The Estimation Tax on Geometric Growth: Fractional Kelly as Edge-Reliability Shrinkage
The growth-optimal (Kelly) leverage for a single risky asset is f* = mu/sigma^2, where mu is the excess drift and sigma^2 the variance. In practice mu is estimated, not known. Working in the continuous-rebalancing geometric-Brownian-motion model where the long-run log-growth rate is exactly g(f) = f*mu - (1/2) f^2 sigma^2, I derive three exact results. (1) An investor who Kelly-bets an unbiased drift estimate suffers an expected geometric-growth loss of exactly Var(mu_hat)/(2 sigma^2), a closed-form "estimation tax" independent of the true edge. (2) Under a Gaussian prior mu ~ N(0, tau^2) and a noisy signal, the expected-growth-maximising leverage is the naive Kelly bet shrunk by the edge reliability rho = tau^2/(tau^2 + s^2); this derives fractional Kelly from log-growth optimisation rather than from a risk-aversion heuristic. (3) There is a sharp threshold: when the signal-noise variance exceeds the true edge variance (s^2 > tau^2), naively Kelly-betting raw estimates has NEGATIVE expected log-growth. All claims are proved in full; no empirical data, backtests, or simulations are reported.