3.2 The Kelly criterion for a binary bet

Motivation

A positive expectancy says a bet is worth taking. It does not say how much to stake. Stake too little and capital grows slowly; stake too much and capital shrinks to zero even though every bet is favourable. Kelly (1956) found the stake that maximises the long-run growth rate of capital.

The growth-optimal fraction#

Setup

A bettor with wealth $W_{n-1}$ stakes a fraction $f \in [0, 1)$ on bet $n$ . With probability $p$ the bet wins and pays $b$ per unit staked (net odds $b{:}1$ ); with probability $q = 1 - p$ the stake is lost. Bets are iid. Then

W_n = W_{n-1}(1 + fX_n), \qquad X_n = \begin{cases} b & \text{w.p. } p \\ -1 & \text{w.p. } q.\end{cases} \tag{3.3}

The expected profit per unit staked is $E[X] = pb - q$ . Define the growth rate

g(f) = E\ln(1 + fX) = p\ln(1 + bf) + q\ln(1 - f). \tag{3.4}

Proposition 3.2binary Kelly fraction

If $pb > q$ , $g$ is strictly concave on $[0, 1)$ and maximised at

f^* = p - \frac{q}{b} = \frac{pb - q}{b}, \tag{3.5}

the expected profit per unit staked divided by the net odds. If $pb \le q$ , the maximum over $[0,1)$ is at $f = 0$ .

Proof

$g'(f) = \dfrac{pb}{1 + bf} - \dfrac{q}{1 - f}$ and $g''(f) = -\dfrac{pb^2}{(1+bf)^2} - \dfrac{q}{(1-f)^2} < 0$ , so $g$ is strictly concave and any stationary point is the unique maximum. Setting $g'(f) = 0$ : $pb(1 - f) = q(1 + bf)$ , so $pb - q = f(pb + qb) = fb$ since $p + q = 1$ . Hence $f^* = (pb - q)/b$ . If $pb \le q$ then $g'(0) = pb - q \le 0$ and concavity gives $g' < 0$ on $(0, 1)$ .∎

At the optimum, $1 + bf^* = p(1+b)$ and $1 - f^* = q(1+b)/b$ , which gives a closed form for the maximal growth:

g(f^*) = p\ln(p(1+b)) + q\ln\frac{q(1+b)}{b} = p\ln p + q\ln q + \ln(1+b) - q\ln b. \tag{3.6}

For even money ( $b = 1$ ) this is $\ln 2 - H(p)$ , where $H(p) = -p\ln p - q\ln q$ is the entropy of the bet in nats. This is Kelly's original "information rate" interpretation: a gambler with a private wire that is right with probability $p$ grows capital at the rate of the channel's capacity.

Asymptotic optimality#

Proposition 3.3asymptotic optimality

Let $W_n(f)$ be wealth after $n$ bets at constant fraction $f$ , driven by the same outcomes for all $f$ . Then (a) $\frac1n\ln(W_n(f)/W_0) \to g(f)$ almost surely; (b) for any $f \ne f^*$ with $f \in [0, 1)$ , $W_n(f^*)/W_n(f) \to \infty$ almost surely; (c) if $g(f) > 0$ then $W_n(f) \to \infty$ a.s.; if $g(f) < 0$ then $W_n(f) \to 0$ a.s., even if $E[X] > 0$ .

Proof

(a) $\ln(W_n/W_0) = \sum_{i=1}^{n}\ln(1 + fX_i)$ is a sum of iid variables with finite mean $g(f)$ (for $f < 1$ they are bounded). Apply the strong law. (b) $\frac1n\ln(W_n(f^*)/W_n(f)) \to g(f^*) - g(f) > 0$ a.s. by (a) and strict concavity, so the log-ratio tends to $+\infty$ . (c) Immediate from (a).∎

Two further facts make the criterion attractive (Breiman 1961; MacLean, Thorp & Ziemba 2010): among all strategies, including ones whose fraction varies with past outcomes, the Kelly strategy asymptotically minimises the expected time to reach a large wealth target, and no other strategy has a higher expected value of $\ln W_n$ for any $n$ . Because $\ln$ is monotone, the Kelly strategy also maximises the median of $W_n$ to first order: the median of $\ln W_n$ is about $ng(f)$ .

Kelly does not maximise expected wealth. $E[W_n] = W_0(1 + fE[X])^n$ is increasing in $f$ and is maximised by $f = 1$ , which goes broke with probability $1 - p^n \to 1$ . The difference between the mean and the median is M02's Corollary 2.3a in discrete form.

A coin and a swing system#

Worked example 3.2.1a coin with an edge

$p = 0.55$ , $b = 1$ : $f^* = 0.10$ . $g(f^*) = 0.55\ln 1.1 + 0.45\ln 0.9 = 0.052421 - 0.047413 = 0.005008$ per bet. Doubling time $\ln 2/g = 138$ bets. In the tests of Exercise m03_kelly (in the app: Exercises → m03_kelly), 2,000 simulated paths of 10,000 bets give mean realised growth within $2\times10^{-4}$ of $g(f)$ at every $f$ on a grid from 0.02 to 0.30, with the maximum at $f = 0.10$ .

Worked example 3.2.2the swing system

In R units, staking a fraction $f$ of capital as $1R$ means a win multiplies wealth by $1 + 2.5f$ and a loss by $1 - f$ : a binary bet with $p = 0.4$ , $b = 2.5$ . Kelly: $f^* = 0.4 - 0.6/2.5 = 0.16$ . The Kelly risk per trade is 16% of capital, with growth $0.4\ln 1.4 + 0.6\ln 0.84 = 0.0300$ per trade. At 1% risk per trade, growth is $0.0038$ per trade, an eighth of the Kelly rate. Why practitioners risk 0.5–1% rather than 16% is the subject of 3.5, 3.7 and 3.9.

Practitioner notes

Kelly sizes the risk, not the position. For the swing system $f^*$ is the fraction of capital lost if the stop is hit, not the fraction invested. A 16% risk with an 8% stop means a position of twice capital.
For a general trade list, the Kelly fraction maximises $\frac1N\sum\ln(1 + fR_i)$ ; kelly_from_R in Exercise m03_sizing solves it by bisection on the derivative.