The Kelly formula and its derivation
The Kelly criterion was developed by John Kelly at Bell Labs in 1956 as a method for maximizing the long-run growth rate of a gambling or investment account. Kelly showed that the optimal betting fraction is derived from maximizing the expected logarithm of wealth, not the expected value of wealth. Maximizing expected value leads to betting everything on any positive-expected-value outcome, which produces ruin with probability approaching 1 over enough trials. Maximizing the log of wealth avoids ruin and produces the fastest sustainable compound growth.
The formula for the optimal Kelly fraction is f* = (bp - q) / b. Breaking this down: b is the net amount won per unit bet (if you bet $1 and win $2 net, b = 2), p is the probability of winning, and q is the probability of losing (q = 1 - p). Equivalently, this can be written as f* = p - q/b, or in a form some find more intuitive: f* = edge / odds, where edge is the expected profit per unit bet (bp - q) and odds is the net payout ratio b. If the formula returns a negative value, the bet has negative expected value and Kelly says not to bet.
A numerical example: suppose a stock has a 60% probability of gaining 30% and a 40% probability of losing 15%. Here p = 0.60, q = 0.40, b = 2.0 (net 30% gain on a 15% risk unit, ratio = 2). Kelly fraction = (2.0 x 0.60 - 0.40) / 2.0 = (1.20 - 0.40) / 2.0 = 0.80 / 2.0 = 40% of capital. Full Kelly in this case recommends allocating 40% of capital to this position. This is the theoretically optimal fraction given those exact probability and payoff estimates.
Kelly is derived from a specific model: binary outcomes (win or lose), known probabilities, known payoffs, and many repeated opportunities with the same parameters. Real investments violate all four assumptions to varying degrees. Returns are not binary; probabilities are estimates subject to significant error; payoffs are distributions, not fixed numbers; and the same setup rarely repeats identically. These violations mean the theoretical Kelly fraction overestimates the true optimal fraction for real investments.
Fractional Kelly and practical application
Full Kelly produces extreme volatility and large drawdowns in practice, even when the probability and payoff estimates are correct. This is because the Kelly fraction is calculated for a specific expected value, but actual results vary around that expectation. When outcomes cluster in the unfavorable tail (which happens regularly even in positive-expected-value situations), full Kelly portfolios can lose 50% or more before recovering. Most investors cannot maintain the discipline required to stay fully invested through those drawdowns.
Fractional Kelly addresses this by using a fraction of the calculated Kelly bet: typically 25% (quarter Kelly), 33% (third Kelly), or 50% (half Kelly). Half Kelly, for example, bets half the theoretically optimal fraction. The cost is meaningful: half Kelly grows at approximately 75% of the full Kelly long-run growth rate. The benefit is that maximum drawdowns are dramatically reduced (roughly halved in percentage terms). Quarter Kelly grows at approximately 44% of the full Kelly rate but with very small drawdowns, suitable for investors who prioritize capital preservation.
Probability estimation error is the most important practical concern in Kelly applications. The Kelly fraction is highly sensitive to the probability estimate: if the true winning probability is 55% but you estimate it at 65%, the Kelly fraction is substantially overstated and you are effectively over-betting. Because investors consistently overestimate their probability of being right (overconfidence is the most documented bias in investment research), the Kelly formula applied to investor estimates of win probabilities will systematically recommend too large a position. Using Kelly at all requires honest, well-calibrated probability estimates, not optimistic ones.
Kelly is most useful in systematic strategies where win rates and payoff ratios can be estimated from historical backtest data or statistical analysis of past signals. In fundamental equity investing, Kelly provides a theoretical framework but should be applied as an upper bound rather than a target: calculate the Kelly fraction from conservative probability estimates, then size the actual position at 25-50% of that fraction as a further safety margin against estimation error.
Limits and alternatives to Kelly sizing
Kelly assumes the investor knows their edge (the probability and payoff of each opportunity) and can bet the same edge repeatedly. In practice, both conditions are imperfect. The investor does not know their true edge in most fundamental investing situations; they have an estimate of it, which may be directionally correct but is far less precise than the formula requires. And investment opportunities are not repeatable in the strict sense: each company faces a unique set of circumstances, and the same quantitative parameters in two different businesses may reflect very different underlying economics.
The Kelly criterion also does not account for correlation between simultaneous positions in a portfolio. If multiple positions are highly correlated (all dependent on the same macro factor, all in the same sector), applying Kelly independently to each position can lead to total portfolio exposure far exceeding what would be optimal at the portfolio level. Portfolio-level risk budgeting constraints should always override individual Kelly calculations when multiple positions are held simultaneously.
Alternatives to Kelly for position sizing include fixed-fractional sizing (risk a fixed percentage of capital on each trade, regardless of estimated edge), equal-weight sizing (allocate equal dollar amounts to all positions), and volatility-adjusted sizing (size each position so its dollar volatility contribution is equal across the portfolio). Each alternative sacrifices some theoretical optimality for simplicity, robustness to estimation error, and resistance to the overconfidence that distorts Kelly estimates. Many experienced investors use Kelly as a sanity check (a position that Kelly would size at 2% should not be sized at 15%) rather than as a primary sizing tool.
Frequently asked questions
What is the Kelly criterion formula?
The Kelly criterion formula is f* = (bp - q) / b, where f* is the optimal fraction of capital to bet, b is the net odds received per unit bet (net payout divided by risk), p is the probability of winning, and q is the probability of losing (1 - p). Equivalently written as f* = edge / odds, where edge = bp - q. A negative result means the expected value is negative and Kelly says not to bet.
Why do investors use fractional Kelly instead of full Kelly?
Full Kelly maximizes long-run compound growth in theory but produces very large drawdowns in practice, because actual results deviate from expected values and because investors routinely overestimate their win probabilities. Half Kelly (50% of the calculated Kelly fraction) grows at roughly 75% of the full Kelly rate but with dramatically smaller drawdowns. Quarter Kelly (25%) grows more slowly but with very small drawdowns. Most practitioners use fractional Kelly as a margin of safety against probability estimation error and against the psychological difficulty of maintaining full Kelly discipline through the inevitable losing streaks.
Can the Kelly criterion be applied to stock investing?
The Kelly criterion can be applied to stock investing as an upper bound on position size, but with significant caveats. The formula requires estimates of win probability and payoff ratio that are inherently uncertain in fundamental equity investing. Overconfident probability estimates produce overstated Kelly fractions. The formula also ignores correlation between simultaneous positions. Most investors who use Kelly in equity investing apply it to conservative probability estimates and then take a fraction (25-50%) of the result, using Kelly as a ceiling rather than a target.
What is the difference between the Kelly criterion and fixed-fractional sizing?
Fixed-fractional sizing risks a constant percentage of current equity on every trade regardless of the perceived edge (typically 1-3% per trade). Kelly sizing varies the fraction based on the estimated edge in each specific opportunity: a high-probability, high-payoff opportunity gets a larger fraction than a low-probability one. Fixed-fractional is simpler, more robust to estimation error, and easier to implement consistently; Kelly is theoretically optimal but requires well-calibrated probability estimates to avoid over-betting and is sensitive to estimation error.
What happens if you bet more than the Kelly fraction?
Betting more than the Kelly fraction (over-betting) reduces the long-run compound growth rate relative to Kelly and increases the severity of drawdowns. Sufficiently large over-bets (more than twice the Kelly fraction) actually produce negative expected log-wealth growth, meaning wealth is expected to decline toward zero over time even in a positive-expected-value game. This is why Kelly is also described as the fraction that avoids ruin while maximizing growth: above Kelly, more risk produces less return, not more.