The Kelly Criterion
Introduction
John L. Kelly, Jr. was an American scientist and researcher who made significant contributions to information theory and telecommunications during the mid-20th century. During his time at Bell Labs, Kelly worked on various projects related to information theory, coding theory, and communication systems. He is best known for his groundbreaking work on the theory of efficient communication and data transmission, which laid the foundation for modern digital communication technologies. In 1956, Kelly published a seminal paper titled “A New Interpretation of Information Rate”, in which he introduced the concept of the “Kelly Criterion” as a method for maximizing the long-term growth of capital in a series of bets or investments. The Kelly Criterion became widely recognized as a powerful tool for risk management and portfolio optimization in gambling and investment contexts.
The Kelly Criterion
The Kelly criterion is a technique or mathematical formula used to determine the optimal size of a series of bets in order to maximize the long-term growth of capital. It was developed by Mathematician, John L. Kelly Jr. in 1956 while working at Bell Labs. The technique seeks to maximize expected growth of assets by optimally investing a fixed fraction of wealth in a series of investments. This idea is particular popular in gambling and investment contexts, where it helps to manage risk and optimize returns.
Suppose the following example. You start with a capital of $50 and you need to flip a biased coin repeatedly. This biased coin will land heads 70% of the time and a 30% of landing a tail. On each flip, you are given a 4-to-6 payout odds heads to tails and a choice on how much to wager. So, if you wager $6, you will profit $4 if it lands heads, otherwise you will lose that $6 while if you wager $3, the profits and loss will be halved. So how much should you bet?
Before we answer that question, we need ask ourselves if we should even flip this coin in the first place. The answer is yes as we do make some capital for every flip (for a $6 wager, we have a 70% chance of winning $4 and a 30% chance of losing the $6). Thus, it averages out to a $1 profit. By performing a fixed bet, for example $5, there are many different possible outcomes which can be visualized by Monte-Carlo Simulation. However, we would want to quantify the betting amount as a point value with a fixed probability so that it is easy to compare across different strategies. Economist will pass the final wealth into the Utility Function and calculate the average utility. The Kelly Criterion does it a different way. For each possible final wealth, we ask ourselves, what growth rate does it imply. Given a final wealth after N flips, if our initial wealth were to grow by Growth Rate (g) for each N flips, what would g be? We can form a simple equation and g the subject.
We used a constant bet of $5 where each path has a simulated probability. By plotting the simulated probability against the Growth Rate, the Kelly Criterion aims to average out all Growth Rates according to the strategy. To maximize our final capital, we need to find a value of average g that maximizes the expected value.
You should bet an amount proportional to how much you have. As you win and your wealth grows, you will bet a larger amount. But you should not bet too much. If you bet your whole capital and it lands a tail, you lose everything. That is poor risk management. On the other hand, if you bet too little then it will take a long time for you to make a decent amount. The Kelly Criterion is to bet a certain fraction of wealth so as to maximize the expected growth of wealth. It is a function of only two things: the probability of heads and the payout odds (in our case it is 4/6). By plotting the expected growth rate against fixed percent wage, we are able to identify a point on the convex function where expected growth rate is maximized. To make things simple, the wealth would be converted to a log scale. A betting fraction of less would be a conservative strategy while increasing wagered fraction, f, adds volatility to returns and decrease the expected returns.
Mathematical Derivation of Kelly
We have to note that Kelly Criterion is really difficult in real world scenarios. This is because we never know the true value of the outcomes, rather, we have an estimate. Using the estimate instead of the true probability, our average growth rates will fall, leading to a lower long run capital. Uncertainty in your estimates leads to less confidence in the bets you place. In investments, the mean and standard deviation are rarely known accurately. A solution to this is to shrink your bets. A common choice is a half Kelly. Note that Risk Management is such a complex subject, there is no hard and fixed rule to this.
Conclusion
John L. Kelly, Jr. passed away on March 2, 1965, at the age of 41. Today, his legacy lives on through the enduring relevance and applications of the Kelly Criterion in finance, gambling, and other fields where risk management and optimization are paramount.
References
[1] The Kelly Criterion — Mutual Information
[2] The Kelly Criterion — Frequently Asked Questions in Quantitative Finance by Paul Wilmott
