Expected utility and Kelly growth

Definition Fix the probability measure $\mathbb{P}$ under which outcomes are evaluated, the physical or a subjective one rather than a pricing measure, a set $\mathcal{A}$ of admissible decisions and, for each $a \in \mathcal{A}$, a terminal wealth $W(a)$ whose values lie in an interval $D$. For a utility function $U$ on $D$, expected-utility maximization chooses $a \in \mathcal{A}$ to maximize

over the decisions for which this expectation exists. An increasing $U$ prefers more wealth to less. If $U$ is concave and $W$ is integrable, Jensen’s inequality gives

so the sure amount $\mathbb{E}^{\mathbb{P}}[W]$ is at least as good as the gamble $W$, which is what risk aversion means. For strictly concave $U$ the inequality is strict unless $W$ is almost surely constant. For a continuous, strictly increasing $U$, the certainty equivalent of $W$ is the sure wealth $\mathrm{CE}$ with $U(\mathrm{CE}) = \mathbb{E}^{\mathbb{P}}[U(W)]$, and a concave $U$ gives $\mathrm{CE} \le \mathbb{E}^{\mathbb{P}}[W]$. Replacing $U$ by $\alpha + c\, U$ with constants $\alpha$ and $c > 0$ leaves every ranking unchanged.

Definition Log utility is $U(W) = \ln W$ for $W > 0$. Its derivatives $U'(W) = 1/W > 0$ and $U''(W) = -1/W^2 < 0$ make it strictly increasing and strictly concave, and its relative risk aversion

is the same at every wealth level. It is undefined for $W \le 0$ and tends to $-\infty$ as $W \to 0$ from above, so a decision under which $W \le 0$ has positive probability is excluded, because its expected log utility is $-\infty$ or undefined. Since $\ln(W_0 X) = \ln W_0 + \ln X$, multiplying every outcome by $W_0 > 0$ adds the constant $\ln W_0$. Decisions whose terminal wealth is $W_0$ times a multiplier $X$ that does not depend on $W_0$ are therefore ranked the same way at every level of wealth, and the best fraction to stake does not depend on current wealth.

Definition Let $R$ be the return per unit staked of one round of an opportunity, and let the stake be the fraction $f$ of current wealth, with unstaked wealth held as cash at zero interest. A fraction $f > 1$ borrows at zero interest and a fraction $f < 0$ takes the opposite side. Wealth then moves from $W$ to $W(1 + fR)$. The positive-wealth constraint requires

and the fractions that satisfy it are the admissible fractions for log utility. They form an interval containing $0$, because $1 + fR$ is affine in $f$. For the binary opportunity with $R = b$ with probability $p$ and $R = -1$ otherwise, where $0 < p < 1$ and $b > 0$, the conditions $1 + bf > 0$ and $1 - f > 0$ give

Position limits, such as a ban on negative stakes or a borrowing limit, shrink this interval further. If $R$ is unbounded below, every $f > 0$ has $\mathbb{P}(R < -1/f) > 0$, so no positive stake is admissible, and if $R$ is unbounded above, no negative stake is. If $1 + R$ is lognormal, the admissible fractions are exactly $0 \le f \le 1$. For these, $1 + fR = (1 - f) + f(1 + R) > 0$, while for $f > 1$ the event $1 + R \le 1 - 1/f$ and for $f < 0$ the event $1 + R \ge 1 + 1/\lvert f \rvert$ both have positive probability.

Definition Let $R_1, R_2, \dots$ be independent copies of $R$, let $f$ be admissible, and rebalance to the fixed fraction $f$ at the start of every round with no costs, so that

The expected log growth of $f$ is $g(f) = \mathbb{E}^{\mathbb{P}}[\ln(1 + fR)]$, in natural log units per round. Taking logs turns the product into a sum, and if $\mathbb{E}^{\mathbb{P}}\bigl[\lvert \ln(1 + fR) \rvert\bigr] < \infty$, the strong law of large numbers gives

almost surely. Hence $W_n \to \infty$ almost surely when $g(f) > 0$, and $W_n \to 0$ almost surely when $g(f) < 0$, and $e^{g(f)}$ is the long-run geometric growth factor per round. When $g(f) = 0$ the strong law alone does not settle the long-run behavior. The mean behaves differently. If $\mathbb{E}^{\mathbb{P}}[\lvert R \rvert] < \infty$, independence gives $\mathbb{E}^{\mathbb{P}}[W_n] = W_0 \bigl(1 + f\, \mathbb{E}^{\mathbb{P}}[R]\bigr)^n$, and Jensen’s inequality gives $e^{g(f)} \le 1 + f\, \mathbb{E}^{\mathbb{P}}[R]$. For the binary opportunity,

Definition The Kelly criterion chooses the admissible fraction $f^{*}$ that maximizes the expected log growth $g$, or, for several opportunities with return vector $R$, the admissible vector $f$ that maximizes $\mathbb{E}^{\mathbb{P}}[\ln(1 + f^{\top} R)]$. This is the same as maximizing the expected log utility of next-round wealth. All expectations in this card are under $\mathbb{P}$. With independent, identically distributed rounds and integrable log multipliers it has two growth properties, which hold inside the probability model and need the true law of $R$. First, let the stake $f_t$ of round $t$ be admissible and depend only on the outcomes of earlier rounds. Since $R_t$ is independent of those outcomes, the conditional mean of $\ln(1 + f_t R_t)$ given them is $g(f_t)$, so

and the fixed fraction $f^{*}$ maximizes expected log terminal wealth at every horizon. Second, if $g(f^{*}) > g(f)$ for a fixed fraction $f$, then on the same sequence of outcomes $W_n(f^{*})/W_n(f) \to \infty$ almost surely, because the log of the ratio divided by $n$ tends to $g(f^{*}) - g(f) > 0$. For small positions there is a quadratic approximation. If $\lvert fR \rvert \le c < 1$ almost surely, Taylor’s theorem gives $\ln(1 + x) = x - x^2/2 + r(x)$ with $\lvert r(x) \rvert \le \lvert x \rvert^3/\bigl(3(1 - c)^3\bigr)$ for $\lvert x \rvert \le c$, so

with

and the quadratic part is maximized at $\mathbb{E}[R]/\mathbb{E}[R^2]$. Replacing $\mathbb{E}[R^2]$ by $\operatorname{Var}(R)$ is a further approximation, close when $\mathbb{E}[R]^2$ is small compared with $\operatorname{Var}(R)$. Neither approximation enforces the positive-wealth constraint. Log-growth formulas for continuously rebalanced diffusion models come from Itô’s formula and assume continuous paths, so they do not account for losses from jumps.

Definition Suppose the rounds are independent, each pays $b > 0$ per unit staked with probability $p$ and loses the stake with probability $1 - p$, where $0 < p < 1$ are the true probabilities, there are no costs, and $f$ ranges over the admissible interval $-1/b < f < 1$. Then

and

so $g$ is strictly concave. Setting $g'(f) = 0$ and multiplying by the positive denominators gives $p b (1 - f) = (1 - p)(1 + bf)$, and the unique maximizer is the Kelly fraction

It lies inside the admissible interval for every such $p$ and $b$, since $1 + bf^{*} = p(1 + b) > 0$ and $1 - f^{*} = (1 - p)(1 + b)/b > 0$, and it is positive exactly when the expected profit $\mathbb{E}^{\mathbb{P}}[R] = pb - (1 - p)$ per unit staked is positive. With $q = 1/(1 + b)$, the win probability that would make the wager fair, $f^{*} = (p - q)/(1 - q)$ and

the relative entropy of the true probabilities with respect to the fair ones, which is positive unless $p = q$. If stakes are restricted to an interval inside the admissible one, concavity makes the best allowed stake the point of that interval nearest $f^{*}$, for example $0$ when negative stakes are not allowed and $f^{*} < 0$.