Kelly Criterion

1. Preliminaries

Definition 1 (Almost Sure Convergence). For a sequence of random variables $X_1, X_2, ..., X_n, ...$ and a random variable $X$, we say that $X_n$ converges almost surely (a.s.) to $X$, denoted by $X_n \xrightarrow{\text{a.s.}} X$ as $n \rightarrow \infty$, if

Theorem 1 (Strong Law of Large Numbers). Let $X_1, X_2, \dots$ be a sequence of independent and identically distributed (i.i.d.) random variables with finite expected value $\mathbb{E}[X] = \mu$. The sample average converges almost surely to the expected value:

Theorem 2 (Central Limit Theorem). Let $X_1, X_2, ...$ be i.i.d. random variables with mean $\mu$ and finite variance $\sigma^2 > 0$. Then the random variable $Z_n = \frac{\sum_{i=1}^n X_i - n\mu}{\sigma\sqrt{n}}$ converges in distribution to a standard normal random variable:

where $\Phi(z)$ is the cumulative distribution function of the standard normal distribution $\mathcal{N}(0,1)$.

Theorem 3 (Jensen’s Inequality). If $\varphi$ is a convex function, then $\varphi(\mathbb{E}[X]) \le \mathbb{E}[\varphi(X)]$. Conversely, if $\psi$ is a concave function (such as $\ln(x)$), then:

Equality holds if and only if $X$ is a constant almost surely or $\psi$ is linear.

2. Problem Setup

Consider a discrete-time repeated game over periods $t = 1, \dots, n$. Let $W_0 > 0$ be the initial capital. Let $R_1, R_2, \dots$ be a sequence of i.i.d. random variables representing the return rate at each step, such that $R_t > -1$ almost surely (to ensure wealth remains positive). Let $f$ be the fixed fraction of wealth wagered in each period (the strategy). The wealth process evolves multiplicatively:

To evaluate if the game is favorable, we could calculate two metrics:

• The Arithmetic Mean: $\mu = \mathbb{E}[R]$. This describes the average wealth across infinite parallel universes (Ensemble Average).
• The Geometric Mean: $g = \mathbb{E}[\ln(1 + f R)]$. This describes the growth rate of a single trajectory over time (Time Average).

3. Expected Return vs. Typical Return

To understand the asymptotic behavior of $W_n$, we must linearize the multiplicative process by taking the natural logarithm.

Let $Z_i = \ln(1 + f R_i)$ be the log-return for period $i$. Since $R_i$ are i.i.d., $Z_i$ are also i.i.d. We can now analyze the average growth rate per period, denoted $G_n$:

By Theorem 1, as $n \to \infty$, the sample average $G_n$ converges almost surely to the true expected value of the log-returns:

Let $g(f) = \mathbb{E}[\ln(1 + f R)]$. Note this is the geometric mean growth rate we defined above. By exponentiating the result, we find the behavior of wealth for a typical investor:

We can therefore draw the conclusion: the long-term capital of any single investor is determined solely by $g(f)$. The arithmetic mean $\mu$ is not directly relevant to the typical outcome because it is dominated by vanishingly rare outlier events that the single investor will likely never experience.

$$\underbrace{\mathbb{E}[\ln(1 + fR)]}_{\text{Expected Log Return } g(f)} \le \ln(\underbrace{1 + f\mu}_{\text{Expected Arithmetic Wealth}}).$$

The term $\ln(1 + f \mu)$ represents the “theoretical” growth if there were no volatility. The difference between the two sides is the volatility drag. This proves mathematically that variance reduces the compound growth rate.

4. Kelly Criterion

Here is when Kelly Criterion ¹ kicks in. Since $W_n$ is driven by $e^{n \cdot g(f)}$, maximizing terminal wealth is equivalent to maximizing the exponent $g(f)$.
Objective:

Since the logarithmic function is concave, we find the maximum by setting the first derivative to zero:

An example of a binary game. Consider the classic coin-toss scenario:

• Win: Return $+b$ (odds) with probability $p$.
• Lose: Return $-1$ (loss of wager) with probability $q = 1-p$.
  The expectation in the derivative becomes:
  $$\mathbb{E}\left[ \frac{R}{1 + f R} \right] = p \left( \frac{b}{1 + fb} \right) + q \left( \frac{-1}{1 - f} \right) = 0.$$
  Solving for $f$:

1. Move the negative term:
   

   $$\frac{pb}{1 + fb} = \frac{q}{1 - f}.$$

2. Cross-multiply:
   

   $$pb(1 - f) = q(1 + fb).$$

3. Expand:
   

   $$pb - pbf = q + qfb.$$

4. Group terms with $f$:

5. Solve for $f^*$:
   

   $$f^* = \frac{pb - q}{b}.$$

   
   This formula, known as the Kelly Criterion, identifies the exact leverage $f^*$ that maximizes the geometric growth rate. Betting more than $f^*$ increases arithmetic return but decreases the log return (due to Jensen’s inequality/volatility drag), eventually leading to a negative growth rate and sure ruin.