Kelly growth rate and doubling time on a 70/30 coin

Question

A coin lands heads with probability 0.7 and you are offered even-money bets, repeatedly, sizing to a fixed fraction of your current bankroll each flip. What fraction is optimal, what growth rate does it achieve, and roughly how many flips to double your money?

Accepted Answer

For an even-money bet the Kelly fraction is the edge: f^ = p - q = 0.7 - 0.3 = 0.4. Bet 40\% of bankroll each flip. The growth rate per flip is g(0.4) = 0.7\ln(1.4) + 0.3\ln(0.6) \approx 0.7(0.3365) - 0.3(0.5108) \approx 0.0823, about 8.2\% per flip, compounding. Wealth grows like e^{0.0823\,n}, so the doubling time is n = \frac{\ln 2}{g} = \frac{0.6931}{0.0823} \approx 8.4 	ext{ flips}. Half Kelly for comparison. At f = 0.2, g(0.2) = 0.7\ln 1.2 + 0.3\ln 0.8 \approx 0.0607, roughly 74\% of the full-Kelly growth but with half the bet size and much smaller swings, doubling in about \ln 2 / 0.0…

Kelly growth rate and doubling time on a 70/30 coin

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