Kelly Criterion Calculator

How to use this calculator

Enter your win probability (the percentage of the time you expect to win) and your win/loss ratio (how much you win on average divided by how much you lose on average). The calculator returns the Full Kelly percentage, Half Kelly, Quarter Kelly, and your expected value per dollar risked.

If the result is 0% or negative, the math is telling you not to take this bet. A negative Kelly means negative expected value — you lose money in the long run regardless of how the bet is structured.

The Kelly Criterion formula

Where:

• f* = fraction of bankroll to bet
• p = probability of winning
• q = probability of losing (1 − p)
• b = win/loss ratio (average win / average loss)

John L. Kelly Jr. derived this formula in 1956 while working at Bell Labs. His original paper was about information theory applied to signal transmission. The gambling and trading applications came later, popularized by Ed Thorp in “Beat the Dealer” (1962) and “The Mathematics of Gambling” (1984).

The formula maximizes the long-run geometric growth rate of your bankroll. That’s different from maximizing expected value: a bet that maximizes expected value (expected value is linear) can have a lower expected geometric growth than the Kelly bet. Kelly finds the sweet spot where you’re growing wealth as fast as possible without risking ruin.

Why Kelly maximizes geometric growth

Here’s the intuition. Suppose you flip a fair coin and win 2x your bet on heads, lose 1x on tails. Win probability = 50%, win/loss ratio = 2. Kelly says bet 25%.

What happens if you bet 100%?

On a win (50% chance): your bankroll doubles to 200. On a loss (50% chance): your bankroll goes to 0.

Expected value after one flip: 0.50 × 200 + 0.50 × 0 = 100. Same as you started. But expected value doesn’t capture ruin.

After two flips: 25% chance of 400, 25% chance of 0 (win-win or loss-first), 50% various outcomes. A significant fraction of paths ends in ruin.

At 25% Kelly betting: after 1,000 flips, the median outcome is dramatically positive. The geometric mean of each flip = (1 + 0.25 × 2)^0.5 × (1 − 0.25)^0.5 = 1.5^0.5 × 0.75^0.5 = 1.061, meaning 6.1% expected growth per flip. That compounds to enormous wealth at 1,000 flips.

The full Kelly fraction is the bet size that maximizes that geometric mean growth rate.

Full Kelly in practice: it’s too aggressive

The math of Full Kelly is correct. The experience of Full Kelly is brutal.

At Full Kelly, your bankroll variance is enormous. Ed Thorp famously described Full Kelly as “one of the most efficient ways to go broke quickly if your edge estimate is slightly wrong.”

Consider: if you think your win probability is 55% but it’s actually 50% (no edge), Full Kelly with a 2:1 win/loss ratio says bet 12.5% of bankroll. Without an edge, every trade erodes your wealth at a steady rate.

The deeper problem is that you never know your true edge precisely. Win probability estimates come from past data, which is finite and imperfect. A sample of 100 trades has a standard error of about 5 percentage points in your win rate estimate. That uncertainty translates directly into uncertainty in your Kelly fraction.

The practical rule: treat Kelly as an upper bound, not a target.

Half Kelly: the practitioner’s standard

Half Kelly (f*/2) has become the default for serious practitioners because it cuts variance dramatically while sacrificing only a modest amount of growth.

The mathematics are elegant here. At Full Kelly, the growth rate is g* = E[ln(1 + f*b)] (geometric mean). At Half Kelly, the growth rate is about 75% of Full Kelly’s rate. The variance is reduced to about 25% of Full Kelly’s variance.

You give up 25% of maximum growth rate but reduce variance by 75%. For most real-world applications, that’s an obvious trade.

Warren Buffett has referenced Kelly-like reasoning in his investment approach. The academic finance literature on Kelly includes work by Thorp, Paul Samuelson, and Nassim Taleb (who prefers fractional Kelly but is skeptical of full Kelly for fat-tailed distributions).

In practical trading, Half Kelly combined with a maximum per-trade cap (e.g., never bet more than 5% of bankroll regardless of Kelly output) is a common conservative implementation.

Estimating your win probability

The win probability input is where most users struggle. Here’s a practical approach.

From historical trades: Count your wins and divide by total closed trades. If you have 200 trades with 110 wins, your estimated win rate is 55%. But the true win rate is somewhere in a range — with 200 samples, you have a 95% confidence interval of roughly ±7 percentage points, so your “55% win rate” really means “probably between 48% and 62%.”

For new strategies: You don’t have data. Be conservative. Assume 45–50% until proven otherwise. Most trading strategies don’t outperform coin-flipping until they’re heavily refined.

Be careful with backtested win rates: Backtested results almost always look better than live results. Overfitting to historical data, not accounting for slippage, and looking-forward bias are all common problems. Apply at least a 5–10 percentage point haircut to any backtested win rate when using it for Kelly sizing.

The Kelly formula is highly sensitive to win probability estimates near 50%. A change from 52% to 48% flips a slightly positive edge to a slightly negative one. Do not treat your historical win rate as a precise, permanent fact.

Estimating your win/loss ratio

Win/loss ratio = average winning trade / average losing trade.

This is a realized number from your trade history, not a planned R:R ratio. These often differ substantially.

You plan 2:1 R:R on every trade. But in practice:

• Some winners get cut short (you exit before target at 1.3:1)
• Some losing trades get stopped out at 1.1× the planned risk (slippage, gap)
• A few home runs exceed the planned target

If your planned R:R is 2:1 but your realized average winner is 1.6× your average loser, use 1.6 in the Kelly formula — not 2.

This is one of the reasons live trading underperforms backtested results even when the strategy hasn’t changed. The planned R:R looks great on paper; the realized R:R is lower due to execution friction and psychological interference.

The Kelly Criterion for portfolio allocation

Kelly extends beyond single-bet sizing. The multi-asset Kelly problem asks: how should you allocate across N assets simultaneously to maximize portfolio growth?

The full multi-asset Kelly solution requires estimating correlations between positions and solving a system of equations. It’s computationally intensive and highly sensitive to input errors.

A simplified version used by some practitioners: size each position independently using single-asset Kelly, then scale down all positions proportionally so total portfolio weight sums to 100% (or less, with cash acting as a buffer). This ignores correlations but produces reasonably sensible allocations in practice.

Thorp’s hedge fund, Princeton Newport Partners, reportedly used Kelly-based sizing from 1969 to 1988, generating approximately 15.1% annual returns net of fees with no down years. He used Half Kelly and emphasized that precise edge estimation was the key challenge.

Kelly and the St. Petersburg Paradox

The Kelly Criterion connects to one of the oldest puzzles in probability theory: the St. Petersburg Paradox.

The paradox: a coin is flipped repeatedly until it lands tails. You receive $2^n where n is the number of flips. The expected value of this game is infinite, yet nobody would pay more than $20–30 to play it.

The resolution: expected value (arithmetic mean) is the wrong thing to maximize. You should maximize expected utility of log wealth — which is exactly what Kelly does.

Kelly’s formula is a special case of maximizing E[ln(wealth)]. This log utility function embeds the insight that a 50% loss requires a 100% gain to recover, so large losses are catastrophically bad even when arithmetic expected value looks positive.

This is why Kelly is the right framework for repeated bets and trading decisions, where you’re compounding wealth over many periods, not just maximizing one outcome.

Limitations and when not to use Kelly

Fat-tailed distributions. Kelly was derived assuming log-normal return distributions. Real financial markets have fat tails — crashes happen more often than a normal distribution predicts. In fat-tailed environments, the optimal fraction is lower than Kelly suggests, because the downside scenarios are more severe than the model accounts for.

Short time horizons. Kelly maximizes long-run geometric growth. In the short run, it can produce devastating drawdowns. If you can’t survive a 50% drawdown emotionally or financially, don’t use Full Kelly even if the math says you should.

Non-ergodic outcomes. If ruin is possible (losing everything), Kelly’s guarantee of survival breaks down unless you’re perfectly precise about inputs and execution. In practice, small errors in win probability and win/loss ratio combined with occasional large adverse moves can deplete capital faster than Kelly’s model predicts.

Unknown or changing distributions. Kelly works when the underlying bet parameters are stable. If you’re trading in regime-changing markets where your edge disappears for months at a time, Kelly’s historical edge estimate can be wildly wrong.

Kelly, position sizing, and the 1% rule: reconciling different frameworks

Traders who use the 1% rule (never risk more than 1% of account per trade) and traders who use Kelly sizing are often at different points on the same spectrum.

At a 55% win rate and 2:1 R:R, Kelly says bet about 17.5% of bankroll. Quarter Kelly is 4.375%. Half Kelly is 8.75%. Even Quarter Kelly is far more than 1%.

The 1% rule is essentially ultra-fractional Kelly. It’s appropriate for:

• New traders who don’t have enough data to estimate their edge reliably
• Strategies with high uncertainty in win rate estimates
• Traders who prioritize durability and longevity over growth-rate optimization
• Discretionary traders who want a simple, consistent rule without complex calculations

Kelly sizing is appropriate for:

• Systematic strategies with large sample sizes (500+ trades) to validate edge
• Quantitative traders who can model their distribution precisely
• Situations where you’re highly confident in your edge estimate

Most professional discretionary traders use fractional Kelly implicitly by applying a conservative fixed percentage rule. They’re not solving the Kelly equation — they’re applying a rule of thumb that behaves like a very conservative Kelly fraction under their actual edge distribution.