Elements · Proposition V of VI

Size is a dial, not a switch.

VWhy it holds

Position size is not all or nothing. Too small and the edge is wasted; too large and one bad stretch ends it. We size below the growth-optimal point, on purpose.

Models take the same discipline — simple enough to trust, rich enough to matter. Too simple underfits; too complex memorises the past.

∎Proof

g(f) = \mu f - \tfrac{1}{2}\sigma^2 f^2, \quad f^* = \frac{\mu}{\sigma^2}, \quad g\!\left(\tfrac{f^*}{2}\right) = \tfrac{3}{4}\,g(f^*), \quad g(2f^*) = 0

Proof · neither greed nor fearIllustrative

Expected log growth of capital against position size, in units of the Kelly fraction f* = μ/σ². Fear — sizing near zero — leaves growth unused. Greed — beyond f* — buys more variance for less growth, and at 2f* grows nothing at all. Half of f* keeps three quarters of the growth for a quarter of the variance. The same curve governs models: too simple underfits, too complex overfits. Exact arithmetic, not data.

Q.E.D.