- Kelly maximizes long-term geometric growth by sizing positions based on edge and odds, not intuition.
- For binary trades the formula is f* = (bp - q)/b. For continuous returns the Gaussian approximation gives f* ≈ μ/σ².
- Estimation error, transaction costs, leverage limits, and correlations make full Kelly aggressive for investors. Use fractional Kelly to control drawdown.
- Multi-asset Kelly becomes a mean variance problem: w = Σ⁻¹μ under the log-utility approximation. Correlations and covariance estimates are critical.
- Practical workflow: estimate edge and variance, compute Kelly, apply fractional scale, impose caps, and monitor re-estimates regularly.
Introduction
The Kelly Criterion is a mathematical method for determining the optimal fraction of capital to risk on a repeated bet or trade to maximize long-term growth. It links probability of success, payoff magnitude, and variance to a single position-size recommendation, and it has been used by gamblers, traders, and portfolio managers for nearly a century.
Why should you care about Kelly if you already use stop losses and diversification? Because position sizing is the lever that converts edge into compounded wealth. Are you risking too little and leaving returns on the table, or too much and courting permanent losses? This article explains the formulas, shows how to estimate inputs in practice, applies Kelly to examples using $AAPL and $NVDA, and gives clear rules for implementation in portfolios.
How the Kelly Formula Works
The classic Kelly formula was derived for a repeated binary wager. If a bet pays b to 1 when you win, the win probability is p, and the loss probability is q which equals 1 minus p, the optimal fraction of capital to wager is:
f* = (b p - q) / b
That formula maximizes the expected log growth of capital. For example if you have p = 0.55 and b = 1.5, then f* = (1.5*0.55 - 0.45)/1.5 which equals 0.25. That implies a 25 percent position size on each independent bet, assuming accurate inputs.
Continuous-return approximation
When returns are modeled as Gaussian and small relative to wealth, the Kelly fraction simplifies to a mean variance ratio.
f* ≈ μ / σ²
Here μ is the expected excess return per unit time and σ² is the variance. If an equity strategy has an expected annual return of 10 percent and a standard deviation of 20 percent, the continuous Kelly suggests f* ≈ 0.10 / 0.04 which equals 2.5 or 250 percent of capital, which requires leverage and carries large drawdowns in practice.
Practical Estimation of Inputs
Kelly's outputs are only as good as its inputs. You must estimate win probability, payoff ratios, expected return, and volatility from data. That requires statistical discipline and awareness of bias. How do you produce robust estimates when returns are noisy and regimes change?
Estimating probabilities and edge
For trade-based strategies you can estimate p as historical win rate and b as average win divided by average loss. Use out-of-sample tests and avoid lookahead bias. For strategies on stocks like $AAPL, compute returns per trade across many cycles and examine distribution tails rather than relying on means alone.
Estimating μ and σ
For continuous sizing use realized excess returns over a benchmark to estimate μ, and use exponentially weighted or shrinkage estimators for σ² to handle time-varying volatility. If you use annualized figures, ensure units match when plugging into the formula.
Handling transaction costs and slippage
Subtract expected trading costs from μ before computing f*. Small changes to μ can materially change Kelly sizes. If your edge is thin after costs, Kelly may recommend zero or negative sizes which means the trade should be avoided.
Fractional Kelly and Risk Controls
Full Kelly maximizes geometric growth asymptotically, but it also produces large volatility and drawdowns that many investors cannot tolerate. A common approach is fractional Kelly which scales the full Kelly fraction by a factor k between 0 and 1. Half-Kelly is widely used since it reduces volatility and roughly doubles the growth half-life relative to full Kelly while cutting peak drawdown significantly.
Why use fractional Kelly
Estimation error is the main reason. If your p and b estimates are noisy, full Kelly is prone to overbets. Fractional Kelly reduces sensitivity to input error and to serial dependence that violates independent bet assumptions. It also aligns with practical constraints such as leverage caps and psychological tolerance for drawdown.
Practical caps and overlays
Implement rules like maximum position size per security, sector concentration limits, and portfolio volatility caps. Combine Kelly sizing with stop loss policies and dynamic risk budgets to ensure positions remain manageable during stress.
Multi-Asset Kelly and Portfolio Allocation
When you have multiple, possibly correlated investments, Kelly optimization generalizes to maximizing expected log wealth. Under small return approximations this becomes a mean variance problem with the solution
w* = Σ⁻¹ μ
Here w* is the vector of portfolio weights, Σ is the covariance matrix of returns, and μ is the vector of expected excess returns. This links Kelly to classical portfolio theory, but the objective remains log utility which targets geometric growth rather than mean return for a given variance.
Example: two-asset Kelly
Imagine you have two strategies, one on $AAPL and one on $NVDA. Suppose expected excess returns are 8 percent and 12 percent respectively. Annual standard deviations are 25 percent and 40 percent respectively. Correlation is 0.6. Construct Σ and μ numerically, invert Σ, and compute w*. The resulting weights may recommend leverage if expected returns are high relative to variance. You should cap leverage and apply fractional Kelly to reflect uncertainty.
Covariance estimation matters
Small errors in Σ inverse can produce extreme weights. Use shrinkage, PCA filtering, or factor models to stabilize estimates. Regularly update covariance estimates and stress test allocations under regime shifts like rising correlations in market stress.
Real-World Examples and Worked Numbers
Concrete examples make Kelly tangible. Below are two scenarios, one trade-level binary case and one portfolio-level continuous case. Both use realistic numbers investors may encounter.
Example 1: Trade-level Kelly with $TSLA strategy
- Suppose a short-term momentum trade on $TSLA wins 56 percent of the time. Average winning trade returns 3 percent, average losing trade is 2 percent. Then b equals average win divided by average loss which is 1.5. p is 0.56 and q is 0.44.
- Apply binary Kelly: f* = (b p - q) / b = (1.5*0.56 - 0.44) / 1.5 = (0.84 - 0.44) / 1.5 = 0.40 / 1.5 = 0.2667.
- Kelly recommends about 26.7 percent of capital per independent trade. In practice you would apply fractional Kelly, for example half-Kelly of 13.3 percent, and limit exposure per ticker to a smaller cap like 5 to 10 percent of portfolio capital.
Example 2: Continuous Kelly for an equity allocation
- Assume an equity factor has expected excess return μ = 6 percent annually and volatility σ = 15 percent annually. Then f* ≈ μ / σ² = 0.06 / 0.0225 = 2.6667 or 267 percent of capital.
- This shows the danger of naive application. A 267 percent allocation implies heavy leverage and very large drawdowns. Most investors scale down to 20 to 50 percent of full Kelly and incorporate leverage constraints.
Common Mistakes to Avoid
- Overreliance on point estimates, which produce extreme Kelly sizes. How to avoid: use shrinkage, bootstrap confidence intervals, and apply fractional Kelly.
- Ignoring correlation across bets. How to avoid: use covariance matrices, factor models, and cap single security exposures.
- Forgetting transaction costs and taxes, which reduce edge. How to avoid: subtract realistic cost estimates from expected returns before computing Kelly.
- Applying full Kelly despite limited capacity or leverage constraints. How to avoid: impose hard caps and use conservative scaling like one quarter to one half Kelly.
- Assuming independence of sequential bets when serial dependence exists. How to avoid: test for autocorrelation and adjust effective sample size when estimating p and μ.
FAQ
Q: How should I choose fractional Kelly scale?
A: There is no universal fraction. Common choices are 0.25, 0.5, or 0.75. Choose based on confidence in your estimates, leverage limits, and drawdown tolerance. Backtest drawdown behavior to select a fraction that matches your risk capacity.
Q: Can I use Kelly for single-stock investments like $AAPL?
A: Yes, but treat single-stock Kelly cautiously. Estimation error and company-specific tail risk make full Kelly risky. Use small fractions, volatility scaling, and position caps to manage idiosyncratic risk.
Q: Does Kelly account for correlations and portfolio effects?
A: Multi-asset Kelly does incorporate correlations through the covariance matrix. However, covariance estimation error can generate unstable allocations. Use factor models and shrinkage to improve stability before computing multi-asset Kelly weights.
Q: Is Kelly suitable for long-term buy and hold investors?
A: Kelly optimizes geometric growth over repeated bets. For long-term buy and hold investors without opportunity to rebalance frequently, Kelly can inform sizing when adding new positions. Still, you should incorporate taxes, transaction costs, and personal liability constraints before applying Kelly-sized allocations.
Bottom Line
The Kelly Criterion links edge and variance to an explicit position size that maximizes long-term geometric growth. It provides a powerful framework for turning statistical advantage into compound returns, but its practical implementation requires humility. Estimation error, transaction costs, leverage limits, and correlations can make full Kelly inappropriate for many investors.
Actionable next steps: estimate your strategy's edge and volatility using robust methods, compute full Kelly, then select a fractional multiplier that fits your risk tolerance. Add practical overlays such as position caps, covariance shrinkage, and ongoing re-estimation. If you want to experiment, simulate Kelly sizing under historical and stress scenarios for $AAPL, $NVDA, and your own holdings before applying live capital.
At the end of the day, Kelly is a tool not a mandate. Use it to structure disciplined position sizing, and combine it with sound risk management to make your edge real and persistent.
