Bet double Kelly and your growth vanishes

Question

You have a genuine edge and each individual bet is positive expected value. A colleague argues, "If every bet is +EV, I should bet as big as I can to maximize profit."

Show why betting twice the Kelly fraction gives zero long-run growth, and beyond that, negative growth, even though every bet is +EV.

Accepted Answer

The trap is confusing expected wealth (which grows with bet size) with typical compounded wealth (which does not). Because wealth multiplies, the right objective is expected log-growth, g(f) = f\mu - 	frac12 f^2\sigma^2. This is a downward parabola in f. It is zero at two points: g(f) = f\left(\mu - 	frac12 f\sigma^2ight) = 0 \;\Longrightarrow\; f = 0 \quad	ext{or}\quad f = \frac{2\mu}{\sigma^2} = 2f^. So growth is positive only between f = 0 and f = 2f^, peaking exactly at Kelly f^ = \mu/\sigma^2. Consequences: - At 2	imes Kelly, g = 0: you compound at zero. Your median wealth goes now…

Bet double Kelly and your growth vanishes

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