Kelly Criterion
In this post, we will discuss the Kelly Criterion, a formula used to determine the optimal size of a series of bets in order to maximize the long-term growth of wealth. We separate the discussion into the discrete and continuous cases.
Binary Return Rates (Discrete Case)
Consider a bet with a binary outcome: it either wins with probability \(p > \frac{1}{2}\) and returns the bet, or it loses with probability \(1-p\) and returns nothing (i.e., odds of 1:1). Suppose you have \(W_0\) dollars initially and you are allowed to play the game $T$ times. We can assume that all \(T\) plays are independent and identical. What will you do?
Try the coin flip experiment: start with 100 dollars and choose a whole-dollar bet before each of 10 flips. Each flip has a probability of winning of \(p = 0.8\). Betting 0 dollars lets you observe a flip without changing your bankroll.
A natural approach may consider betting a fixed fraction \(f\) of their wealth on each play. Let \(W_t\) denote the wealth after the \(t\)-th play. Then we have the recurrence relation
\[W_{t+1} = \begin{cases} W_t(1+f) & \text{with probability } p, \\ W_t(1-f) & \text{with probability } 1-p. \end{cases}\]If we aim to maximize the expected wealth after $T$ plays, we need to maximize the expected value of the final wealth, i.e., maximize the \(\mathbb{E}[W_T].\) In this way, conditioning on the outcomes of the previous plays, we can express the expected wealth after $T$ plays as
\[\begin{align*} \mathbb{E}[W_T | W_{1}, W_{2}, \dots, W_{T-1}] &= \mathbb{E}[W_T | W_{T-1}] \\ &= p W_{T-1}(1+f) + (1-p) W_{T-1}(1-f) \\ &= W_{T-1} \bigl( 1 + f(2p-1) \bigr). \end{align*}\]According to the tower property of conditional expectation (also known as the law of iterated expectations), we have
\[\mathbb{E}[W_T] = \mathbb{E}\bigl[ \mathbb{E}[W_T | W_{1}, W_{2}, \dots, W_{T-1}] \bigr] = \mathbb{E}\bigl[ W_{T-1} \bigl( 1 + f(2p-1) \bigr) \bigr].\]Applying this iteratively, we obtain
\[\mathbb{E}[W_T] = W_0 \bigl( 1 + f(2p-1) \bigr)^T.\]To maximize \(\mathbb{E}[W_T]\) with respect to the fraction \(f\), we need to maximize
\[\bigl( 1 + f(2p-1) \bigr)^T.\]Since the function \(x \mapsto x^T\) is monotonically increasing for \(x > 0\), it suffices to maximize the inner term:
\[1 + f(2p-1).\]The maximum is achieved by setting \(f = 1\), i.e., betting the entire wealth on each play. However, this strategy is extremely risky and can lead to ruin if a loss occurs early.
The chance to be wiped out before the \(T\)-th play is given by
\[(1-p) + p(1-p) + p^2(1-p) + \dots + p^{T-1}(1-p) = 1 - p^T.\]For example, if we only have a small edge, i.e., \(p = 0.51\), and we bet the entire wealth on each play. Then there is about 93.2% chance of being wiped out before the 5-th play and 99.9% chance of being wiped out before the 10-th play! This is clearly an unacceptable level of risk for most investors.
This motivates considering the logarithm of wealth instead, leading to the classical Kelly Criterion. Instead of maximizing the expected wealth, we maximize the expected logarithm of wealth:
\[\mathbb{E}[\log W_T].\]Conditioning on the outcomes of the previous plays, we can express the expected logarithm of wealth after $T$ plays as
\[\begin{align*} \mathbb{E}[\log W_T | W_{1}, W_{2}, \dots, W_{T-1}] &= \mathbb{E}[\log W_T | W_{T-1}] \\ &= p \log \bigl( W_{T-1}(1+f) \bigr) + (1-p) \log \bigl( W_{T-1}(1-f) \bigr) \\ &= \log W_{T-1} + p \log(1+f) + (1-p) \log(1-f). \end{align*}\]Applying the tower property of conditional expectation iteratively, we obtain
\[\mathbb{E}[\log W_T] = \mathbb{E}[\log W_0 + T \bigl( p \log(1+f) + (1-p) \log(1-f) \bigr)] = \log W_0 + T \bigl( p \log(1+f) + (1-p) \log(1-f) \bigr).\]To maximize the expected logarithm of wealth, we need to maximize \(p \log(1+f) + (1-p) \log(1-f).\)
\[f^* = \arg\max_f \left[ p \log(1+f) + (1-p) \log(1-f) \right].\]Taking the derivative with respect to \(f\) and setting it to zero, we get
\[\frac{p}{1+f} - \frac{1-p}{1-f} = 0.\]Solving for \(f\), we obtain the classical Kelly fraction:
\[f^* = 2p - 1.\]This makes intuitive sense: if the probability of winning is slightly greater than 50%, we should bet a fraction of our wealth proportional to our edge. The Kelly fraction balances the trade-off between growth and risk, avoiding the extreme risk of betting the entire wealth.