How Position Sizing under Edge Variance works

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)