The Kelly criterion is a formula-based approach to investing and gambling. For each investment or bet, the individual allocates funds as a percentage of the entire portfolio. The Kelly criterion was created by researcher John Kelly in 1956 as a means of analyzing long-distance telephone signal noise. In more recent times, the formula has been adopted by the gambling and investment industries as means of wise resource allocation to a bet or investment.
Aspect
Explanation
Purpose
The Kelly Criterion is used to determine the optimal bet size or investment allocation in situations with uncertain outcomes, where each choice has associated probabilities and payoffs. It aims to maximize the long-term growth of capital while considering risk.
Formula
The core formula for the Kelly Criterion is: ( f^* = \frac{{bp – q}}{{b}} ) Where: – ( f^* ) represents the fraction of capital to be wagered or invested. – ( b ) represents the odds received on the bet (the payoff-to-stake ratio). – ( p ) represents the probability of winning the bet. – ( q ) represents the probability of losing the bet (( q = 1 – p )).
Optimal Bet Size
The Kelly Criterion provides the optimal fraction of capital to bet or invest, ensuring that long-term capital growth is maximized. It suggests betting more when there is a positive expected value (EV) and less when the EV is negative.
Positive vs. Negative EV
– Positive EV: When the expected value of a bet or investment is positive (( bp – q > 0 )), the Kelly Criterion suggests allocating a fraction of capital greater than zero. This means there is an edge, and the bet or investment is favorable. – Negative EV: When the expected value is negative (( bp – q < 0 )), the Kelly Criterion suggests not betting at all (( f^* = 0 )). Betting in such situations would lead to long-term capital erosion.
Fractional Betting
The Kelly Criterion often results in fractional bet sizes, which means not risking the entire capital on a single bet or investment. This approach helps protect capital during losing streaks while allowing for optimal growth during winning streaks.
Risk of Ruin
The Kelly Criterion considers the risk of losing one’s entire capital (( q )) if the optimal bet size is consistently applied. To avoid the risk of ruin, some practitioners use a fraction of the calculated ( f^* ) (e.g., half Kelly or quarter Kelly) to reduce the impact of potential losses.
Applications
The Kelly Criterion is applied in various domains, including: – Gambling: Professional gamblers use it to size bets in games like poker, blackjack, and sports betting. – Investing: Investors use it to determine portfolio allocations and position sizing. – Information Theory: The Kelly Criterion has connections to information theory and the concept of maximizing the growth rate of wealth over time.
Variations
Several variations of the Kelly Criterion exist, such as the Full Kelly (using the entire ( f^* )), Half Kelly (using half of ( f^* )), and Fractional Kelly (using a fraction like 1/4 or 1/2). These variations help balance aggressive growth with risk mitigation.
Criticisms
The Kelly Criterion assumes perfect knowledge of probabilities and payoffs, which may not be realistic in practice. It also doesn’t account for the impact of transaction costs, taxes, or external factors that can affect outcomes. Additionally, it may result in volatile bet sizes, which some find uncomfortable.
Adaptations
In real-world scenarios, individuals and investors often adapt the Kelly Criterion to their risk tolerance and constraints. They may use fractional Kelly strategies, incorporate safety margins, or combine the Kelly strategy with other risk management techniques.
Conclusion
The Kelly Criterion is a powerful tool for optimizing decision-making in situations involving uncertainty and financial stakes. While it has limitations and may require adjustments in practical applications, it provides a structured approach to balancing risk and reward, aiming for long-term capital growth.
Table of Contents
For example:
A blackjack player determining how much of their bankroll to wager on the next hand.
A stock investor deciding on the percentage of their portfolio that should be devoted to speculative resource stocks.
A real-estate investor questioning how much of their portfolio to devote toward condominiums in Miami Beach.
In each example, the goal of the gambler or investor is to grow their capital.
According to the Kelly criterion, the amount of money invested should be proportional to the knowledge of the bet or investment itself.
Key components of the Kelly criterion formula
The Kelly criterion is expressed by the formula:
Here:
f* = the total percentage of wealth that should be risked.
p = the historical probability of a win. Ideally, the p-value needs to be above 0.50 or 50%.
b = the decimal odds minus 1, otherwise known as the amount that could potentially be won or lost. For investors, calculate the b value by dividing the total number of trades yielding a positive amount by the total number of trades.
q = the probability of failure (i.e. 1-p).
When interpreting the f*value, multiply it by 100 to get the percentage of total funds that should be risked. For example, if f* = 0.17 then the total percentage of funds allocated should be 17%.
Kelly criterion limitations
Earlier, it was stated that knowledge of the bet was proportional to the amount of money invested in that bet.
This begs the question: how is knowledge obtained and what does it constitute?
The Kelly criterion defines knowledge as the perceived edge, itself defined as a betting advantage gained from exploiting bookmaker margins or possessing proprietary knowledge.
However, it is important to note that the formula is not foolproof. While it does define the point of maximum portfolio growth, the f* value is calculated from real-world probabilities that are best estimates only. Invariably, factors that influence the probability of winning are complex, obscured, or hard to define.
To reduce variance, many choose to estimate returns that are 30 to 50% of the calculated f* value. This is known as a safety margin, where risks are assumed to be higher than stated.
Other limitations of the Kelly criterion include:
A focus on growth stocks. As growth stock profits are continually re-invested to encourage exponential growth, Kelly’s formula favors growth stock investors. Investors wishing to make smaller, more consistent income-stock profits may find the model too aggressive.
Higher margin for error. Given that the f* value is the point of maximum potential growth, a figure only slightly higher brings substantially more risk, variance, and decreased profit. This means that undisciplined gamblers or investors prone to greed could suffer larger losses by ignoring the stated f* value.
Examples of the Kelly Criterion
Blackjack Player: A blackjack player decides how much of their bankroll to wager on the next hand based on their perceived edge in the game.
Stock Investor: An investor determines the percentage of their portfolio that should be allocated to speculative resource stocks, considering the potential gains and risks involved.
Real Estate Investor: A real-estate investor contemplates how much of their portfolio should be devoted to condominiums in Miami Beach, taking into account market conditions and potential returns.
Case Studies
Mutual Funds:
Description: An investor is contemplating investing in several mutual funds based on their past performance.
Using the Kelly Criterion: For each fund, they can estimate the probability of the fund outperforming the market average. By plugging in the potential returns and the probabilities into the Kelly formula, they can determine the optimal allocation for each mutual fund in their portfolio.
Art Collection:
Description: An art collector believes certain artworks will appreciate over time based on the artist’s reputation or historical trends.
Using the Kelly Criterion: They can estimate the probability of each art piece appreciating in value over a certain time frame and the potential rate of return. Using the formula, they can decide how much to invest in each piece relative to their total art collection budget.
Bond Investments:
Description: An individual wants to invest in corporate bonds with varying degrees of risk.
Using the Kelly Criterion: They can assess the likelihood of each bond defaulting versus the expected yield. With these probabilities and potential returns, they can allocate their funds optimally among the bonds.
Agriculture/Farming:
Description: A farmer is deciding how much of their land to allocate for different crops based on expected market prices and yields.
Using the Kelly Criterion: The farmer can assess the probability of each crop yielding a good harvest based on historical data and current conditions. Considering the potential profit from each crop, they can determine the optimal land allocation.
E-commerce Business:
Description: An online retailer is deciding how much inventory to stock for various products based on projected sales.
Using the Kelly Criterion: By estimating the probability of each product selling out based on past sales data and potential profit margins, the retailer can decide on optimal inventory levels for each product.
Music Industry:
Description: An indie music producer wants to invest in promoting several artists based on their potential to become popular.
Using the Kelly Criterion: The producer can estimate the probability of each artist becoming a hit based on early feedback and potential returns from streaming and sales. This helps in allocating the promotional budget optimally among the artists.
Wine Investment:
Description: An investor believes certain wines will appreciate over time due to their vintage and rarity.
Using the Kelly Criterion: They can estimate the probability of each wine appreciating in value based on historical data and expert opinions. This helps in deciding how much to invest in each wine relative to their total wine investment budget.
Peer-to-peer Lending:
Description: An individual is considering lending money to various borrowers on a peer-to-peer lending platform.
Using the Kelly Criterion: By assessing the creditworthiness of each borrower and the potential interest returns, the lender can decide the optimal amount to lend to each borrower.
Antique Collection:
Description: An antique collector believes certain items will appreciate based on their age, rarity, and historical significance.
Using the Kelly Criterion: By estimating the probability of each item appreciating and the potential rate of return, they can determine how much to invest in each antique relative to their total budget.
Startup Angel Investing:
Description: An angel investor is looking to invest in several startups based on their potential growth.
Using the Kelly Criterion: They can assess the likelihood of each startup succeeding based on market research and potential returns from an exit (like an acquisition or IPO). This helps in deciding the optimal investment allocation for each startup.
Key Highlights of the Kelly Criterion:
Resource Allocation Formula: The Kelly criterion is a formula-based approach to resource allocation in gambling and investment. It suggests investing a certain percentage of the entire portfolio based on the perceived edge of the bet or investment.
Origin: The Kelly criterion was developed by researcher John Kelly in 1956 while analyzing long-distance telephone signal noise. It has since been widely adopted in the gambling and investment industries.
Components of the Formula: The formula is expressed as f* = (bp – q) / b, where f* represents the total percentage of wealth to be risked, p is the historical probability of a win, b is the decimal odds minus 1, and q is the probability of failure (1-p).
Interpreting f Value:* The f* value, once calculated, indicates the percentage of total funds that should be risked. For example, if f* = 0.17, then 17% of the total funds should be allocated to the bet or investment.
Knowledge and Perceived Edge: The Kelly criterion emphasizes that the amount of money invested should be proportional to the knowledge of the bet or investment. Knowledge is defined as the perceived edge, gained from exploiting bookmaker margins or possessing proprietary knowledge.
Limitations: The Kelly criterion relies on real-world probabilities, which are often best estimates and subject to uncertainty. To reduce variance, many use a safety margin, estimating returns lower than the calculated f* value. The formula may favor growth stock investors, and a slight deviation from the f* value can lead to significant risk and decreased profit.
Key takeaways
The Kelly criterion is a mathematical formula that guides gambling and investment decisions by way of risk-managed resource allocation.
The Kelly criterion argues that the background knowledge of an investment or bet is directly proportional to the amount of money that should be allocated.
While useful in a variety of scenarios, the Kelly criterion is nevertheless based on mostly unquantifiable probabilities that increase variance. As a result, a conservative approach to calculating investment percentages should be taken.
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Systems thinking is a holistic means of investigating the factors and interactions that could contribute to a potential outcome. It is about thinking non-linearly, and understanding the second-order consequences of actions and input into the system.
Vertical thinking, on the other hand, is a problem-solving approach that favors a selective, analytical, structured, and sequential mindset. The focus of vertical thinking is to arrive at a reasoned, defined solution.
Maslow’s Hammer, otherwise known as the law of the instrument or the Einstellung effect, is a cognitive bias causing an over-reliance on a familiar tool. This can be expressed as the tendency to overuse a known tool (perhaps a hammer) to solve issues that might require a different tool. This problem is persistent in the business world where perhaps known tools or frameworks might be used in the wrong context (like business plans used as planning tools instead of only investors’ pitches).
The Peter Principle was first described by Canadian sociologist Lawrence J. Peter in his 1969 book The Peter Principle. The Peter Principle states that people are continually promoted within an organization until they reach their level of incompetence.
The straw man fallacy describes an argument that misrepresents an opponent’s stance to make rebuttal more convenient. The straw man fallacy is a type of informal logical fallacy, defined as a flaw in the structure of an argument that renders it invalid.
The Streisand Effect is a paradoxical phenomenon where the act of suppressing information to reduce visibility causes it to become more visible. In 2003, Streisand attempted to suppress aerial photographs of her Californian home by suing photographer Kenneth Adelman for an invasion of privacy. Adelman, who Streisand assumed was paparazzi, was instead taking photographs to document and study coastal erosion. In her quest for more privacy, Streisand’s efforts had the opposite effect.
As highlighted by German psychologist Gerd Gigerenzer in the paper “Heuristic Decision Making,” the term heuristic is of Greek origin, meaning “serving to find out or discover.” More precisely, a heuristic is a fast and accurate way to make decisions in the real world, which is driven by uncertainty.
The recognition heuristic is a psychological model of judgment and decision making. It is part of a suite of simple and economical heuristics proposed by psychologists Daniel Goldstein and Gerd Gigerenzer. The recognition heuristic argues that inferences are made about an object based on whether it is recognized or not.
The representativeness heuristic was first described by psychologists Daniel Kahneman and Amos Tversky. The representativeness heuristic judges the probability of an event according to the degree to which that event resembles a broader class. When queried, most will choose the first option because the description of John matches the stereotype we may hold for an archaeologist.
The take-the-best heuristic is a decision-making shortcut that helps an individual choose between several alternatives. The take-the-best (TTB) heuristic decides between two or more alternatives based on a single good attribute, otherwise known as a cue. In the process, less desirable attributes are ignored.
The concept of cognitive biases was introduced and popularized by the work of Amos Tversky and Daniel Kahneman in 1972. Biases are seen as systematic errors and flaws that make humans deviate from the standards of rationality, thus making us inept at making good decisions under uncertainty.
The bundling bias is a cognitive bias in e-commerce where a consumer tends not to use all of the products bought as a group, or bundle. Bundling occurs when individual products or services are sold together as a bundle. Common examples are tickets and experiences. The bundling bias dictates that consumers are less likely to use each item in the bundle. This means that the value of the bundle and indeed the value of each item in the bundle is decreased.
The Barnum Effect is a cognitive bias where individuals believe that generic information – which applies to most people – is specifically tailored for themselves.
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The ladder of inference is a conscious or subconscious thinking process where an individual moves from a fact to a decision or action. The ladder of inference was created by academic Chris Argyris to illustrate how people form and then use mental models to make decisions.
The Six Thinking Hats model was created by psychologist Edward de Bono in 1986, who noted that personality type was a key driver of how people approached problem-solving. For example, optimists view situations differently from pessimists. Analytical individuals may generate ideas that a more emotional person would not, and vice versa.
Second-order thinking is a means of assessing the implications of our decisions by considering future consequences. Second-order thinking is a mental model that considers all future possibilities. It encourages individuals to think outside of the box so that they can prepare for every and eventuality. It also discourages the tendency for individuals to default to the most obvious choice.
Lateral thinking is a business strategy that involves approaching a problem from a different direction. The strategy attempts to remove traditionally formulaic and routine approaches to problem-solving by advocating creative thinking, therefore finding unconventional ways to solve a known problem. This sort of non-linear approach to problem-solving, can at times, create a big impact.
Bounded rationality is a concept attributed to Herbert Simon, an economist and political scientist interested in decision-making and how we make decisions in the real world. In fact, he believed that rather than optimizing (which was the mainstream view in the past decades) humans follow what he called satisficing.
The Dunning-Kruger effect describes a cognitive bias where people with low ability in a task overestimate their ability to perform that task well. Consumers or businesses that do not possess the requisite knowledge make bad decisions. What’s more, knowledge gaps prevent the person or business from seeing their mistakes.
Occam’s Razor states that one should not increase (beyond reason) the number of entities required to explain anything. All things being equal, the simplest solution is often the best one. The principle is attributed to 14th-century English theologian William of Ockham.
The Mandela effect is a phenomenon where a large group of people remembers an event differently from how it occurred. The Mandela effect was first described in relation to Fiona Broome, who believed that former South African President Nelson Mandela died in prison during the 1980s. While Mandela was released from prison in 1990 and died 23 years later, Broome remembered news coverage of his death in prison and even a speech from his widow. Of course, neither event occurred in reality. But Broome was later to discover that she was not the only one with the same recollection of events.
The bandwagon effect tells us that the more a belief or idea has been adopted by more people within a group, the more the individual adoption of that idea might increase within the same group. This is the psychological effect that leads to herd mentality. What in marketing can be associated with social proof.
Gennaro is the creator of FourWeekMBA, which reached about four million business people, comprising C-level executives, investors, analysts, product managers, and aspiring digital entrepreneurs in 2022 alone | He is also Director of Sales for a high-tech scaleup in the AI Industry | In 2012, Gennaro earned an International MBA with emphasis on Corporate Finance and Business Strategy.
