5.7 Kelly Criterion & Bet Sizing

Basic Prerequisites: Ergodicity & Statistical Mechanics (multiplicative growth, log-utility) and Calculus & KKT Optimization (maximising a concave function). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

The Kelly criterion answers the one question Markowitz deliberately leaves open: how much to bet on an edge, not just what to hold. Given a strategy with a positive expected edge, there is a unique fraction of capital that maximises the long-run growth rate $g=\mathbb{E}[\ln W]$ - the compounded, time-average return you actually live with. Bet too little and you leave growth on the table; bet too much and the volatility drag $-\frac{1}{2}{\sigma }^2{f}^2$ turns compounding negative; past a critical fraction ruin becomes almost sure even with a genuine edge.

Layered on the mean-variance machinery of this pillar: every Sharpe ratio $S$ is worth $S^2/2$ of growth, and the continuous Kelly fraction $f^{\ast }=(m-r)/s^2$ is, in exact form, the tangency (maximum-Sharpe) portfolio scaled by risk - so Kelly is the growth-optimal objective that MPT’s quadratic utility approximates. This folder is the pillar’s deployment layer: it takes a validated edge and sizes the position safely.

The one-sentence essence. “Maximise $\mathbb{E}[\ln W]$, not $\mathbb{E}[W]$: the full-Kelly fraction $f^{\ast }$ is the unique growth-optimiser, and because $f^{\ast }$ is estimated - and overbetting is punished far more severely than underbetting - practice sizes at a fraction of Kelly.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). All formulas below are the same object maximised in 04 · Kelly Criterion (foundations) and transcribed from Thorp (2006) and Kelly (1956); the check-column numbers were re-executed and reproduced exactly from the verified corpus (§3).

Notation: $f$ fraction of capital bet, $p$ win prob, $q=1-p$, $b$ win per unit staked, $a$ loss per unit staked, $m$ instantaneous drift, $s^2$ variance rate, $r$ riskless rate, $S=(m-r)/s$ Sharpe ratio, $W$ wealth.

The asymmetry that justifies fractional Kelly. Half Kelly ($c=\frac{1}{2}$) keeps $c(2-c)=\frac{3}{4}$ of the growth rate with half the volatility - give up 25% of growth to halve risk. Because the growth function is concave and asymmetric around $f^{\ast }$, an overbet of $\Delta f$ costs more than an underbet of the same size, and overbetting past $f_c$ destroys the account (see 04 · Fractional Kelly & Ruin).

3. Computational Implementation - the sizing engine

Stdlib only. Compute the Kelly fraction, the growth function, and simulate what happens at full, half, and double Kelly when the parameter is estimated correctly vs. wrong.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Parameter error → overbetting ruin. $f^{\ast }$ is computed from estimated $p,m,s$; if the edge or variance is mis-estimated, the “optimal” fraction is already an overbet. In the worked example, a $+1\sigma$ error in $p$ flips a growing strategy into a 73%-ruin losing one - the practitioner’s #1 reason to use fractional Kelly.
2. Log-utility is the only correct objective. Maximising $\mathbb{E}[W]$ means “bet everything”, which maximises expected wealth and guarantees ruin. Kelly maximises $\mathbb{E}[\ln W]$ precisely to avoid that trap.
3. Fat tails break the Gaussian formula. $g\approx f\mu -frac12{\sigma }^2{f}^2$ understates the tail; a heavy-loss outcome makes the exact log-optimal $f^{\ast }$ smaller than the Gaussian approximation, so sizing on the Gaussian number overbets.
4. The discrete/continuous divide. $f^{\ast }=p-q$ assumes even-money, win-or-lose-all; leverage/unbalanced payout requires $f^{\ast }=(m-r)/s^2$ or $m/(ab)$. Mixing the two mis-sizes badly when $f^{\ast }>1$ (borrowable).

5. References

• Kelly, J. L. jr.: A New Interpretation of Information Rate, Bell System Technical Journal 35(4):917–926 (1956)
• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006; repr. in MacLean–Thorp–Ziemba 2011)
• MacLean, Thorp & Ziemba (eds.): The Kelly Capital Growth Investment Criterion: Theory and Practice (World Scientific, 2011)
• MacLean, Ziemba & Blazenko: “Growth versus Security in Dynamic Investment Analysis” (Management Science 1992)
• Breiman, L.: Optimal Gambling Systems for Favorable Games, Proc. 4th Berkeley Symposium (1961)

6. Connected Graph Bridges

• Foundational base: Ergodicity & Statistical Mechanics · Probability & Measure Theory
• Sibling topic: Modern Portfolio Theory & Mean–Variance (continuous Kelly = scaled tangency portfolio) · Risk Parity
• Sub-pages (in-folder): 01 From Zero · 02 The Kelly Formula · 03 Growth & Optimality · 04 Fractional Kelly & Ruin · 05 Failure Modes & Practice · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05