Kelly allocation to a volatile asset

Question

Each period an asset either rises by a factor of $1.5$ or falls by a factor of $0.7$ , each with probability $0.5$ . You can put a fraction $f$ of your capital into it and hold the rest in cash (no interest), rebalancing each period.

What fraction maximizes long-run growth, and why not go all in?

Accepted Answer

Investing fraction f, your wealth multiplies by 1 - f + 1.5f = 1 + 0.5f on an up move and 1 - f + 0.7f = 1 - 0.3f on a down move. So this is a bet with win fraction 0.5 and loss fraction 0.3. Maximize g(f) = 0.5\ln(1 + 0.5f) + 0.5\ln(1 - 0.3f). Set g'(f) = \frac{0.25}{1 + 0.5f} - \frac{0.15}{1 - 0.3f} = 0 \Rightarrow 0.25(1 - 0.3f) = 0.15(1 + 0.5f) \Rightarrow 0.10 = 0.15f \Rightarrow f = 	frac{2}{3}. Invest f^ = 	frac{2}{3} of your capital. The general rule for a win fraction a and loss fraction c is f^ = \frac{pa - qc}{ac} = \frac{0.5(0.5) - 0.5(0.3)}{0.5 \cdot 0.3} = \frac{0.10}{0.15} = …

Kelly allocation to a volatile asset

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