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The Kelly Criterion — Separate Loss Risk From Position Allocation
The binary Kelly fraction measures equity lost on a losing bet. Translate a separately chosen risk limit into position size using the entry-to-stop distance, and assess uncertainty, drawdowns, and joint exposure before sizing.
The binary Kelly fraction describes the equity lost on a losing bet. Turning that figure into a trading position requires a separate decision about acceptable loss and the distance from entry to stop.
The Kelly criterion asks how to size repeated bets to maximize long-run compound growth under a specified model. In the simple binary model, each win earns +bR and each loss costs −1R, with win probability p and loss probability q = 1 − p. Its optimal fraction is f* = p − q/b. For the hypothetical values p = 0.55 and b = 1.5, the result is f* = 0.25. Here, 25% is the fraction of equity lost on a losing bet. It is not an instruction to invest 25% of equity in an asset.
That distinction changes the sizing decision. For a linear position, a separately chosen planned loss risk r and an entry-to-stop distance d, both expressed as fractions, give a notional allocation of approximately w = r/d. A 1% planned loss with a stop 2.5% from entry implies 40% of equity in notional exposure before fees, slippage, gaps, and financing costs. The binary Kelly calculation alone does not choose that 1% limit or ensure the stop will contain the loss.
Kelly maximizes expected log wealth within its assumptions. It does not set a maximum acceptable drawdown, and an estimate from past trades is neither a proven optimum for future trading nor a safe ceiling. Before using a Kelly estimate, identify what a loss means, test how uncertain the return distribution is, and set limits for the account and its simultaneous positions.
Kelly Maximizes Expected Log Growth Under Its Model
The Kelly criterion does not maximize the expected profit of one bet. In the binary model, expected profit is positive when pb > q, or p > 1/(1+b). A win rate above 50% is the relevant threshold only at even odds, where b = 1. If expected one-bet profit were the only objective and the stake were restricted to between zero and all available equity, a positive edge would favor the largest permitted stake. Expected log growth instead accounts for the compounding damage from losses.
For our hypothetical 55% win rate and 1.5 payoff, expected log growth per bet is g(f) = 0.55 ln(1+1.5f) + 0.45 ln(1−f). At full Kelly, f = 0.25, it is 0.0456926; at twice that fraction, f = 0.50, it is −0.0041275. These are expected logarithmic growth values using natural logs, not expected simple percentage returns. Twice Kelly produces slightly negative growth in this example; a zero-growth point at exactly twice Kelly is not a universal rule.
A market selloff helps explain why the loss model matters, but cannot establish a strategy's Kelly fraction. In Binance BTCUSDT spot daily data, using UTC candles, May 19, 2021 opened at 42,849.78 USDT and reached 30,000, USDT about 30% below the open. The April 14 high had been 64,854 USDT. These prices show the scale of the market move; account losses would also depend on exposure, leverage, entries, exits, and fills. The fall does not by itself move a strategy to the right of its Kelly peak. Check whether the modeled losses cover such moves before trusting the sizing calculation.
Full Kelly Can Produce Drawdowns Beyond an Account's Tolerance
Full Kelly maximizes expected log growth under the specified assumptions; it does not maximize drawdown. Nevertheless, the associated position sizes can produce substantial losses. In the example with 25% of current equity lost per losing bet, three consecutive losses leave 0.75³ = 42.1875% of the starting equity: a 57.8125% drawdown if that sequence begins at an equity peak. The frequency of such drawdowns depends on the outcome distribution, dependence between trades, and observation horizon. A Kelly fraction alone cannot tell you the likely maximum drawdown.
The objective also has no explicit penalty for a trader's discomfort at an interim loss. Two paths that end with the same wealth have the same terminal log wealth, even if one suffered a much deeper drawdown along the way. That does not make the paths equally tolerable. If the trader stops during a decline, or margin requirements force liquidation, the intended compounding process changes. Set an acceptable drawdown before deciding how much of an estimated Kelly fraction to use.
The November 2022 FTX crisis provides another stress period to examine. Binance BTCUSDT spot opened at 21,299.37 USDT on November 6, reached 17,166.83 USDT on November 8, and 15,588 USDT on November 9, using UTC daily candles. The last low was about 27% below the November 6 open over the November 6–9 date range. Those prices do not establish what a Kelly-sized account lost. From a position sizing standpoint, replay the strategy's actual exposure and fills through that period and compare the resulting drawdown with its operating limit.
The One-Line f* Requires a Binary Model and Reliable Inputs
The formula f* = p − q/b is exact for the specified two-outcome model with a fixed gain of bR and loss of 1R. In trading, estimated win rates and average payoffs may change, and actual outcomes usually span more than two values. Knowing a win rate and an average win-to-loss ratio does not, by itself, supply the full return distribution needed for an exact Kelly solution.
Input error can still be illustrated within the binary model. At b = 1.5, a win probability of 50% gives f* ≈ 0.1667, rather than the 25% calculated from 55%. Using 25% would therefore overbet relative to that revised model. The growth cost depends on the size of the error and the shape of the return distribution; it should not be described as a fixed penalty for any small mistake. Costs, execution differences, and selection bias can also widen the gap between backtest and live (backtest vs live). Check their effect on both the frequency and size of wins and losses.
During the August 5, 2024 selloff, Binance BTCUSDT spot fell from a UTC daily open of 58,161 USDT to a low of 49,000, USDT about 16%. This is a useful execution stress case, not proof of any particular system's loss or a single cause for the selloff. If stops fill worse than modeled, realized losses can exceed 1R, changing the distribution used for sizing. With p held fixed in the binary approximation, a smaller b implies a smaller Kelly fraction. Re-measure risk-reward expectancy from actual fills, including costs, and retain the range of outcomes when assessing growth and downside risk.
Half Kelly Reduces Exposure, but Does Not Guarantee Safety
Fractional Kelly scales down a model's growth-optimal allocation. This sacrifices some modeled growth in exchange for less exposure to individual outcomes. Half Kelly is one such choice; it is not a universally appropriate fraction or evidence that the underlying estimate is sound.
For the hypothetical 55% win rate and fixed 1.5 payoff, half Kelly uses f = 0.125. Its expected log growth is 0.0344285 per bet, or about 75.348% of the full-Kelly value. This comparison belongs to this particular binary model.
| Type | Equity fraction lost on a loss | Expected log growth per bet | Per-bet simple-return standard deviation |
|---|---|---|---|
| Full Kelly | 0.25 | 0.0456926 | 1× |
| Half Kelly | 0.125 | 0.0344285 | 0.5× |
In this model, simple account returns take the values +bf and −f, giving a standard deviation of f(b+1)√(pq). Halving f therefore halves the standard deviation of the per-bet simple return exactly. It does not guarantee that equity-curve volatility or maximum drawdown will be halved. Those outcomes depend on compounding, the sequence and dependence of returns, and the horizon. Half of an overestimated Kelly fraction can also remain above the true growth optimum.
For a practical risk of ruin comparison, define failure and a horizon explicitly—for example, the probability of reaching a 30% peak-to-trough drawdown within the next 250 trades. Compare full and fractional sizing under the same assumptions, with uncertainty and adverse execution included. In the ideal bounded binary model with a fixed 0 ≤ f < 1, equity cannot reach exactly zero in finitely many bets; that mathematical fact says little about a margin call or an intolerable drawdown. Do not label a fractional allocation safe without testing the failure condition that matters to the account.
Changing Payoffs Require More Than One Average Kelly Input
General Kelly optimization can use a full return distribution and multiple assets. The limitation belongs to the simple binary formula: a win rate and average payoff cannot generally recover the exact optimal fraction when winning and losing outcomes vary. Separate market conditions may also have different distributions, so a pooled estimate can hide risks in the current setup.
A higher payoff ratio increases the binary model's optimal fraction only if win probability is held fixed. In trading, a more distant target may also reduce the probability of reaching it. Fixed risk per trade is an operational way to control planned losses as stop distances change; it does not automatically make each trade Kelly-optimal. Choose the planned loss limit first, then use the entry-to-stop distance to calculate exposure.
For example, consider a separately chosen 1% equity risk limit around the period when Binance BTCUSDT spot reached a UTC daily high of 73,777 USDT on March 14, 2024. If a technically justified stop is 3% from the entry price, a linear position has notional exposure of about 33.3% of equity: 1% ÷ 3%. A stop 1.5% from entry implies about 66.7%. A stop placed 3% below a structural low does not imply a 3% entry-to-stop distance. These calculations exclude costs and losses beyond the planned stop, and a tighter stop can change the strategy's outcomes. Treat the 1% limit as a separate policy choice, not a result dictated by Kelly.
A 55% Win Rate and 1.5 Payoff Illustrate the Sizing Steps
The following is a hypothetical example, not a claim that a live strategy has established these parameters. Assume each win earns exactly 1.5R, each loss costs exactly 1R, and the win probability is 55%. Real trading needs separate evidence for the entry rule, outcome distribution, and execution assumptions.
- Full-Kelly calculation:
f* = 0.55 − 0.45/1.5 = 0.25. The model's optimum loses 25% of current equity on a losing bet. It is not a safe trading ceiling or a 25% notional allocation. - Illustrative operating risk: Half Kelly is
0.125. For this example, choose a separate planned loss limit of 1% of equity. That is1% ÷ 25% = 4%, or one twenty-fifth of full Kelly; quarter Kelly would be 6.25% of equity. Kelly does not establish that 1% is suitable for a particular account. - Entry and position size: Enter only when the strategy's independently tested entry condition is satisfied. Kelly supplies no entry signal. With the stop 2.5% from entry, 1% planned risk implies 40% notional exposure; with a 1.5% stop, it implies about 66.7%, before costs and adverse execution.
- Stop and target: For a long position, a stop 2.5% below entry and a target 3.75% above entry produce a planned
1.5Rpayoff. Exit at the strategy's stop or target, while recognizing that actual fills can change the realized payoff. - Review before increasing risk: Fifty or more live trades and an estimated risk of ruin below 1% do not automatically justify raising planned risk from 1% to 1.5%. Specify the failure threshold and horizon, review costs, data outside the development sample, estimation uncertainty, and portfolio limits before considering any increase.
- Illustrative review trigger: A live win rate below 50% or payoff ratio below 1.2 may trigger a review and a reduction to 0.5% planned risk under a separately chosen operating rule. Those thresholds and the 0.5% figure are not outputs of Kelly; a deteriorating or uncertain edge may instead justify pausing entries.
The sizing step that must remain explicit is the conversion from planned loss to notional exposure. In this example, the account risks a planned 1% per trade under a chosen limit, equal to 4% of the hypothetical full-Kelly fraction. Neither that arithmetic nor the past estimates guarantees the next loss will stay within the planned amount.
Simultaneous Positions Need a Joint Risk Model
Independence makes a repeated-bet example simpler, but general Kelly optimization is not restricted to independent assets. A portfolio calculation can account for their joint returns and correlations. The error is to compute isolated fractions for several positions and assume those fractions remain appropriate when all positions are held together.
A BTC long and an ETH long can lose at the same time. The period following Binance BTCUSDT spot's UTC daily high of 69,000 USDT on November 10, 2021, and the November 2022 FTX crisis are useful periods for checking such joint losses. They do not establish that all altcoins moved identically. Three highly correlated positions are not necessarily equivalent to three times the same bet; that equivalence requires identical proportional outcomes. Nominal planned losses add if all stops fill as assumed, while gaps and slippage can make the actual combined loss larger.
If a separately chosen portfolio limit is 1% of equity in aggregate planned losses, divide that budget among simultaneous positions rather than allocating 1% to each. Check joint downside scenarios as well as historical correlations, because average correlation alone does not describe every crash scenario. Before adding another symbol, calculate what the account would lose if the positions reached their stops together and what adverse fills could add. Size against that combined exposure and the account's operating limits.
Sources: Kelly's 1956 paper, pp. 919–920 sets out fixed-fraction betting and log growth; Rotando and Thorp's stock-market treatment extends the discussion to investment returns; Thorp's description of fractional Kelly explains the risk tradeoff. The BTC prices cited here use Binance BTCUSDT spot daily candles in UTC.