The Kelly Criterion

People who are thinking about investing in stocks usually start by asking which stocks they should buy. But there is another question that is just as important. How much of their money should they invest? Even if someone finds a promising investment opportunity, putting too much money into it can lead to serious losses. On the other hand, investing too little might mean missing out on potential profits. One way to understand this problem is through the Kelly criterion. The Kelly criterion is a mathematical principle that uses the probabilities of winning and losing to calculate how much money to risk in order to maximize long-term growth. The Kelly criterion originated in a paper published in 1956 by John Larry Kelly Jr., an American mathematician and researcher at Bell Labs. Interestingly, Kelly was not originally studying the stock market. He was investigating the relationship between information and probability in communication systems. At the time, one challenge in communication technology was that noise could interfere with signals, making information less reliable. Kelly wanted to explain mathematically how valuable information could be when it helped someone predict an uncertain event. His research led to a principle about how much money a person should risk when the probabilities of different outcomes are known. To understand the Kelly criterion, imagine a simple game involving a special coin. This coin has a 60% chance of landing on heads and a 40% chance of landing on tails. If the coin lands on heads, you win an amount equal to the money you bet. If it lands on tails, you lose your entire bet. For example, suppose you bet $100. If the coin lands on heads, you receive your original $100 plus another $100 in profit, giving you a total of $200. If it lands on tails, you lose the $100 you bet. Because heads is more likely than tails, the game gives you an advantage. But how much money should you bet? Suppose you have $1,000. Since you have a greater chance of winning than losing, would betting all $1,000 be the best choice? If you bet everything and win, your money doubles to $2,000. However, if you lose, you have nothing left. Even when the odds are in your favor, one unlucky outcome can wipe out all your money and prevent you from playing again. On the other hand, betting a very small amount means you cannot fully benefit from the advantage the game offers. Kelly developed a mathematical way to determine what fraction of your money would produce the highest long-term growth rate. In a game like this one, where you gain an amount equal to your bet when you win and lose your entire bet when you lose, the calculation is relatively simple. You subtract the probability of losing from the probability of winning. The probability of winning is 60%, and the probability of losing is 40%. Subtracting 40% from 60% gives 20%. According to the Kelly criterion, you should therefore bet 20% of your available money on each round. If you have $1,000, that means betting $200 and keeping the remaining $800 for future opportunities. An important detail is that you do not keep betting 20% of your original $1,000. Instead, you bet 20% of however much money you currently have. Suppose you win the first round after betting $200. Your total money increases to $1,200. On the second round, you bet 20% of $1,200, which is $240. If you lose that round, your money falls to $960. Even though you won once and lost once, you now have less money than when you started. This example shows why equal percentage gains and losses do not necessarily cancel each other out. If you invest $1,000 and lose 50%, you are left with $500. To get back to $1,000, you need a 100% return on the remaining money, not a 50% return. This is why avoiding large losses is an important part of investing. The Kelly criterion is not designed to maximize the money you make from a single bet. Instead, it identifies the fraction that theoretically maximizes long-term compound growth when you repeatedly take opportunities with favorable odds. Compounding means that money earned from previous gains can also contribute to future growth. However, following the Kelly criterion does not guarantee that you will make money. Even if a coin has a 60% chance of landing on heads, it can still land on tails several times in a row. A person following the Kelly criterion can lose money during such a streak. The criterion does not eliminate risk. It calculates a long-term growth strategy under specific mathematical assumptions, including accurate knowledge of probabilities and payouts. So can the Kelly criterion also be used in the stock market? A stock represents a small ownership share in a company. Investors buy stocks because they believe a company may become more valuable in the future. If the stock price rises, they may be able to sell their shares for a profit. But if the company performs poorly or economic conditions change, the stock price may fall, causing investors to lose money. The basic idea behind the Kelly criterion can also be applied to stock investing. However, unlike the coin game, real stock investments do not come with clearly known probabilities of winning or losing. The size of potential gains and losses is also uncertain. A stock might rise 2% one day and fall 5% another day. For this reason, investors cannot simply use the coin game's formula to decide how much money to put into a stock. Applying the Kelly criterion to actual investments requires considering different possible returns and the probabilities associated with them. There is another problem. An investor's probability estimates may be wrong. Someone might believe a stock has a 60% chance of rising when the actual probability is only 40%. If the probability estimate is inaccurate, the investment amount calculated using the Kelly criterion may also be inappropriate. To reduce this risk, some investors use a more conservative approach called fractional Kelly. Instead of investing the full amount suggested by the calculation, they invest only a portion of it. For example, if the Kelly criterion suggests risking 20%, someone using half-Kelly would risk just 10%. This may reduce the theoretical long-term growth rate, but it can also reduce losses when the original estimates turn out to be wrong. Of course, these percentages do not apply to every stock or investment. The 20% and 10% figures in these examples are based on a hypothetical game. They are not recommendations to invest those percentages of your money in a particular stock. For someone who has never invested before, understanding risk is more important than calculating a precise investment percentage. Stock prices can fall at any time, and investments offering the possibility of high returns may also carry substantial risks. Investing money needed for everyday expenses or putting all your savings into a single company can create serious financial difficulties if the investment performs poorly. The Kelly criterion teaches an important lesson: investing more money does not automatically make an investment decision better. Even when you believe your chances of success are high, you must consider the possibility of failure and preserve enough money to take advantage of future opportunities. Ultimately, the Kelly criterion is not a formula that tells you which stocks will rise. It is a principle that helps explain how much money to put at risk when outcomes are uncertain. Successful investing is not only about making a large profit from one decision. It is also about managing losses and preserving the ability to continue investing over time.