Kelly Sizer Worked Examples
Full Kelly maximizes long-run growth on paper but assumes perfect edge knowledge, which no practitioner has. The scenarios below are designed to show how much size drops when you move to half- or quarter-Kelly, and why that discount is worth taking. The Kelly formula for a binary bet is applied directly to each scenario; the half- and quarter-Kelly columns are the sizes practitioners actually use to survive estimation error and cut volatility.
Worked Examples
See the inputs and outcome together
Each scenario keeps the starting point, the outcome, and the actual lesson in one place so the page reads like a decision notebook, not a data dump.
- 1
Baseline: small edge, even payoff
A slight edge with symmetric wins and losses. This is the canonical case where a modest probability advantage produces a modest bet.
Full Kelly 10.0% of bankroll. Half Kelly 5.0%. Quarter Kelly 2.5%.
Win probability (p)
55%
Payoff ratio (b)
1.0
A 55/45 edge at even money is worth only a tenth of bankroll at full Kelly. Because the fraction is linear in your probability estimate, mistaking a true 53 percent edge for 55 percent overbets you by two thirds. That sensitivity is why desks run quarter Kelly here, sizing 2.5 percent and treating the formula as a ceiling, not a target.
- 2
Same edge, better payoff
Keep the 55 percent win rate but improve the reward-to-risk so each win pays 1.5 times each loss.
Full Kelly 25.0% of bankroll. Half Kelly 12.5%. Quarter Kelly 6.25%.
Win probability (p)
55%
Payoff ratio (b)
1.5
Raising the payoff from 1.0 to 1.5 more than doubles the Kelly fraction. Payoff asymmetry moves the recommended size faster than small changes in win probability do.
- 3
Stronger edge, even payoff
Return to even-money payoff but assume a stronger 60 percent win rate.
Full Kelly 20.0% of bankroll. Half Kelly 10.0%. Quarter Kelly 5.0%.
Win probability (p)
60%
Payoff ratio (b)
1.0
At even money the full-Kelly fraction equals 2p minus 1, so a 60 percent edge maps cleanly to 20 percent. The linearity makes this case a useful sanity check on the tool.
- 4
Sub-50 percent edge with a big payoff
A strategy that wins less than half the time but pays 2 times on wins, like a trend follower with frequent small losses.
Full Kelly 17.5% of bankroll. Half Kelly 8.75%. Quarter Kelly 4.375%.
Win probability (p)
45%
Payoff ratio (b)
2.0
Winning under half the time is still positive expectancy when the payoff is large enough, and Kelly sizes it at 17.5 percent. The catch is psychological: a 45 percent hit rate means losing streaks of five or six are routine, so the quarter-Kelly 4.4 percent is what keeps you in the seat through the drawdown that full Kelly would make unbearable.
Patterns
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Sources & References
- A New Interpretation of Information Rate — J. L. Kelly Jr., Bell System Technical Journal (1956)
- The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market — Edward O. Thorp (2007)
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