The Coin That Pays Fifty-Five Percent: Why Bigger Bets Lose
Posted: Thu Oct 29, 2026 10:25 am
The Coin That Pays Fifty-Five Percent: Why Bigger Bets Lose
Scenario: you can bet on a coin that lands your way 55 percent of the time, paying even money. You will bet a fixed fraction of your bankroll each flip. Which fraction grows your money fastest? A larger bet seems better, since you have an edge.
The growth rate per flip is the expected logarithm of your wealth multiple: p times ln(1 plus f) plus q times ln(1 minus f), where p is 0.55, q is 0.45, and f is the fraction wagered.
At f of 5 percent: 0.55 times ln(1.05), which is 0.55 times 0.04879, giving 0.02683. Plus 0.45 times ln(0.95), which is 0.45 times minus 0.05129, giving minus 0.02308. Net: plus 0.00375 per flip.
At f of 10 percent: 0.55 times 0.09531 is 0.05242. Plus 0.45 times minus 0.10536 is minus 0.04741. Net: plus 0.00501.
At f of 20 percent: 0.55 times 0.18232 is 0.10028. Plus 0.45 times minus 0.22314 is minus 0.10041. Net: about zero.
At f of 30 percent: 0.55 times 0.26236 is 0.14430. Plus 0.45 times minus 0.35667 is minus 0.16050. Net: minus 0.0162.
So betting 30 percent of your bankroll on a favorable coin loses money over time. The growth peaks near 10 percent, the Kelly fraction, which for even-money bets is 2p minus 1, or 10 percent here.
Lesson for traders: a positive edge does not justify any size. Oversizing turns an edge into a slow loss. Practical risk is much lower than Kelly.
Exercise: compute the growth rate at 2 percent and compare with 10 percent.
Scenario: you can bet on a coin that lands your way 55 percent of the time, paying even money. You will bet a fixed fraction of your bankroll each flip. Which fraction grows your money fastest? A larger bet seems better, since you have an edge.
The growth rate per flip is the expected logarithm of your wealth multiple: p times ln(1 plus f) plus q times ln(1 minus f), where p is 0.55, q is 0.45, and f is the fraction wagered.
At f of 5 percent: 0.55 times ln(1.05), which is 0.55 times 0.04879, giving 0.02683. Plus 0.45 times ln(0.95), which is 0.45 times minus 0.05129, giving minus 0.02308. Net: plus 0.00375 per flip.
At f of 10 percent: 0.55 times 0.09531 is 0.05242. Plus 0.45 times minus 0.10536 is minus 0.04741. Net: plus 0.00501.
At f of 20 percent: 0.55 times 0.18232 is 0.10028. Plus 0.45 times minus 0.22314 is minus 0.10041. Net: about zero.
At f of 30 percent: 0.55 times 0.26236 is 0.14430. Plus 0.45 times minus 0.35667 is minus 0.16050. Net: minus 0.0162.
So betting 30 percent of your bankroll on a favorable coin loses money over time. The growth peaks near 10 percent, the Kelly fraction, which for even-money bets is 2p minus 1, or 10 percent here.
Lesson for traders: a positive edge does not justify any size. Oversizing turns an edge into a slow loss. Practical risk is much lower than Kelly.
Exercise: compute the growth rate at 2 percent and compare with 10 percent.