Kelly Fraction Under Parameter Uncertainty

The Kelly fraction maximizes long-run growth if the edge is what you think it is. You estimated the edge from a finite sample, and the growth penalty for betting too much is steeper than the penalty for betting too little, so a noisy estimate should be bet at a fraction of full Kelly — the usual “half Kelly” advice, made precise. In the continuous case the answer has a closed form: shrink full Kelly by t² ÷ (1 + t²), where t is the t-statistic of your edge estimate. A Sharpe of 1 over four years (t = 2) says bet 80% of Kelly; the same Sharpe over one year says bet half. The discrete-bet case is solved numerically. Defaults are editable; everything runs in your browser. The variance this sizing fights is visible in the equity-curve simulator.

How to read this

Full Kelly. For a bet that pays b per unit risked with win probability p, the growth-optimal fraction of bankroll is f = p − (1 − p) ÷ b. For a continuous return stream with mean μ and volatility σ, it is leverage f = μ ÷ σ². Both maximize the expected log of wealth, which is the only objective that makes “long-run growth” well defined. Growth is zero again at exactly twice Kelly: the curve is asymmetric, and over-betting is punished faster than under-betting is.

Why uncertainty shrinks the bet. You do not bet Kelly(p); you bet Kelly(p̂), and p̂ is noisy. Because the growth curve is concave and steeper on the right, a bet that is sometimes too large and sometimes too small loses more on the over-bets than it gains on the under-bets, so the strategy “bet m × Kelly(p̂)” has its highest expected growth at some m below one. In the continuous case the expected growth is SR² × (m − m² (1 + 1/t²) ÷ 2), with t = SR × √years, which peaks at m* = t² ÷ (1 + t²). The discrete case is the same calculation done by quadrature over the sampling distribution of the win rate. This is the Baker–McHale framing of fractional Kelly: not a taste for safety, but the optimal response to an estimate.

The two probabilities. P(edge ≤ 0) is the chance your sample came from a zero-edge process, the usual one-sided test. P(over-betting) is more useful for sizing: the chance that the true Kelly is below half your bet, which is the condition for your bet to have negative expected growth. It is uncomfortably large for short track records even when the edge is probably real.

Drawdowns. For a continuous Kelly bettor at fraction m of full Kelly, the probability of ever falling to a fraction x of the starting bankroll is x^(2/m − 1): full Kelly has a 50% chance of a 50% drawdown at some point, half Kelly has 12.5%, quarter Kelly 1.6%. Discrete bets, fat tails and serial correlation all make the real numbers worse than this; the equity-curve simulator shows the discrete version as a fan.

What it does not fix. The win rate or Sharpe you typed may itself be the best of many trials, which no amount of shrinkage corrects: the deflated Sharpe calculator comes first. Estimation of the edge is also only one of the seven ways a high Sharpe fails to become money; costs and capacity set the leverage you can actually run before Kelly is consulted. And if the edge is regime-dependent, the right sample size is the number of regimes you have seen, not the number of bets.

Formulas: Kelly (1956); Thorp’s drawdown result for continuous-time fractional Kelly; the shrinkage follows Baker & McHale, “Optimal Betting Under Parameter Uncertainty: Improving the Kelly Criterion” (Decision Analysis, 2013), with the true parameter set at the point estimate and the bettor’s estimate drawn from its sampling distribution. Normal CDF via the Hart/West rational approximation.