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Worked examples, in Risk & Portfolio Construction

Position Sizing Under Edge Variance: Examples

The key insight is how much sizing shrinks when you account for uncertainty in the edge estimate, even when the point estimate stays fixed. These scenarios apply three approaches to the same edge: the standard continuous Kelly fraction (edge mean over outcome variance), the Bayesian form from Browne and Whitt (which inflates the denominator by the squared standard deviation of the estimate), and a conservative form that subtracts one standard deviation before sizing. The divergence as estimation noise grows is the point.

4 examples. Published May 26, 2026. By AI Fin Hub Research.

Worked examples

See the inputs and outcome together

Each scenario keeps the starting point, the outcome, and the actual lesson in one place so the page reads like a decision notebook, not a data dump.

  1. 1

    Known edge: all three methods agree

    You are certain of the edge, so the estimate has zero standard deviation. This is the textbook deterministic Kelly case and the baseline for everything below.

    Deterministic Kelly 1.00, Bayesian Kelly 1.00, conservative Kelly 1.00.

    Edge mean

    0.04

    Edge std dev

    0.00

    Outcome variance

    0.04

    Kelly fraction

    1.0 (full)

    With edge mean equal to outcome variance the full Kelly fraction is exactly 1.0, and zero estimation noise collapses all three methods onto the same number. Any divergence below is purely the price of uncertainty.

  2. 2

    Modest estimation noise

    Same edge and same outcome variance, but you now admit the edge estimate has a standard deviation of 0.02. The point estimate is unchanged.

    Deterministic Kelly 1.00, Bayesian Kelly 0.99, conservative Kelly 0.50.

    Edge mean

    0.04

    Edge std dev

    0.02

    Outcome variance

    0.04

    Kelly fraction

    1.0 (full)

    Bayesian Kelly barely moves because the estimation variance (0.0004) is tiny next to outcome variance (0.04). The conservative method halves the bet because subtracting one standard deviation cuts the edge from 0.04 to 0.02. The minus-one-sigma rule is far more punishing than the variance-penalty rule at low noise.

  3. 3

    Estimation noise equal to the edge

    The standard deviation of the edge estimate now equals the edge mean itself, so a one-standard-deviation move would wipe the edge to zero. Quarter Kelly is applied.

    Deterministic Kelly 1.00, fractional Bayesian 0.24, fractional conservative 0.00.

    Edge mean

    0.04

    Edge std dev

    0.04

    Outcome variance

    0.04

    Kelly fraction

    0.25 (quarter)

    The conservative method sizes to zero: subtracting one standard deviation leaves no edge, so it refuses the bet entirely. Bayesian Kelly still bets a quarter of 0.96 because it treats the noise as a denominator penalty rather than a hard cutoff. This is the case that separates a hard cutoff from a graceful shrink.

  4. 4

    Bigger edge, bigger swings, half Kelly

    A higher-conviction signal: a 6 percent edge but with a larger per-bet outcome variance and moderate estimation noise, sized at half Kelly as most desks would.

    Deterministic Kelly 0.67, fractional Bayesian 0.33, fractional conservative 0.17.

    Edge mean

    0.06

    Edge std dev

    0.03

    Outcome variance

    0.09

    Kelly fraction

    0.5 (half)

    Raising outcome variance from 0.04 to 0.09 pulls deterministic Kelly down to two thirds, before any uncertainty discount. The half-Kelly Bayesian bet of 0.33 and conservative bet of 0.17 bracket the range a risk committee would actually argue over.

Patterns

Continuous Kelly is edge mean over outcome variance, so when those two are equal the full fraction is exactly 1.0.
Bayesian Kelly penalizes estimation noise smoothly by adding it to the denominator; the minus-one-sigma method penalizes it as a hard haircut on the edge.
When the edge standard deviation equals the edge mean, the conservative method sizes to zero while Bayesian Kelly still bets.
Outcome variance moves the bet before any uncertainty discount, so confirm it first when a recommended size surprises you.

Run the numbers next

  • 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.

Sources

  1. Portfolio Choice and the Bayesian Kelly Criterion , Browne, S. and Whitt, W., Advances in Applied Probability (1996)
  2. The Kelly Criterion and the Stock Market , Rotando, L. M. and Thorp, E. O., American Mathematical Monthly (1992)

Keep the topic connected

  • Learn Kelly Criterion

    What the Kelly criterion is, when full Kelly blows up, and why most working quants size at half- or quarter-Kelly.

  • Learn Expected Shortfall (CVaR)

    Expected shortfall: the average loss given a VaR breach. Why regulators are migrating from VaR and what ES catches that VaR misses.

  • Learn Drawdown

    Drawdown explained: peak-to-trough decline, why max drawdown alone is misleading, and the recovery math that actually matters.