Kelly when a loss only costs part of the stake

Question

Consider a trade where a win pays net odds $b$ times what you risk, but a loss only forfeits a fraction $a$ of the stake (a stop-loss caps the damage). With win probability $p$ :

Derive the optimal Kelly fraction, then evaluate it for $p = 0.5,\ b = 1,\ a = 0.5$ .

Accepted Answer

Betting fraction f of bankroll, a win multiplies wealth by (1 + bf) and a partial loss by (1 - af). Maximize expected log growth: g(f) = p\,\ln(1 + bf) + q\,\ln(1 - af), \qquad q = 1 - p. Differentiate and set to zero: g'(f) = \frac{pb}{1 + bf} - \frac{qa}{1 - af} = 0 \;\implies\; pb(1 - af) = qa(1 + bf). Expand: pb - pbaf = qa + qabf \implies pb - qa = abf(p + q) = abf, so \boxed{f^ = \frac{pb - qa}{ab}.} Setting a = 1 recovers the familiar f^ = (pb - q)/b, a good check. Plug in p = 0.5,\ q = 0.5,\ b = 1,\ a = 0.5: f^ = \frac{0.5 \cdot 1 - 0.5 \cdot 0.5}{1 \cdot 0.5} = \frac{0.5 - 0.25}{0.5}…

Kelly when a loss only costs part of the stake

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