Methodology: Position Sizing under Edge Variance

Worked example

Running the shipped position-sizing-edge-variance engine on the input below produces exactly this output. Continuous integration recomputes it against the engine bundle on every build, so these numbers cannot drift from the code.

Input

{
  "tool": "position-sizing-edge-variance",
  "edge_mean": 0.04,
  "edge_stddev": 0.02,
  "outcome_variance": 0.04,
  "kelly_fraction": 0.25
}

Output

{
  "deterministicKelly": 1,
  "fractionalDeterministic": 0.25,
  "bayesianKelly": 0.9900990099009902,
  "fractionalBayesian": 0.24752475247524755,
  "conservativeKelly": 0.5,
  "fractionalConservative": 0.125,
  "cvar5": 0.09221350532214989
}

Frequently asked questions

What does the Position Sizing under Edge Variance methodology page document?

Browne-Whitt Bayesian-Kelly derivation, CVaR formula, and references for the AI Fin Hub Position Sizing under Edge Variance calculator. It states the formulas, assumptions, data sources, limitations, and reproducibility steps behind the Position Sizing under Edge Variance, in the Finance category.

When was the Position Sizing under Edge Variance methodology last reviewed?

This methodology was last reviewed on 2026-05-08. The matching tool is at https://aifinhub.io/position-sizing-edge-variance/.

Are the Position Sizing under Edge Variance numbers reproducible?

Yes. This page embeds a worked example whose output is the verbatim result of running the shipped position-sizing-edge-variance engine on a fixed input; the embedded JSON is recomputed and diffed against the engine in CI, so the numbers cannot drift from the code.

Setup

You have a posterior distribution over your per-bet edge μ. Its mean is your point estimate, its standard deviation σ_μ is the credible-interval half-width. The bet outcome itself has variance σ²_outcome.

Three Kelly variants

f_det    = μ / σ²_outcome                              ← deterministic
f_bayes  = μ / (σ²_outcome + σ²_μ)                     ← Bayesian (Browne-Whitt 1996)
f_cons   = (μ − σ_μ) / σ²_outcome                      ← lower-bound estimate

bet_size = max(0, f) · kelly_multiplier                ← practitioner damping

Bayesian Kelly is the principled penalty: when σ_μ → 0 it collapses to deterministic Kelly; when σ_μ is large the recommended bet shrinks.

Conditional Value at Risk (CVaR)

For a bet of size f with normally-distributed PnL N(f·μ, f²·σ²_outcome), the 5%-CVaR is:

CVaR_5% = −f·μ + f·√(σ²_outcome) · φ(z_0.05) / 0.05

where φ is the standard normal density and z_0.05 ≈ 1.645. CVaR > 5% of bankroll on a single bet is generally too risky for fractional-Kelly sizing.

References

Browne, S., Whitt, W. (1996). "Portfolio choice and the Bayesian Kelly criterion." Advances in Applied Probability 28(4): 1145–1176. DOI: 10.1017/S0001867800027750.
Rotando, L. M., Thorp, E. O. (1992). "The Kelly criterion and the stock market." American Mathematical Monthly 99(10): 922–931. DOI: 10.2307/2324484.
MacLean, L. C., Thorp, E. O., Ziemba, W. T. (eds.) (2010). The Kelly Capital Growth Investment Criterion. World Scientific. ISBN: 978-981-4293-49-5.
Rockafellar, R. T., Uryasev, S. (2000). "Optimization of conditional value-at-risk." Journal of Risk 2(3): 21–41.

Limitations

Assumes normal posterior on μ. Heavier-tailed posteriors need a corresponding adjustment to the denominator term.
Continuous-bet formulation. For discrete win/lose Kelly, the multiplicative form differs — use the Fractional Kelly Sizer for that case.
CVaR formula assumes Gaussian PnL; fat-tailed PnL would require empirical CVaR.

External resources

Kelly Capital Growth Investment Criterion (MacLean, Thorp, Ziemba 2011)