5.7.1 The Kelly Criterion from Zero

Basic Prerequisites: 03 · Multiplicative Growth (log returns, volatility drag).

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1. Intuition & Practical Objective

This page builds the why of Kelly with no prior quantitative knowledge needed. The objective is one idea: betting maximises expected wealth, but expected wealth is the wrong thing to maximise - the thing that decides whether you get rich or go broke is the growth rate of your bankroll, $\mathbb{E}[\ln W]$, and it has a unique optimal size.

Start with the dumbest question: if I have an edge, should I bet as much as I can? Imagine a coin that lands heads 55% of the time. Every bet: win and you double your stake, lose and you lose it. The expected value per dollar staked is $0.55(1)-0.45(1)=+0.10$, a 10% edge. If expected value were the objective, you’d bet everything, every round: $E[W]$ after $n$ rounds is $(1+f(2p-1){)}^n$, which grows fastest at $f=1$ (bet it all).

But betting everything is certain death. One loss out of a long enough run and you have a zero bankroll that never recovers - and with $p=0.55$ you will eventually hit a loss. Expected wealth loves this; you do not. The resolution is to maximise not $\mathbb{E}[W]$ but $\mathbb{E}[\ln W]$ - the growth rate that your own single path through time actually compounds at.

Three steps:

1. Bet zero and you never grow - bet too much and volatility eats you. Compounding is geometric: $W_n=W_0(1+f{)}^S(1-f{)}^F$, so the per-round growth rate is $g(f)=p\ln (1+f)+q\ln (1-f)$, which is concave - it goes up then down, and past a critical fraction it turns negative even though the edge is positive. There is a middle sweet spot.
2. The growth rate is the only number that survives compounding. $E[W]$ is an average across parallel worlds; $g(f)$ is the average along your one real path. For multiplicative wealth these differ - the entire lesson of Ergodicity & Statistical Mechanics.
3. The optimum is computable, not a vibe. Maximising a concave function of $f$ gives a closed-form maximiser. That maximiser is the Kelly fraction $f^{\ast }$, unique and explicit.

Takeaway. The right objective is the growth rate $g=\mathbb{E}[\ln W]$, not expected wealth. Its maximiser $f^{\ast }$ is Kelly’s famous bet size - and it is systematically smaller than “bet as much as the edge suggests”, because volatility drag punishes overbetting harder than it rewards boldness.

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2. Mathematical Ground Truth & Derivations

The concave growth function. Bet a fixed fraction $f$ on each fair-odds coin toss with $p>1/2$, $q=1-p$. After $n$ trials with $S$ heads and $F=n-S$ tails,

$$W_n=W_0(1+f{)}^S(1-f{)}^F⟹\boxed{g(f)=p\ln (1+f)+q\ln (1-f)}$$

Derivation by maximisation. This is concave: $g^{\prime \prime }(f)=-p/(1+f{)}^2-q/(1-f{)}^2<0$, so the critical point is the unique global max. Set $g^{\prime }(f)=\frac{p}{1+f}-\frac{q}{1-f}=0$:

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

Why $\mathbb{E}[\ln W]$ and not $\mathbb{E}[W]$. The expected terminal wealth at full Kelly grows like $E[W]\propto (1+2p-1{)}^n$, maximised at $f=1$. But $\ln W$ - the growth you actually live with - is maximised at $f^{\ast }=p-q$, which for $p=0.55$ is only $0.10$. The gap between “maximise the mean” and “maximise the median/typical path” is precisely the Jensen gap of multiplicative growth from the foundations folder.

The critical fraction $f_c$. Since $g(0)=0$, $g$ rises to a positive peak at $f^{\ast }$, then falls through zero at a unique $f_c>f^{\ast }$. For any $f>f_c$ the growth rate is negative and ruin is almost sure. For $p=0.55$, $f_c=0.1987$ - just $2\times$ the full-Kelly fraction is already a losing strategy.

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3. Computational Implementation - the growth-rate landscape

Stdlib only. Sweep the growth function, then simulate what actually happens at the Kelly fraction vs. betting everything.

Read the first block: the growth rate is positive only between $f=0$ and $f_c\approx 0.199$; the maximum is at $f=0.10$, and beyond $f\approx 0.2$ the edge turns into a guaranteed-loss machine. The second block is the punchline: betting everything turns your $+10\%$ edge into almost-sure ruin, while betting the Kelly fraction (a fifth as much) grows a bankroll $2.7\times$ over 200 bets with zero ruin in 40,000 trials.

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4. Failure Modes & First-Principles Breakdowns

1. “Expected value says bet it all.” $\mathbb{E}[W]$ is maximised by all-in, but it is the ensemble average across parallel worlds - not what one investor’s single path compounds. Maximising $\mathbb{E}[W]$ when wealth multiplies is the classic non-ergodicity trap (02 · Ensemble vs Time Averages).
2. Reading "$f_c=0.199$" as a safety margin. It is not: $f_c$ is the boundary of certain ruin, not a target. Any estimate error that pushes the deployed fraction above $f_c$ - and overbetting is cheaper to do accidentally than underbetting - destroys the account deterministically.
3. Treating this as a growth maximiser for what to bet, not whether to bet. Kelly only sizes a position whose edge is real. If $p\le 1/2$ there is no positive $f^{\ast }$ and the whole discussion is moot - sizing does not manufacture a signal.

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5. References

• Kelly, J. L. jr.: A New Interpretation of Information Rate, Bell System Technical Journal 35(4) (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 (2011)

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6. Connected Graph Bridges

• Base: Multiplicative Growth & Volatility Drag · Ensemble vs Time Averages
• Continue: 02 · The Kelly Formula · Index Hub