Kelly criterion formula for position sizing (and why full Kelly fails)

The formula, and what each part is

For a bet that pays R units per unit risked, with win probability W, the growth-optimal fraction of capital is f = W − (1 − W) / R.

Two consequences fall straight out of it. If the payoff ratio R is 1 and the win rate is exactly 50%, then f = 0.5 − 0.5 = 0: the formula tells you not to bet at all. A coin flip with no edge has no optimal bet size, because any positive size is a slow leak.

And the fraction is very sensitive to small changes in W. At R = 1, going from a 50% win rate to a 55% win rate takes the optimal fraction from 0% to 10% of capital. Five percentage points of win rate, a tenfold change in the answer.

A worked example

Suppose a strategy wins 55% of the time and wins and loses the same average amount (R = 1). The formula gives f = 0.55 − 0.45 = 0.10, so 10% of capital per trade.

Ten percent of capital per trade sounds aggressive for a strategy whose edge is a 55/45 split, and it is. It is also the number that grows the account fastest if the 55% is real, permanently, and known exactly. Every one of those three conditions is false in practice.

Change the shape instead: a trend-following system that wins 35% of the time but wins three units when it wins. Then f = 0.35 − 0.65/3 = 0.133, so about 13% of capital. Notice that a 35% win rate produces a larger Kelly fraction than a 55% win rate here. Win rate alone tells you almost nothing about position size; it is the pair that matters.

Why full Kelly is usually the wrong answer

Kelly assumes W and R are known constants. You never know them. You estimate them from a past sample, and that sample has an error bar around it.

This is not a small technicality. Take a strategy whose true win rate is 52% at R = 1. The true optimal fraction is 4% of capital. Now suppose your backtest of 400 trades showed 55% instead — an entirely ordinary estimation error. You apply f = 10%, more than double the optimal size. On the true 52% process, that position size produces a negative expected log growth rate. The strategy has an edge and you still lose money, because you bet the wrong size.

Push the error further and it gets worse monotonically: believing 58% when the truth is 52% means betting 16% per trade against a negative growth rate. The further past the optimum you go, the faster the account bleeds, and there is no point at which the formula warns you, because it thinks the 58% is true.

What fractional Kelly actually costs you

Betting half the Kelly fraction does not halve your growth. Near the optimum the growth curve is flat, so a second-order loss in growth buys a first-order reduction in volatility.

Concretely: at a 52% win rate with R = 1, full Kelly grows at 0.0008 in log terms per trade. Half Kelly grows at 0.0006, which is 75% of the growth, while the size of your swings halves. At a 40% win rate with R = 2, half Kelly retains 76% of the growth; at 35% with R = 3, it retains 77%. That is the trade: give up a quarter of your compounding to halve the amplitude of the ride, and to buy a large margin of safety against having overestimated your edge.

Many professional sizing rules are quarter Kelly or less, which is an admission of exactly this uncertainty rather than a disagreement with the mathematics.

Why sizing, not edge, is what actually kills accounts

It is tempting to think the account dies because the strategy stopped working. Often the strategy's edge was simply too small for the size being traded.

Simulate a strategy with no edge at all — a 50/50 coin at R = 1 — and bet a fixed fraction of current capital per flip, 500 flips, calling a 50% drawdown a ruin. At 1% per flip, about 0.3% of paths hit that drawdown. At 2%, it is about 16%. At 5%, about 71%. At 10%, about 94%.

The edge was zero in all four cases. The only thing that changed was the size, and it moved the outcome from 'nothing happens' to 'almost certain ruin'. Sizing is not a refinement you apply after finding an edge. It is the variable that decides whether the edge gets a chance to compound.

With a genuinely small edge the picture shifts but does not change shape: a true 52% win rate at 2% per trade ruins about 4% of paths over 500 trades, at 4% about 28%, at 8% about 68%, and at 20% about 98%. A real edge does not make large size safe. It only moves the threshold.

What to do instead of calculating Kelly precisely

Estimate the fraction, then divide it. If your edge estimate is uncertain — and it always is — a quarter to a half of the computed value is a more defensible starting point than the value itself.

Work from the out-of-sample win rate and payoff ratio, never the in-sample ones. The in-sample pair is the one the optimiser already selected for being flattering, so it is biased in exactly the direction that makes Kelly overbet.

Check the sensitivity rather than the point estimate. Recompute f using a win rate one standard error below your estimate. If the answer collapses to near zero, you do not have enough evidence to size on, whatever the point estimate says.

Remember what the formula optimises. It maximises the growth rate of capital. It does not know about your rent, your drawdown tolerance, or the probability that you abandon the system at the worst moment. Those are constraints on the answer, not inputs to it.

What Kelly does not tell you

It does not tell you whether you have an edge. It takes W and R as given; if they are wrong, the output is a precise answer to the wrong question.

It does not account for correlated positions. If you apply it independently to five strategies that all bet on the same thing, you have not sized five bets at f each — you have sized one bet at roughly five times f. That is a portfolio-level question and it needs a portfolio-level answer.

It does not handle non-stationarity. A win rate measured over the last three years is an average over regimes, and the next regime is not obliged to resemble the average.

None of this makes the formula useless. It makes it a limit rather than a target: the fraction you should never exceed, computed from numbers you should not fully trust.