41.2x	0.02x	22.3%	98.1%	100.0%	72.8%
Double Kelly	20.0%	1.4x	about 0	48.3%	99.9%	100.0%	88.5%

Three things stand out.

The simulation agrees with Thorp. The chance of ever falling to half was 46.4% at full Kelly and 12.0% at half Kelly, against his infinite-horizon values of 50% and 12.5%.
Full Kelly’s drawdowns are brutal even with a strong edge. Every single path had a drawdown of at least 50%, and the median worst drawdown was 89%. Few traders, and almost no clients, would keep following a system through that.
Mean and median part company. At double Kelly, the expected ending equity is 1.04 to the 500th power, about 330 million times the start, because each trade returns 4% on average. That expectation is carried by paths so rare that even 20,000 simulations barely sample them: the simulated average is unstable from run to run, while the median was a stable 1.4x and nearly half of all paths lost money. This is Kelly’s original point: maximizing expected wealth and maximizing typical wealth are different objectives.

Worked example 3: when the edge is estimated

The signal above was assumed to win exactly 40% of the time. In reality you estimate the win rate from a track record. With 200 trades at an observed 40%, the standard error is sqrt(0.4 x 0.6 / 200) = 3.5 percentage points, so a 95% interval for the true win rate runs from about 33% to 47%.

Suppose you size at the Kelly fraction for the estimate (10%), half of it (5%), or a quarter (2.5%), and the true win rate is different:

True win rate	True Kelly fraction	Growth at 10% risk	Growth at 5% risk	Growth at 2.5% risk
34%	1.0%	-0.00755	-0.00145	-0.00012
35%	2.5%	-0.00467	+0.00002	+0.00062
36%	4.0%	-0.00179	+0.00148	+0.00136
37%	5.5%	+0.00108	+0.00295	+0.00210
38%	7.0%	+0.00396	+0.00442	+0.00284
40%	10.0%	+0.00971	+0.00735	+0.00433
42%	13.0%	+0.01547	+0.01028	+0.00581

If the true win rate is 36%, still inside the confidence interval and still a positive-expectancy system (0.08R per trade), betting the estimated full Kelly produces negative growth: the account shrinks over time despite a genuine edge. Growth at 10% risk turns negative below a true win rate of 36.6%; at 5% risk, below 35.0%; at 2.5% risk, below 34.2%. This is the quantitative case Thorp makes in his section on fractional Kelly: estimates of the mean “are more likely to be too high than too low,” partly because “systems that worked may be partly or entirely based on data mining,” so it is “wise to assume” the true mean is below the estimate and to bet less than the estimated Kelly fraction.

Worked example 4: when a few losses exceed the stop

The binary formula assumes every loss is exactly -1R. Real trades are messier: weekend gaps, news spikes and slippage on stops mean that some losses are much larger than planned. Kelly is very sensitive to that tail, far more sensitive than the average trade suggests.

Keep the same signal, but suppose that 3% of all trades (one loser in twenty) gap through the stop and lose 3R or 5R instead of 1R. Maximizing the expected log growth numerically over the three outcomes:

Loss profile	Expectancy	Kelly fraction	Maximum log growth per trade	Growth if still sized at 10%	Growth if sized at 5%
All losses -1R	0.20R	10.0%	0.00971	0.00971	0.00735
3% of trades lose 3R	0.14R	5.9%	0.00411	0.00217	0.00401
3% of trades lose 5R	0.08R	2.7%	0.00108	-0.00792	0.00026

Turning one loser in twenty into a 3R gap cuts the expectancy by 30% but cuts the Kelly fraction by 41% and the best achievable growth rate by 58%. With 5R gaps, a trader still sizing at the “clean” Kelly fraction of 10% shrinks the account over time, even though the trades have positive expectancy on average. Half the clean Kelly fraction (5%) is close to full Kelly in the 3R case, and close to double Kelly, with almost no growth, in the 5R case. This is the same lesson as Worked example 3 from a different direction: the part of the distribution that a short track record is least likely to show, the rare large loss, is the part that matters most for sizing. It is also why fat-tailed strategies, including those that hold losers or average down, deserve much smaller fractions than their win rate and average trade imply.

What Kelly says about the common 1% rule

Kelly also gives a way to read conventional advice. In this binary setup, the Kelly fraction is roughly expectancy divided by the variance of R (the continuous approximation m / s^2 gives 0.093 here, against the exact 0.10).

Edge (win +2R / lose -1R)	Expectancy	Full Kelly risk	1% risk as a fraction of Kelly
34% win rate	0.02R	1.0%	1.0 (full Kelly)
35% win rate	0.05R	2.5%	0.4
40% win rate	0.20R	10.0%	0.1

For a strategy with a modest real edge, 1% risk per trade is already in fractional Kelly territory. For a strategy with a tiny edge, 1% is full Kelly, with all the drawdowns that implies. For a strategy with no edge, any positive risk is overbetting: Kelly is zero.

What can go wrong in real trading

Losses can exceed 1R. Kelly for a stop-loss trade assumes the stop is honoured. Gaps, weekend opens and illiquid moments break that assumption. Thorp uses the October 1987 crash as his example (the S&P 500 fell about 20% on October 19, 1987; his survey gives 23%, close to the Dow’s 22.6% fall); in currencies, the Swiss franc shock of 2015 moved EUR/CHF far beyond any normal stop. A rare loss of 3R or 5R changes the Kelly fraction sharply.
Correlated signals are one bet. Long EUR/USD, long GBP/USD and short USD/CHF opened together are largely the same US dollar trade. Kelly applies to the combined exposure, so three 2% risks on correlated pairs can amount to something close to one 6% risk.
Leverage hides size. A small margin deposit can carry a large notional position. Kelly is about the loss at the stop relative to equity, not the margin used; see leverage and margin in forex.
The edge decays. Copying a signal adds slippage and spread, and a strategy’s edge can shrink after it becomes popular. Each cut in expectancy cuts the Kelly fraction roughly proportionally.
Martingale is the opposite of Kelly. Fixed fractional sizing reduces size after losses. Martingale and grid systems increase it, which is why they can look stable for months and then fail completely (see martingale and grid strategies explained).
Drawdowns are psychological as well as financial. A plan that requires sitting through an 89% drawdown is not a plan most people execute. Thorp’s own conclusion, in the case study that closes his survey, was that fractional Kelly is “prudent to the extent the results of the Kelly calculations reflect uncertainties.”

The short version

Fixed fractional sizing risks a constant share of equity per trade. Kelly (1956) showed that the fraction maximizing long-run growth is f* = p - q / b for a binary bet, and Thorp showed that half Kelly keeps 75% of the growth while cutting the chance of ever losing half your capital from 1/2 to 1/8. Our 500-trade simulation of a 40% win rate, 2:1 signal matched those figures and showed a median worst drawdown of 89% at full Kelly. MacLean, Thorp and Ziemba stress that overbetting raises risk while lowering growth, and that errors in the estimated edge dominate everything else. Because a track record of a few hundred trades leaves the win rate uncertain by several points, the practical Kelly fraction is a fraction of the estimated one, and a strategy without a demonstrated edge has a Kelly fraction of zero.