Kelly Criterion Calculator

The classic Kelly formula for a simple bet with two outcomes (win or lose) is:

f* = (bp − q) / b

Where:

• f* — The optimal fraction of your bankroll to bet (the Kelly fraction)
• b — The “odds” or payoff ratio. If you risk $100 and stand to win $150, then b = 150 / 100 = 1.5. This is the ratio of the net win amount to the net loss amount.
• p — The probability of winning
• q — The probability of losing, which is simply 1 − p

Worked example: Suppose you have a coin flip bet where you win $200 if heads and lose $100 if tails, and the coin is fair (50/50).

• b = 200 / 100 = 2.0
• p = 0.50
• q = 0.50
• f* = (2.0 × 0.50 − 0.50) / 2.0 = (1.0 − 0.50) / 2.0 = 0.25

Kelly says you should bet 25% of your bankroll on each flip. With a $10,000 bankroll, that means a $2,500 bet. Any more than 25% reduces your long-term growth rate, and betting above 50% (2x Kelly) would actually cause your bankroll to shrink over time despite the positive expected value.

Important note: If f* comes out negative, it means you have no edge — the expected value is negative and Kelly says do not bet at all. If f* is greater than 1, it technically suggests using leverage, but in practice this is extremely risky and not recommended.

Half-Kelly means betting exactly half of the amount the full Kelly formula recommends. If full Kelly says bet 20% of your bankroll, half-Kelly says bet 10%. Quarter-Kelly takes it even further — bet just 5% in this example.

While full Kelly is theoretically optimal for maximizing long-term growth, there are several compelling reasons why most professionals prefer fractional Kelly:

• Smoother ride — Full Kelly produces enormous volatility. A full Kelly bettor can expect to lose 50% or more of their bankroll at some point before recovering. Half-Kelly retains approximately 75% of the growth rate of full Kelly while cutting the variance roughly in half. For most people, this tradeoff is well worth it.
• Estimation error protection — The Kelly formula assumes you know the exact probability of winning and the exact payoff ratio. In the real world, these are estimates. If you overestimate your edge by even a small amount, full Kelly can turn into over-betting, which destroys wealth. Fractional Kelly provides a crucial margin of safety against errors in your probability estimates.
• Psychological sustainability — Even with a proven edge, large drawdowns cause emotional distress that leads to poor decision-making. Half-Kelly keeps drawdowns manageable enough that most people can stick with the strategy long-term.
• Real-world constraints — In investing (as opposed to pure gambling), you face transaction costs, taxes, liquidity constraints, and correlated bets. All of these make fractional Kelly more practical than full Kelly.

The growth rate tradeoff: If full Kelly has an expected growth rate of g, then half-Kelly has a growth rate of approximately 0.75g, and quarter-Kelly has a growth rate of approximately 0.4375g. You retain a disproportionately large share of the growth while dramatically reducing risk. This makes fractional Kelly one of the best risk-reward tradeoffs in all of finance.

Over-betting — wagering more than the full Kelly fraction — is one of the most counterintuitive and dangerous mistakes in bankroll management. Even with a guaranteed positive edge, betting too much will cause your bankroll to shrink over time and eventually approach zero.

Why over-betting leads to ruin:

• Geometric vs. arithmetic returns — A 50% loss followed by a 50% gain does not bring you back to even. $100 → $50 → $75. You are down 25%. This asymmetry compounds over many bets. Over-betting amplifies this asymmetry to devastating effect.
• The 2x Kelly boundary — Betting exactly twice the Kelly fraction produces a zero long-term growth rate. Your bankroll essentially goes nowhere over time. Betting more than 2x Kelly produces a negative growth rate — guaranteed long-term ruin.
• Volatility drag — Higher bet sizes mean larger swings. These swings create “volatility drag” where the geometric mean of returns falls further and further below the arithmetic mean. Full Kelly already experiences significant volatility drag; anything above it becomes self-destructive.

Concrete example: Imagine a coin flip that pays 2:1 (win $200 or lose $100). Kelly says bet 25%. If instead you bet 60% of your $10,000 bankroll:

• Win: $10,000 + $12,000 = $22,000 (up 120%)
• Loss: $10,000 − $6,000 = $4,000 (down 60%)
• After one win and one loss (in any order): $22,000 × 0.4 = $8,800. You lost money despite winning half the time on a favorable bet.

This is the core lesson of the Kelly criterion: having an edge is not enough. You must size your bets correctly or the edge works against you. The growth curve chart in this calculator shows this effect visually — the 2x Kelly line trends downward even when the other fractions are compounding upward.

Applying the Kelly criterion to stock market investing requires some adaptation from the simple bet framework, but the core principle remains the same: size your positions based on your edge and the odds.

Adapting Kelly for continuous distributions:

• Simple approach — Use the basic Kelly formula by defining “win” as the trade hitting your target and “loss” as hitting your stop loss. Estimate the probability of each based on your analysis or historical win rate. This is a practical approximation most traders use.
• Continuous Kelly formula — For normally distributed returns, the Kelly fraction simplifies to: f* = (expected return − risk-free rate) / variance of returns. This version works better for portfolio allocation where returns are not binary.
• Multi-asset Kelly — The generalized Kelly criterion can allocate across multiple correlated assets simultaneously, accounting for the covariance between positions. This is how quantitative hedge funds often implement it.

Practical considerations for stock investors:

• Estimate conservatively — In the stock market, you rarely know your true edge with precision. Always use conservative probability estimates and apply fractional Kelly (half or quarter) to protect against overconfidence.
• Correlated positions — Stocks are not independent bets. If you hold five tech stocks, a market crash hits them all simultaneously. Kelly must account for correlations, or you will be effectively over-betting.
• Combine with stop losses — Using Kelly alongside defined stop losses creates a clean framework: the stop loss defines your “loss amount,” your target defines your “win amount,” and your conviction level defines the win probability.
• Position limits — Most risk managers cap any single position at 5–10% of the portfolio regardless of what Kelly recommends. Use Kelly as a guide, not an absolute rule.

The Kelly criterion is particularly useful for concentrated portfolio investors who want a disciplined framework for deciding how much to allocate to high-conviction ideas versus diversified positions.

Like any mathematical model, the Kelly criterion makes simplifying assumptions that do not perfectly reflect reality. Understanding these limitations is essential for using Kelly wisely.

Key assumptions:

• Known probabilities — Kelly assumes you know the exact probability of winning. In practice, probabilities are always estimates. The further your estimate is from reality, the worse Kelly performs — which is the primary reason to use fractional Kelly.
• Independent bets — The basic formula assumes each bet is independent of the others. In financial markets, positions are often correlated, which means the effective position size is larger than each individual Kelly calculation suggests.
• Infinite time horizon — Kelly maximizes growth over an infinite number of bets. Over short horizons, Kelly-sized bets can produce severe drawdowns that are psychologically or practically unacceptable.
• Binary outcomes — The classic formula assumes you either win a fixed amount or lose a fixed amount. Real investments have continuous outcome distributions with varying magnitudes of wins and losses.
• Infinitely divisible capital — Kelly assumes you can bet any fractional amount. In practice, you cannot buy 0.37 of a share (on most platforms) or make bets in arbitrary amounts.

Practical limitations:

• Drawdown tolerance — Full Kelly can produce 50–80% drawdowns even with a significant edge. Most investors and institutions cannot tolerate this level of volatility.
• Transaction costs and taxes — Kelly does not account for the costs of trading. Frequent rebalancing to maintain Kelly-optimal sizes can eat into returns.
• Utility and risk preferences — Kelly is optimal only if your goal is maximizing long-run wealth. If you have different risk preferences (such as avoiding catastrophic losses), Kelly may be too aggressive even at half-Kelly.

Despite these limitations, the Kelly criterion remains one of the most rigorous and widely-used frameworks for position sizing. The key is to use it as a ceiling on how much to bet, not a floor, and to always apply a fractional reduction.

The expected geometric growth rate (also called the Kelly growth rate or log-optimal growth rate) is the rate at which your bankroll is expected to compound over many repeated bets. It is the metric that the Kelly criterion is specifically designed to maximize.

The formula for expected geometric growth:

g(f) = p × ln(1 + f × b) + q × ln(1 − f)

Where:

• g(f) — The expected geometric growth rate for a given fraction f
• f — The fraction of bankroll wagered
• b — The payoff ratio (win/loss)
• p, q — The probability of winning and losing
• ln — The natural logarithm

Why geometric matters more than arithmetic:

• Compounding is multiplicative — When you reinvest returns, each bet multiplies your bankroll. The geometric mean of these multipliers determines your actual long-run wealth. A strategy with a high arithmetic expected return but high variance can have a low or even negative geometric return.
• The Kelly fraction maximizes g(f) — Taking the derivative of g(f) and setting it to zero yields exactly the Kelly formula. This is not a coincidence — it is the mathematical proof that Kelly sizing is optimal for wealth growth.
• Diminishing returns beyond Kelly — As you increase f beyond the Kelly fraction, g(f) decreases. At 2x Kelly, g(f) drops to zero. Beyond that, it goes negative. This is why over-betting is so destructive.

In the growth simulation chart on this calculator, each line represents the expected bankroll multiplier using the geometric growth rate for that Kelly fraction. The full Kelly line grows fastest, but has the most variance. The half-Kelly line sacrifices a quarter of the growth rate in exchange for dramatically smoother compounding.