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Position Sizing under Edge Variance

Bayesian-Kelly bet sizing when your edge is itself uncertain. Compare deterministic Kelly, Bayesian-adjusted, and conservative lower-bound versions.

By AI Fin Hub Research. Published . Corrections.

Runs in your browser. Nothing you enter is uploaded, and no account or API key is needed.

Education, not investment advice. Past performance does not predict future results. How we check our numbers.

Inputs

Expected edge per bet μ (%)

Your best estimate of the mean return per bet.

Uncertainty of the edge σ_μ (%)

Standard error of μ: how far the true edge could plausibly sit from your estimate.

Per-bet outcome variance σ²

Variance, not std-dev: 0.040 is a 20.0% per-bet std-dev.

Kelly multiplier (%)

Fraction of full Kelly you actually bet; 25% is quarter-Kelly.

How this differs from raw Kelly

Raw Kelly assumes you know μ exactly. In practice μ is estimated. Bayesian Kelly (Browne & Whitt 1996) penalizes that uncertainty by adding it to the denominator — bet less when you're less sure.

Bayesian-adjusted bet size

24.8%

of bankroll. 1.0% smaller than deterministic Kelly (25.0%), because edge uncertainty σ_μ = 2.0% adds σ_μ² to the variance in the denominator.

Sizing comparison

Deterministic

25.00%

μ assumed exact

Bayesian

24.75%

penalizes σ_μ

Conservative (μ − σ)

12.50%

lower-CI version

Tail risk of recommended bet

CVaR (5%)

9.22%

expected loss in worst 5% of outcomes

Bayesian Kelly before damping

99.0%

full Kelly; you bet 25% of it

Formulas

f_det    = μ / σ²_outcome
f_bayes  = μ / (σ²_outcome + σ²_μ)        ← penalty for edge uncertainty
f_cons   = (μ − σ_μ) / σ²_outcome         ← lower-bound estimate
CVaR_5%  = -f·μ + f·√σ²_outcome · φ(z)/α     (f = damped Bayesian size)

How to use it

  1. Enter your expected edge per bet μ and its uncertainty σ_μ (the standard error of that estimate), both in percent.
  2. Enter the per-bet outcome variance σ² (a variance: 0.04 is a 20% per-bet standard deviation) and the Kelly multiplier you actually bet.
  3. Read the Bayesian-adjusted bet size as a share of bankroll and how much smaller it is than deterministic Kelly.
  4. Compare the deterministic, Bayesian and conservative (μ − σ_μ) sizes, each after your Kelly multiplier.
  5. Check the 5% CVaR of the recommended bet: the expected loss in the worst 5% of outcomes under a normal approximation.

Questions people ask

How is this different from Kelly?

Deterministic Kelly sizes a bet at μ / σ², assuming the edge μ is known exactly. The Bayesian version adds the uncertainty of the edge to the denominator, μ / (σ² + σ_μ²), and the conservative version sizes on μ − σ_μ. All three are then scaled by your Kelly multiplier.

What's the role of edge uncertainty?

It shrinks the bet. In the Bayesian size σ_μ² is added to the outcome variance, so the haircut is small when σ_μ is small relative to the per-bet volatility and large when your edge estimate is as noisy as the bets themselves. The conservative size falls to zero once σ_μ reaches μ.

How do I estimate edge SE?

From your trade log: the standard error of the mean return per trade is the sample standard deviation divided by √n, or the spread of bootstrap means if returns are far from normal. The tool does not take a trade log; you enter the standard error directly.

What if I don't have a trade log yet?

Enter a prior. For paper or untested strategies, set σ_μ to half the edge or more, which pulls the conservative size down sharply, and tighten it as live results accumulate.

How does this compare to fixed fractional sizing?

Fixed fractional bets the same share of bankroll every time, whatever the edge. This tool sizes one bet profile from its edge, edge uncertainty and variance; to size trades with different conviction, run each profile through it separately.

Guides, formulas and worked examples

  • Calculators Kelly Criterion Calculator

    Size positions with full, half, quarter, or eighth Kelly and stress-test the choice with a drawdown Monte Carlo simulator. Client-side. Private by default.

  • Calculators Drawdown-Recovery Markov Simulator

    Time to recover from an N% drawdown given monthly Sharpe + skew + kurtosis. Cornish-Fisher Monte Carlo, percentile distribution of recovery months.

  • Calculators Deflated Sharpe Ratio Calculator

    Bailey & López de Prado deflated Sharpe — corrects observed Sharpe for selection bias across K trials. Reports deflated Sharpe, PSR (probability of skill).

  • 8 min read Conviction-Scaled Kelly Bet Sizing

    Full Kelly is brutally unforgiving of over-estimation. Quarter-Kelly with a conviction-tier mapping and a per-trade cap is the defensible default.

  • 10 min read Risk Parity vs Kelly: When Each Sizing Framework

    Risk parity and Kelly solve different problems. Risk parity wins when correlations are stable and edge is noisy; Kelly wins when edge is concentrated.

  • 6 min read Edge-Variance Sizing for LLM Signals

    Deterministic Kelly on a 2.5% edge says 55.6% of bankroll. The conservative lower-bound posterior says 15.6%. For LLM signals, conservative wins.

All articles

Use it from code

The same calculation as a JavaScript module you can import. It runs where you import it, with no request, key or rate limit.

import { compute } from "https://aifinhub.io/engines/position-sizing-edge-variance.js";

Input and output contract and the guide for agents.