5.7.2 The Kelly Formula

Basic Prerequisites: 01 · From Zero and Calculus & Optimization (concave optimisation, bisection).

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1. Intuition & Practical Objective

Page 01 established the objective: maximise the growth rate $g(f)=p\ln (1+f)+q\ln (1-f)$. This page turns that objective into formulas you can compute - the numeric answer to “how much of my capital goes on this position?” There are three standard cases, and a practitioner must know which one applies:

1. Even-money discrete bets (win pays $b=1$, lose pays $a=1$): the famous $f^{\ast }=p-q$.
2. Unequal-payoff discrete bets (e.g. $2:1$ payout, or losing more than you stake): $f^{\ast }=m/(ab)$ with margin $m=bp-aq$.
3. Continuous / securities (drift $m$, variance $s^2$, riskless rate $r$, can lever): $f^{\ast }=(m-r)/s^2$.

Alongside each optimum sits the critical fraction $f_c$ - the point where the growth rate crosses zero and ruin becomes certain. The practical punchline: because $f^{\ast }$ is estimated, and overbetting toward $f_c$ is catastrophic, the numbers you must memorise are not just $f^{\ast }$ but the whole $g(f)$ curve and where it dies.

Takeaway. $f^{\ast }$ shifts from $p-q$ (coin toss) to $m/(ab)$ (asymmetric discrete) to $(m-r)/s^2$ (securities). Get the case right: in a leveraged securities setting the continuous $f^{\ast }$ can exceed $1$ - you borrow - while the discrete formula wrongly caps you at your own wealth.

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2. Mathematical Ground Truth & Derivations

2.1 Even-money discrete: $f^{\ast }=p-q$

Win $+1$ unit per unit staked with prob $p$, lose $1$ with prob $q=1-p$. Growth rate $g(f)=p\ln (1+f)+q\ln (1-f)$. Maximise:

$$\frac{dg}{df}=\frac{p}{1+f}-\frac{q}{1-f}=0\Rightarrow f^{\ast }=p-q,$$

with $g^{\prime \prime }(f)=-\frac{p}{(1+f{)}^2}-\frac{q}{(1-f{)}^2}<0$ so $f^{\ast }$ is the unique global max. At $p=0.55$: $f^{\ast }=0.10$, $g(f^{\ast })=0.005008$ per round.

Critical fraction. Solve $g(f_c)=0$. For $p=0.55$, bisection gives $f_c=0.1987$. Note $f_c$ is only $\approx 1.99\times f^{\ast }$ - a $2\times$ sizing error already straddles ruin.

2.2 Unequal payoffs: $f^{\ast }=m/(ab)$

Win $b$ per unit with prob $p$, lose $a$ per unit with prob $q$. Then $g(f)=p\ln (1+bf)+q\ln (1-af)$ and setting $g^{\prime }(f)=0$:

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

For $a=1$ (lose exactly your stake) this reduces to $f^{\ast }=(bp-q)/b$; for even money ($a=b=1$) to $p-q$. Example $b=2,a=1,p=0.4$: $m=2(0.4)-0.6=0.2$, $f^{\ast }=0.2/2=0.10$ - same $f^{\ast }$ as an even-money $p=0.55$ game but a very different $g(f)$.

2.3 Continuous / securities: $f^{\ast }=(m-r)/s^2$

For a continuously-rebalanced portfolio with instantaneous drift $m$, variance rate $s^2$ and riskless $r$, the growth rate is the concave quadratic (Thorp §7.1):

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

where $S=(m-r)/s$ is the Sharpe ratio. A Sharpe ratio $S$ is worth $S^2/2$ of growth - the clean bridge to the mean-variance language of MPT, because $f^{\ast }=(m-r)/s^2$ is exactly the excess-return-to-variance ratio the tangency portfolio uses (see 06 · Advanced Extensions). The critical fraction solves $g_{\infty }(f_c)=0$, a quadratic whose positive root for the worked numbers is $f_c=5.427$.

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3. Computational Implementation - the three formulas, and $f_c$ by bisection

Stdlib only: closed-form $f^{\ast }$ for each case, plus bisection for the critical fraction of the discrete game.

Two things worth noting in the output. First, the discrete $f^{\ast }$ is bounded by $1$ (you cannot bet more than your bankroll on even money) while the continuous $f^{\ast }=2.22$ demands leveraging 2.2× capital - a completely different regime. Second, the continuous $f_c=5.43$ is far above $f^{\ast }=2.22$, a much bigger safety gap than in the discrete game ($f_c=0.199$ vs $f^{\ast }=0.10$), because volatility drag scales like $f^2$.

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4. Failure Modes & First-Principles Breakdowns

1. Using the wrong formula for the regime. Plugging the even-money $p-q$ into a leveraged/securities sizing decision is wrong: the continuous case admits $f^{\ast }>1$ (borrow), and the unequal case requires the $m/(ab)$ margin form - for a losing payout $a>1$, $f^{\ast }$ shrinks because you can lose more than you stake.
2. Ignoring $f_c$. $f^{\ast }$ is the optimum; $f_c$ is the cliff. Because $g(f)$ is asymmetric and concave, an overbet costs more than an underbet, and any $f>f_c$ is a guaranteed-loss strategy regardless of how real the edge is.
3. Assuming a stable edge. All three formulas take $p,m,s$ as known numbers. They are estimated; an overestimated $p$ or underestimated $s$ inflates $f^{\ast }$ - the theme of 04 · Fractional Kelly & Ruin and 05 · Failure Modes.
4. A zero/negative edge has no Kelly. If $m\le r$ or $bp-aq\le 0$, there is no positive $f^{\ast }$; the correct bet is $0$. Kelly never manufactures an edge.

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5. References

• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• Kelly, J. L. jr.: A New Interpretation of Information Rate, BSTJ 35(4) (1956)
• Breiman, L.: Optimal Gambling Systems for Favorable Games, Proc. 4th Berkeley Symposium (1961)

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Deeper formulation: 04 · Kelly Criterion (foundations) (the same $f^{\ast }$ and $f_c$, growth-criterion treatment)
• Continue: 03 · Growth & Optimality
• Theory: Calculus & KKT Optimization