Where the Kelly criterion comes from

John L. Kelly Jr., a researcher at Bell Labs, published the idea in 1956 in the Bell System Technical Journal, in a paper titled A New Interpretation of Information Rate. He asked how a gambler receiving tips over a noisy private wire should size his bets to make his money grow fastest, and showed that the answer is to maximize the expected logarithm of wealth. The mathematician Edward Thorp later applied the idea to blackjack and then to markets, and it has been part of every serious debate on position sizing since.

The logarithm matters because returns compound. A gain of 50% followed by a loss of 50% leaves 1.5 × 0.5 = 0.75 of the starting capital, a 25% loss, even though the average of +50% and −50% is zero. Growth depends on the product of results, not on their average, so a large loss hurts more than an equal gain helps. Kelly sizing finds the bet that balances the two; beyond it, the losses start to dominate. The same arithmetic drives compound interest, in reverse when you lose.

The formula: f* = W − (1 − W) / R

For a bet that wins b times the stake with probability p and loses the stake with probability q = 1 − p, Kelly’s fraction is f* = p − q ÷ b, the form usually quoted for betting. In trading terms, p is the win rate W, b is the payoff ratio R (average win ÷ average loss) and the stake is the amount lost when the stop is hit, so f* = W − (1 − W) ÷ R.

Two details change how you use it. First, f* is the share of the account you put at risk, the loss at the stop, not the size of the position: risking 10% of a $50,000 account with a $2.00 stop means $5,000 at risk and 2,500 shares, a $125,000 position at $50 a share. Second, the formula can be rewritten as f* = E ÷ R, the expectancy per trade in R divided by the payoff ratio, so a zero or negative expectancy gives a zero or negative fraction: no edge, no bet. The same formula in four forms:

  • f* = W − (1 − W) ÷ R, with W the win rate and R = average win ÷ average loss.
  • f* = p − q ÷ b, the same formula in betting notation.
  • f* = E ÷ R, the expectancy per trade in R divided by the payoff ratio.
  • Position size = account × f ÷ (entry − stop), the arithmetic of the position size calculator.

A worked example

Hypothetical: a setup with 300 logged trades shows a hit rate of 45%, an average win of 2R and an average loss of 1R, after costs. The account is $50,000.

  1. Payoff ratio: R = 2R ÷ 1R = 2.
  2. Expectancy: E = 0.45 × 2 − 0.55 × 1 = +0.35R per trade.
  3. Full Kelly: f* = 0.45 − 0.55 ÷ 2 = 0.175, the same as E ÷ R = 0.35 ÷ 2. That is 17.5% of the account, $8,750 at risk on every trade.
  4. Position: with an entry at $50.00 and a stop at $47.50, the risk is $2.50 a share, so $8,750 ÷ $2.50 = 3,500 shares, a $175,000 position: 3.5 times the account, only possible with leverage.
  5. Half Kelly: 8.75%, or $4,375 at risk; quarter Kelly: about 4.4%, or $2,187.50.
  6. Fixed 1% risk: $500 at risk and 200 shares, less than a seventeenth of full Kelly.

Why full Kelly is too aggressive for traders

Kelly’s result assumes that you know W and R exactly, that each trade is an independent repeat of the same bet and that losses stop at the planned amount. Trading breaks all three, and the growth curve punishes errors unevenly: betting half the Kelly fraction keeps about three quarters of the growth, while betting twice the Kelly fraction brings long-run growth down to about zero, and betting more shrinks the account even with a positive expectancy.

Estimation error is the main danger. At R = 2, the Kelly fraction is 1.5 × W − 0.5, so every 5 points of win rate move it by 7.5 points, as the drawing below shows. If you estimate 45% from a lucky sample and the truth is 40%, the true Kelly fraction is 10%, your 17.5% is 1.75 times too large, and the expected growth per trade falls by more than half, with much bigger swings. Other reasons to stay well below full Kelly:

  • Drawdowns: in the standard continuous approximation, a full-Kelly bettor has about a 50% chance of seeing the account halve at some point, and about a 10% chance of seeing it fall to a tenth.
  • Streaks: 10 losses in a row at 17.5% risk leave 0.825^10 ≈ 14.6% of the account, a loss of about 85%.
  • Gaps and slippage: a stop does not cap the loss, and a gap can make a real loss larger than 1R.
  • Correlation: several open trades in related markets act like one bigger bet.
  • Changing edges: a win rate measured last year may not hold this year.

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On the left, a line of the Kelly fraction against the estimated hit rate for R = 2 (average win 2R, average loss 1R): zero up to about 33%, then rising, with points at 40% → 10%, 45% → 17.5% in red and 50% → 25%, and a note reading 5 points of hit rate = 7.5 points of Kelly. On the right, a card titled you think 45%, it is really 40%: you risk 17.5%, true Kelly 10%, you bet 1.75× Kelly, growth per trade: less than half, with 0.0046 vs 0.0097 underneath.

Half Kelly and quarter Kelly

Because the growth curve is flat near its peak, betting less than Kelly costs little growth and removes a lot of risk. In the example above, half Kelly (8.75%) keeps about 76% of the maximum growth rate and quarter Kelly (about 4.4%) about 45%, while the swings in the account shrink in proportion to the bet.

The odds of a deep drawdown fall even faster. In the same continuous approximation, betting a fraction λ of Kelly gives a probability of x^(2/λ − 1) that the account ever falls to a share x of its current value. For halving, that is 50% at full Kelly, 12.5% at half Kelly and under 1% at quarter Kelly. These figures assume the edge is known exactly, which in trading it never is, so they flatter even fractional Kelly.

Even quarter Kelly is 4.4% per trade in this example, well above what most risk guidelines allow, which is why many traders treat fractional Kelly as an upper bound and size by a small fixed fraction instead.

Fixed-fractional risk: the practical version

Fixed-fractional sizing risks the same small share of current equity on every trade, typically 0.5% to 2%, and sets the position size from the stop: shares = (account × risk %) ÷ (entry − stop). The position size calculator does exactly this. Because the risk is a share of current equity, positions shrink after losses and grow after gains, which slows a drawdown without any extra rule.

The drawing below shows why the small numbers matter. After 10 losses in a row, an account risking 1% per trade is down about 9.6%; at 2%, about 18.3%; at 5%, about 40%; at the full Kelly of the example, about 85%. Halving the account takes 69 straight losses at 1%, 35 at 2% and only 4 at 17.5%. Add a cap on total open risk and a daily or weekly loss limit, as described in risk management, and keep the rules in writing in your trading plan.

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Five bars show an account that starts at 100 after 10 losses in a row, by risk per trade: 95.1 at 0.5%, 90.4 at 1%, 81.7 at 2%, 59.9 at 5% and, in red, 14.6 at 17.5%, labeled full Kelly in the example. Under each bar, the number of straight losses that halves the account: 139, 69, 35, 14 and 4.

Risk of ruin and drawdown recovery

Risk of ruin is the probability of losing so much that you cannot or will not continue: a margin call, an account below a broker’s minimum, or simply a loss you could not live with. With fixed-fractional sizing the account never mathematically reaches zero, so in practice traders define ruin as a level, such as a 50% drawdown, and size so that the chance of reaching it is very small. That chance depends on the edge, the risk per trade and how reliable the estimates are, which is why no formula replaces small size.

Recovery arithmetic is unforgiving: after a loss of d, you need a gain of 1 ÷ (1 − d) − 1 to get back to even. A 10% loss needs +11.1%, a 25% loss +33.3%, a 40% loss +66.7%, a 50% loss +100% and a 75% loss +300%. Deep drawdowns also take time to repair, which tests discipline, the subject of trading psychology.

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Pairs of bars under the title gain needed to get back to even = 1 ÷ (1 − loss) − 1: a red loss bar below the line and a green bar above it for the gain needed to recover, on the same scale: −10% and +11%, −25% and +33%, −40% and +67%, −50% and +100%, and −75% and +300%, the last green bar broken because it is too tall to draw.

Never size by Kelly with an unproven edge

Kelly multiplies whatever error is in its inputs. Feed it an edge estimated from 40 trades, an optimized backtest or a hopeful guess, and it returns a large, confident number built on noise. Before Kelly enters the discussion, the edge needs a large sample, realistic costs and a test against random entries, as explained in trading expectancy and backtest overfitting.

If you do use it, use it conservatively: plug in the low end of your expectancy range rather than the point estimate, then take a fraction of the result. In the worked example, 300 trades with a standard deviation of about 1.49R give a standard error of about 0.09R, so a cautious expectancy is about 0.35 − 2 × 0.09 ≈ +0.17R. That gives f = 0.17 ÷ 2 ≈ 8.5%, and a quarter of that is about 2.1%, at the top of the usual fixed-fractional range. Treat that as a ceiling, not a target.

Common Kelly mistakes

The formula is simple; the ways to misuse it are many:

  • Treating f* as the position size instead of the amount at risk to the stop.
  • Using a win rate and payoff from a small sample or an optimized backtest.
  • Leaving out costs, which lower both the expectancy and the Kelly fraction.
  • Applying the full fraction to several correlated trades at once, which multiplies the real risk.
  • Raising size after a winning streak because the edge feels stronger.
  • Using leverage to reach a Kelly-sized position, as in the worked example, where full Kelly needed 3.5 times the account; see leverage and margin.