Fractional Kelly Formula
Fractional Kelly stakes a fixed multiple c of the full Kelly fraction, so the bet size is c times f*. Because the growth rate is approximately quadratic near the optimum, betting half-Kelly keeps roughly three-quarters of the maximum growth while cutting the variance of growth in half. It is the standard defense against the fact that the true edge is never known precisely.
Formula
Copy the exact expression or work through it step by step below.
f_frac = c x f*
Approx growth at fraction c of Kelly: g(c) ~ g_max x (2c - c^2)
where f* is the full Kelly fraction and 0 < c <= 1 Variables
c
Kelly multiplier
The fraction of full Kelly to stake, between 0 and 1. Common choices are 0.5 (half-Kelly) and 0.25 (quarter-Kelly). Lower c trades growth for smoother equity and shallower drawdowns.
f*
Full Kelly fraction
The growth-maximizing fraction from the Kelly criterion, p - q/b for a binary bet or mu/sigma^2 in the continuous case.
f_frac
Fractional Kelly stake
The actual fraction of capital wagered, the product of the multiplier and full Kelly.
2c - c^2
Growth retention factor
The share of maximum log-growth captured at multiplier c, from the quadratic approximation of the growth curve near its peak. At c = 0.5 it equals 0.75; at c = 1 it equals 1.
Step By Step
- 1
Compute the full Kelly fraction from your edge estimate.
A setup yields full Kelly f* = 0.20.
- 2
Choose a multiplier c reflecting your confidence in the edge and your drawdown tolerance.
Use half-Kelly: c = 0.5.
- 3
Multiply to get the fractional stake.
0.5 x 0.20 = 0.10, a 10% stake.
- 4
Note the growth-versus-volatility trade by evaluating the retention factor 2c - c^2.
At c = 0.5 the retention factor is 2(0.5) - 0.25 = 0.75, so about three-quarters of maximum growth remains.
Worked Example
Applying half-Kelly to a setup with full Kelly of 20%
Full Kelly f*
0.20
Multiplier c
0.5 (half-Kelly)
Fractional stake = c x f* = 0.5 x 0.20 = 0.10. Growth retention = 2c - c^2 = 2(0.5) - (0.5)^2 = 1.0 - 0.25 = 0.75. Variance of growth scales with c^2 relative to a linear growth term, so halving c cuts the volatility contribution far faster than the growth.
Half-Kelly stake of 10% capturing about 75% of the maximum growth rate while roughly halving growth volatility. The asymmetry is the point: you give up a quarter of the theoretical growth to gain a large reduction in drawdown risk and a wide margin of safety against having overestimated the edge.
Common Variations
Try These Tools
Run the numbers next
Position Sizing under Edge Variance
Bayesian-Kelly bet sizing when your edge is itself uncertain. Compare deterministic Kelly, Bayesian-adjusted, and conservative lower-bound versions.
Risk-Adjusted Returns Calculator
Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.
Sources & References
- The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market — Edward O. Thorp, Handbook of Asset and Liability Management (2006)
- The Kelly Capital Growth Investment Criterion — MacLean, Thorp, Ziemba (eds.), World Scientific (2011)
Related Content
Keep the topic connected
Kelly Criterion Formula
The Kelly criterion formula: the bet fraction maximizing long-run log growth from win probability and payoff odds. The optimal sizing rule.
Expectancy Formula
The expectancy formula: win rate times average win minus loss rate times average loss. The expected profit per trade in R-multiples, with an example.
Maximum Drawdown Formula
The maximum drawdown formula: the largest peak-to-trough decline in an equity curve. The worst loss from a prior high, with a worked example.
Kelly Criterion
What the Kelly criterion is, when full Kelly blows up, and why most working quants size at half- or quarter-Kelly.