OptiNod Academy

The Kelly Criterion — Why the Mathematically Optimal Size Is Almost Always Too Large in Practice

Kelly's \"optimal\" points to the single fraction that maximizes long-run compound growth. To stand exactly on that point you have to know your win rate and payoff ratio precisely, but in trading both are estimates, so even a slightly generous figure turns straight into over-betting. That is why, in practice, traders cut it down to half-kelly or less.

> Kelly's "optimal" points to the single fraction that maximizes long-run compound growth. To hit that fraction exactly you have to know your win rate and payoff ratio precisely, but in trading both are always estimates.

The Kelly criterion is a betting fraction derived in 1956 by John Kelly of Bell Labs while he was studying the information transmission rate of communication channels. It asks what percentage of your capital you should stake on a single bet to grow that capital fastest over the long run, and the answer comes out in a single line. With the win rate as p, the loss rate as q=1−p, and the payoff ratio b (the ratio of what you win when you win to what you lose when you lose), the optimal fraction is f* = p − q/b. For a 55% win rate and a payoff ratio of 1.5, f* = 0.55 − 0.45/1.5 = 0.25 — that is, staking 25% of your capital on a single bet.

Popularly, this formula is taken as "the mathematically proven optimal bet size." Once the number 25% comes out, people read it as the fastest route to bet at that fraction. But what Kelly maximizes is the expected value of log capital — the compound growth rate. It is not the *expected return* of a single bet. This point is almost never mentioned.

Read it this way and the cost shows up immediately in the equity curve. An account betting at the full-Kelly fraction has the highest growth rate, but its volatility and drawdowns both blow out to the extreme. And, crucially, if the p and b you fed into f* were even slightly more generous than the real values, that 25% turns straight into over-betting. When you calculate your own system's Kelly fraction, treat that number not as a target but as a ceiling you must never exceed.

**The asymmetry where growth collapses sharply past full-Kelly**The Kelly curve climbs gently to its peak, but past the peak the growth rate falls quickly below zero, leaving over-betting exposed.

What Kelly Maximizes Is the Compound Growth Rate

The Kelly criterion does not solve the problem of maximizing the expected profit of a single bet. If all you wanted was to maximize expected profit, the answer would be to stake your entire capital every time the win rate clears 50%. What Kelly looks for is the fraction that grows the account fastest when the same bet is repeated indefinitely — the fraction that maximizes the expected value of log capital.

This distinction matters because under compounding a single large loss drags down the entire growth rate. Recovering a 50% loss requires a 100% gain. That is why the Kelly peak is asymmetric left to right. To the left of the peak (under-betting) the growth rate falls gently, but to the right of the peak (over-betting) volatility losses accumulate and the growth rate drops quickly into negative territory. Under simple Bernoulli-bet assumptions, doubling the bet fraction to twice full-Kelly takes the long-run growth rate to nearly zero. For the 55% win rate / 1.5 payoff system above, the per-bet growth rate of about 0.0457 at full-Kelly falls to −0.004 at twice that fraction.

A BTC example shows this asymmetry clearly. On May 19, 2021, BTC fell about 30% in a single day, from a daily open of roughly $42,850 to an intraday low of $30,000. The prior high of $64,854 on April 14 makes it deeper still from the top. An account leveraged near the full-Kelly fraction would be pushed by this one day into the steep right-hand stretch of the growth curve. If you have calculated a Kelly fraction, first confirm that the fraction sits at the peak. Bet past the peak and the long-run growth rate actually shrinks.

Full-Kelly's Drawdowns Run Deeper Than Most Traders Can Endure

The full-Kelly fraction maximizes the growth rate at the cost of pushing the equity curve's drawdown to the extreme. Run full-Kelly and a maximum drawdown that erases more than half the account is not rare. The exact expected drawdown depends on the bet distribution and trade frequency, but the direction — that full-Kelly presupposes deep drawdowns — is clear. Even standing exactly on the peak of the growth curve, the equity curve still swings this hard.

The reason the drawdown runs this deep is that the Kelly criterion does not count the psychological pain a trader feels as a cost. The formula contains only the expected value of log capital. If a path that goes through a 60% drawdown before eventually printing a new high and a path that climbs gently through a 20% drawdown arrive at the same endpoint, the Kelly criterion does not distinguish between them. To a trader, though, the two paths are entirely different experiences. If you doubt the system and bail out during the 60% drawdown stretch, the long-run growth assumption collapses in that instant.

The week of the FTX collapse in November 2022 makes this difference clear. BTC fell about 27% in three trading days, from a November 6 open of roughly $21,300 to a November 8 low of $17,167 and a November 9 low of $15,588. An account using full-Kelly sizing would see tens of percent evaporate in this stretch alone, and with an earlier drawdown stacked on top, it sinks into a drawdown deep enough that holding the system through it becomes psychologically hard. So before deciding whether to apply the Kelly fraction as is, first check from a position sizing standpoint whether you can endure the drawdown that fraction will produce.

The One-Line f\* Is Only Optimal When p and b Are Exact

The biggest trap in the Kelly criterion is that it assumes the p and b inside the formula are known constants. For casino roulette this holds, because the probabilities are fixed by the rules. In trading, the win rate and payoff ratio are values estimated from past data, and there is no guarantee they will stay the same in the future.

Estimation error is dangerous because the f* formula amplifies input error. If you took the win rate as 55% but the real value is 50%, the q/b term grows and the calculated fraction comes out larger than the true optimum. On top of that, since the right side of the Kelly peak is steep, even slightly overshooting the peak shaves a large amount off the growth rate. Estimate generously and you pay the price in the steep stretch of the growth curve. Backtest win rates almost always come out more generous than live trading, because the gap between backtest and live (backtest vs live) reduces the payoff ratio b.

The August 5, 2024 yen-carry liquidation plunge illustrates this well. BTC fell about 16% in a single day, from a daily open of roughly $58,161 to a low of $49,000. Even for a system that had estimated its usual payoff ratio at 1.5, in a gap-down move like this the size of a single loss comes out larger than assumed, due to stop-loss slippage. When the real size of a loss grows, b shrinks, and when b shrinks, the appropriate Kelly fraction must shrink too. Re-measure your own system's risk-reward expectancy against live fill prices, then feed that value into the f* formula.

Half-Kelly Is the Practical Standard Because It Keeps Almost All the Growth While Cutting Volatility in Half

In practice, few managers run full-Kelly as is. Most use half-Kelly (half of full-Kelly) or less. Cut the fraction in half and, thanks to the asymmetry of the peak curve, the growth rate you give up is small while the volatility you shed is large.

Here are the numbers. Since the Kelly peak is flat near the top, halving the bet fraction does not reduce the long-run growth rate much. For the 55% win rate / 1.5 payoff system above, half-Kelly keeps about 75% of full-Kelly's growth rate.

|---|---|---|---|

| Full-Kelly | 0.25 | 0.0457 | Baseline |

| half-Kelly | 0.125 | 0.0344 | About half |

By contrast, equity-curve volatility scales roughly in proportion to the bet fraction, so it falls nearly by half (drawdowns tend to shrink even more than volatility). You give up a quarter of the growth rate in exchange for cutting volatility in half. Factor in estimation error and the trade becomes even more favorable. Even if you estimated p and b generously and full-Kelly had in fact overshot the peak, half-Kelly is still very likely a safe fraction that falls short of the peak.

Here we have to revisit what Kelly means by "optimal." The Kelly fraction points to the spot where the growth rate is maximized, and this value is meaningful only when the win rate and payoff ratio you entered are exact. Full-Kelly alone can produce a drawdown deep enough to make you stop trading. So in practice traders do not treat that fraction as a target but only as a ceiling, and operate at half-Kelly or below. Calculate your own system's risk of ruin at the Kelly fraction and at half-Kelly separately, and the difference between the two is your margin of safety.

**The difference in equity curves between full-Kelly and half-Kelly**Half-Kelly gives up part of the final growth rate but greatly reduces drawdown and volatility, raising the odds of surviving in practice.

When the Payoff Ratio Differs From Bet to Bet, a Single Kelly Fraction Does Not Hold

The Kelly criterion assumes every bet has the same p and b. In trading, the stop distance and target differ from trade to trade, so b changes every time. The b of a trade with a tight stop and a distant target and the b of a trade with a wide stop and a near target are entirely different values.

This matters because, if you lump the per-trade b into a single average and compute one Kelly fraction, you depart from the appropriate fraction on every individual trade. On trades with a large b you under-bet, and on trades with a small b you over-bet. In practice, you do not apply the Kelly fraction literally as a fixed bet size of your capital; you use it as the basis for setting one unit of risk per trade. Fix the loss risk of a single trade at a set percentage of capital, then back out the position size from the stop distance, and the per-trade risk stays constant even as b changes.

For example, say you fix per-trade risk at 1% of capital. In the stretch where BTC printed its all-time high of $73,777 on March 14, 2024, if you place the stop 3% below the prior structural low, the position size comes to about 33% of capital (1% ÷ 3%). For the same 1% risk, tightening the stop to 1.5% doubles the position to about 66%. It fits practice better to treat the Kelly fraction as the higher-level basis that decides whether to lower this per-trade risk of 1% to 0.5% or raise it to 1.5%. If your stop distance differs from trade to trade, do not read the Kelly fraction as a position size; read it as a ceiling on the per-trade risk percentage.

A Setup That Applies Kelly to a 1.5 Payoff, 55% Win Rate System

Let me work one of the most common applications all the way through. This is the procedure for applying Kelly to a trend-following system that, after backtest and live checks, has secured a live-basis 55% win rate and a 1.5 payoff ratio. Every number must be a value measured against live fill prices.

Full-Kelly calculation: f* = 0.55 − 0.45/1.5 = 0.25. Full-Kelly says to set per-trade risk at 25% of capital; this value is not applied to actual betting but kept only as a ceiling.
Practical fraction: half-Kelly = 0.25 × 0.5 = 0.125. Accounting for estimation error, cut further from here and fix per-trade risk at 1% of capital. 1% is a conservative fraction, roughly 4% of full-Kelly's 25%.
Position size: Back it out by dividing the 1% per-trade risk by the stop distance. If the stop is 2.5% away from the entry, the position is 40% of capital (1% ÷ 2.5%); if the stop is 1.5%, about 66%.
Taking profit: Exit when the target payoff ratio 1.5R is reached. If the distance from entry to stop is 2.5%, the target sits at entry +3.75% (2.5% × 1.5).
Condition to scale up: Only when the win rate and payoff ratio measured over 50 or more live trades hold at the input values and the risk of ruin is under 1%, raise per-trade risk one step from 1% to 1.5%.
Invalidation: If the live win rate drops below 50% or the payoff ratio falls below 1.2, immediately lower per-trade risk to 0.5% and re-measure the inputs.

The core of this setup is that the full-Kelly number of 25% is never used in actual betting. That number exists only as a ceiling that must not be crossed, and the actual operation takes place at a quarter of it.

The Kelly Fraction Must Be Cut Again for Trade Independence and the Number of Concurrent Positions

Kelly's last assumption is that each bet is independent of the others. With coin flips, the previous result has no effect on the next. In markets, when you hold several positions at once, those positions often move together in the same direction.

This is why the Kelly fraction has to be cut further. Hold a BTC long and an ETH long at the same time and the two trades are not independent. In the decline that began after BTC printed its $69,000 high on November 10, 2021, and through the FTX collapse stretch of 2022, altcoins moved almost identically to BTC. Apply a Kelly fraction that assumes independence to each trade, and the actual combined risk is larger than the calculated value. Stake full-Kelly on each of three highly correlated positions and you have effectively staked three times full-Kelly on a single trade.

In practice, you manage the sum of risk across all positions held at once within a single Kelly limit. If you have set per-trade risk at 1%, size individual positions smaller so the combined risk of highly correlated positions does not exceed that limit. Measure the correlations among your holdings in advance and you can see how many positions are effectively one bet. When you enter several symbols on the chart at once, before adding up their individual Kelly fractions, first check the probability that those symbols crash together on the same day. The higher that probability, the more you have to reduce the combined Kelly, and only a fraction cut down that way is usable in practice.