Formula, in Risk & Portfolio Construction
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.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
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
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Sources
- 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)
Keep the topic connected
- Formula 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.
- Formula 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.
- Formula 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.
- Learn Kelly Criterion
What the Kelly criterion is, when full Kelly blows up, and why most working quants size at half- or quarter-Kelly.