M.9.4 The Kelly Criterion

Basic Prerequisites: 03 · Multiplicative Growth and Calculus & Optimization (maximising a concave function).

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1. Intuition & Practical Objective

If the right objective is not the ensemble mean but the time-average growth rate $g=\mathbb{E}[\ln W]$, then there is a unique fraction of your capital that maximises it. That fraction is the Kelly criterion. It is the answer to the sizing question - not what to bet, but how much - and it is the operational core of this entire folder.

The intuition has three layers:

1. Bet too little and you leave growth on the table (your money sits idle).
2. Bet too much and volatility drag eats your growth - past a critical fraction, compounding turns negative even with a genuine edge, and ruin becomes certain.
3. There is a sweet spot between the timid and the bold, and it is computable in closed form.

Kelly (1956) discovered it in information theory; Thorp (2006) turned it into the practical discipline used by blackjack teams, sports bettors, and (fractionally) by quant funds. The objective is log utility: maximise $\mathbb{E}[\ln W]$. Growth-optimal, geometric-mean-maximising, capital-growth - these all name the same thing.

Takeaway. The optimal bet fraction is $f^{\ast }=(bp-q)/b$ (discrete) or $f^{\ast }=(m-r)/s^2$ (continuous). It is the unique maximiser of $\mathbb{E}[\ln W]$; it is not the fraction that maximises expected wealth (that is “bet everything”, which ruins you).

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2. Mathematical Ground Truth & Derivations

2.1 The coin-tossing case (Thorp §2)

Bet a fixed fraction $f$ on each independent trial; win $+1$ unit per unit staked with probability $p$, lose $1$ with probability $q=1-p$. After $n$ trials with $S$ successes and $F=n-S$ failures,

$$X_n=X_0(1+f{)}^S(1-f{)}^F⟹G_n(f)=\frac{1}{n}\ln \frac{X_n}{X_0}=\frac{S}{n}\ln (1+f)+\frac{F}{n}\ln (1-f).$$

The expected growth coefficient is

$$\boxed{g(f)=p\ln (1+f)+q\ln (1-f)}$$

Maximise: $g^{\prime }(f)=\frac{p}{1+f}-\frac{q}{1-f}=\frac{p-q-f}{(1+f)(1-f)}=0\Rightarrow$

$$\boxed{f^{\ast }=p-q}$$

and $g^{\prime \prime }(f)=-\frac{p}{(1+f{)}^2}-\frac{q}{(1-f{)}^2}<0$, so the maximum is unique. Its value is

$$g(f^{\ast })=p\ln p+q\ln q+\ln 2>0(\text{for }p>\frac{1}{2}).$$

The critical fraction $f_c$. $g(0)=0$, $g$ rises to $g(f^{\ast })>0$, then falls through zero at a unique $f_c>f^{\ast }$ with $g(f_c)=0$. For $f>f_c$ the growth rate is negative: ruin is almost sure. Also ${\lim }_{f\to 1}g(f)=-\infty$.

2.2 Unequal payoffs (Thorp §2, §7)

Win $b$ units per unit staked with probability $p$, lose $a$ with probability $q$. Then $g(f)=p\ln (1+bf)+q\ln (1-af)$ and

$$f^{\ast }=\frac{bp-aq}{ab}=\frac{m}{ab}(m\equiv bp-aq>0).$$

For $a=1$ this is $f^{\ast }=(bp-q)/b$; for even money ($a=b=1$), $f^{\ast }=p-q$.

2.3 Continuous / securities form (Thorp §7.1)

For continuous trading with instantaneous drift $m$, variance $s^2$ and riskless rate $r$, the growth rate of the rebalanced portfolio is

$$\boxed{g_{\infty }(f)=r+f(m-r)-\frac{1}{2}{s}^2{f}^2}⟹\boxed{f^{\ast }=\frac{m-r}{s^2}}$$

and the maximum growth rate is

$$g_{\infty }(f^{\ast })=\frac{(m-r{)}^2}{2s^2}+r=\frac{S^2}{2}+r,$$

where $S=(m-r)/s$ is the Sharpe ratio. A Sharpe ratio $S$ is worth $S^2/2$ of growth - the clean bridge between the mean-variance language of Pillar 5 and the growth language of this folder.

2.4 Fractional Kelly: the $c(2-c)$ law

Scale the bet to $f=c{f}^{\ast }$. For $r=0$,

$$g_{\infty }(c{f}^{\ast })=\frac{m^2}{s^2}c\left(1-\frac{c}{2}\right),\frac{g_{\infty }(c{f}^{\ast })}{g_{\infty }(f^{\ast })}=c(2-c),\frac{\mathrm{Sdev}(G_{\infty }(c{f}^{\ast }))}{\mathrm{Sdev}(G_{\infty }(f^{\ast }))}=c.$$

So half Kelly ($c=\frac{1}{2}$) keeps $c(2-c)=\frac{3}{4}$ of the growth rate with only half the risk - the asymmetry that makes fractional Kelly so attractive: you give up 25% of growth to halve volatility. (Section 7.3 of Thorp makes the case quantitatively.)

2.5 Why Kelly is “optimal” - Thorp’s Theorem 1

For the coin game with $g(f)>0$:

1. If $g(f)>0$, $X_n\to \infty$ a.s.
2. If $g(f)<0$, $X_n\to 0$ a.s. (ruin).
3. If $g(f)=0$, $\mathrm{limsup}{X}_n=\infty$ and $\mathrm{liminf}{X}_n=0$ a.s. (oscillation).
4. Dominance: for any essentially different strategy $\Phi$, $X_n({\Phi }^{\ast })/X_n(\Phi )\to \infty$ a.s. - Kelly beats every other strategy asymptotically.
5. Time-optimality: Kelly minimises the expected time to reach any fixed goal.
6. If probabilities change per trial, choose $f_i^{\ast }=p_i-q_i$ each trial.

The strategy that maximises $\mathbb{E}[\ln X_n]$ is thus asymptotically optimal by both growth and goal-reaching criteria - the two reasons Kelly is the reference sizing rule.

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3. Computational Implementation - solving and stress-testing Kelly

Stdlib only. First solve for $f^{\ast }$, $f_c$ and the growth at scaled fractions; then simulate fixed-fraction betting to watch overbetting destroy the account.

Read the third block: at $c=2$ (double Kelly, $f=0.12$) the growth rate has collapsed to $\approx 0$ - right at the empirically observed $f_c=0.1197$ - and 10.7% of accounts are already ruined. At $c=3$, growth is negative and 56% are wiped out. The theoretical $c(2-c)$ law (capped at $c=2$) is visible in the $g/g^{\ast }$ column: half Kelly keeps 0.88 of the growth here because $r>0$ slows the fall; in the $r=0$ case it is exactly $c(2-c)=0.75$.

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4. Failure Modes & First-Principles Breakdowns

1. Maximising $\mathbb{E}[X_n]$ instead of $\mathbb{E}[\ln X_n]$. Betting the whole bankroll maximises expected wealth and guarantees ruin (Thorp Theorem 1(ii)). “Bold” is optimal only in a world where you do not have to live with the outcome.
2. Overbetting past $f_c$. Any $f>f_c$ (here $0.1197$) gives a negative growth rate despite a positive edge. Overbetting is punished far more severely than underbetting: $g(f)$ is asymmetric around $f^{\ast }$, so an error of $+\delta$ costs more than $-\delta$.
3. Using the wrong $f^{\ast }$ formula. Win/lose asymmetry requires $f^{\ast }=m/(ab)$, not $p-q$. Continuous/levered settings require $f^{\ast }=(m-r)/s^2$, which can exceed $1$ (borrow) - plugging the discrete formula into a leverage decision mis-sizes badly.
4. Treating Kelly as a point estimate. $f^{\ast }$ is computed from estimated $p,m,s$; if the edge is overestimated, the “optimal” fraction is already an overbet. This is the practitioner’s #1 reason to run fractional Kelly (page 06).

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5. References

• Kelly, J. L. jr.: A New Interpretation of Information Rate, BSTJ 35(4):917–926 (1956)
• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• MacLean, Thorp & Ziemba (eds.): The Kelly Capital Growth Investment Criterion (World Scientific, 2011)
• Breiman, L.: Optimal Gambling Systems for Favorable Games, Proc. 4th Berkeley Symposium (1961)

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6. Connected Graph Bridges

• Back: 03 · Multiplicative Growth
• Forward: 05 · Ruin & Drawdown · 06 · Advanced Extensions · Index Hub
• Base: Constrained Optimization (concave maximisation) · Probability & Measure Theory
• Applications: Portfolio Optimization · Inventory Management & Skewing