Notes on "Fortune's Formula"
Gambler’s Ruin: Over time, gambling tends to transfer money from the gambler to the house.
Martingale Strategy: Rather than preventing loss, this strategy actually accelerates the gambler’s ruin.
The Kelly Criterion
The best strategy is one that offers the highest compound return consistent with no risk of going broke. The Kelly system ensures the bettor stays in the game long enough for the law of large numbers to work, allowing positive-expectation bets to yield their expected returns over time.
Fraction to bet = Edge/Odds
The edge is how much you expect to win, on average, assuming you could make this wager over and over with the same probabilities. It is a fraction because the profit is always in proportion to how much you wager.
Odds is not necessarily a good measure of probability. Odds are set by market forces, by everyone else’s beliefs about the chance of winning. These beliefs may be wrong. In fact, they have to be wrong for the Kelly gambler to have an edge.
Example: The odds for A are 5 to 1 (odds = 5), and you believe A has a 1-in-3 chance of winning.
Expected value: 1/3 x $600 = $200 (for a $100 bet)
Net profit: $200 - $100 = $100
Edge: 100/100 = 1
Kelly criterion: 1/5 = 0.2 = 20% of your bankroll
If your edge is 0 or negative, you should not bet. If you know the outcome is certain, your edge equals the odds, and the Kelly criterion suggests betting everything (100%).
Bachelier’s “A theory of speculation” and Random walk
The unpredictability of stock prices makes them somewhat predictable in a statistical sense.
Likened to a crowd of drunks all wandering randomly from a lamppost, the exact path of any individual is unpredictable. Each step the drunk takes is random and independent of previous steps. There is no way to predict the exact path or final position of any individual drunk after a certain number of steps. [Unpredictability at the micro level]
While individual paths are unpredictable, the average behavior of a large group of drunks (or repeated trials of the same drunk) is highly predictable in a statistical sense. [Predictability at the macro level]
The overall distribution of the crowd will remain centered on the lamppost. That’s because nothing is “pushing” the wandering drunks in any particular direction. All directions are the same to them. Over time, the crowd diffuses outward in all directions. [No directional bias]
The average distance from the lamppost after a given number of steps (or time intervals) increases with the square root of time, not linearly.
This statistical predictability, despite individual randomness, is the foundation for understanding phenomena like stock price movements, where prices fluctuate unpredictably in the short term but follow predictable statistical properties (e.g., distribution, variance) over time.
EMH and Random walk are closely related in that if markets are efficient, then price changes should be random. A random walk in prices is a necessary condition for an efficient market, but it is not sufficient. Random price movements can result from factors unrelated to informational efficiency, such as noise trading, market frictions, or bubbles. True market efficiency requires that prices reflect all available information, not just that their movements appear random.
St. Petersburg Paradox
It describes a lottery game where a fair coin is tossed repeatedly until it lands heads, and the payout doubles with each tails before the first heads. The expected value of the game is infinite, but in reality, people are only willing to pay a small amount to play. This mismatch between mathematical expectation and actual behavior is the paradox.
Bernoulli’s logarithmic utility
Daniel Bernoulli (1738) introduced the concept of expected utility, arguing that rational decision-making under uncertainty should maximize the expected value of a utility function, not simply wealth. He proposed logarithmic utility to account for the diminishing marginal utility of money. That means the utility (subjective value) of wealth increases with wealth, but at a decreasing rate.
This means that when you account for diminishing returns in the value of money, the infinite expected value/utility of the game becomes finite, even though the expected monetary value is infinite.
Bernoulli's Adjustment to the St. Petersburg Paradox With log utility, the maximum a rational person would pay to play the St. Petersburg game is much lower than the infinite expected value, aligning with observed behavior.
Risky ventures should be evaluated using the geometric mean of outcomes rather than the arithmetic mean.
The geometric mean embodies risk aversion by accounting for diminishing marginal utility—each additional unit of wealth is valued less, so large losses have a greater negative impact than equivalent gains provide in positive value. Maximizing expected log utility, as Bernoulli proposed, is mathematically equivalent to maximizing the geometric mean of outcomes, which discourages highly risky bets and aligns with the cautious preferences of risk-averse individuals. [Risk aversion]
Buying insurance can improve the long-term growth rate of wealth for a relatively poor merchant, even if the insurance seems overpriced. The insurance company can also benefit by selling that same insurance policy, as the risk is small relative to its wealth and the premiums add to its growth. Both parties, despite being on opposite sides of the transaction, are improving their own geometric mean of outcomes, which is the key to maximizing long-term wealth.
In the context of repeated bets or investments where returns compound, the geometric mean accurately measures the typical long-term growth rate. Unlike the arithmetic mean, which can exaggerate the value of risky ventures by ignoring variability and the compounding effect of losses, the geometric mean reflects the true rate at which wealth grows or shrinks over time, offering a more realistic assessment of long-term outcomes. [Multiplicative processes]
Example
The Kelly criterion and Logarithmic utility
The Kelly criterion prescribes betting a fraction of wealth that maximizes the long-term growth rate of wealth, which is achieved by choosing the one with the highest geometric mean of outcomes. [binary outcomes] When the possible outcomes are not all equally likely, you need to weight them according to their probability. Maximizing this expected log return is mathematically equivalent to maximizing the weighted geometric mean of possible outcomes. [probabilistic outcomes]
Following the Kelly criterion is equivalent to acting as if you have logarithmic utility for wealth. This means you value each additional unit of wealth less than the previous one. Log utility ensures risk aversion and avoids strategies that could lead to ruin, emphasizing sustainable long-term growth.
Markowitz vs Kelly
Markowitz’s mean-variance optimization seeks to balance expected return (arithmetic average return) against risk (variance), producing a set of “efficient” portfolios that offer the best trade-off between these two factors for a single investment period.
Unlike mean-variance analysis, which leaves the choice to personal preference, Latané’s geometric mean criterion/ Kelly criterion/ Logarithmic utility identifies a single “best” portfolio: a portfolio with the highest geometric mean of the probability distribution of outcomes, as computed from current means, variances, and other statistics. The returns and volatility of your investments will change with time. When they do, you should adjust your portfolio accordingly, again with the sole objective of attaining the highest geometric mean. This approach is especially relevant for investors with a long-term perspective when compounding is involved.
In risk-free scenarios, the arithmetic and geometric means are identical. When risk enters the picture—such as with volatile stocks—the arithmetic mean can be misleading. A stock with a higher average return but much higher volatility may have a lower geometric mean, meaning it could be a worse long-term investment.
The “Wheel of Fortune” Analogy The Kelly criterion that prescribes maximizing geometric mean growth of wealth (CAGR) is only meaningful when profits are reinvested/ compounded. If winnings are not reinvested, maximizing the arithmetic mean is preferable.
Non-compounding: Choosing the highest arithmetic mean (expected value) is optimal because each round is independent; the law of large numbers ensures average winnings approach the expected value.
$1 bet per week for 52 weeks on a game with an arithmetic mean return of $1.75, and pocket any gains; expected total winnings after a year will be $91 ($1.75 x 52), regardless of the volatility or risk of ruin in any single bet.
Compounding: The geometric mean dominates because returns build on each other, and volatility or risk of ruin has a much greater impact on long-term wealth.
Start with $1 and reinvest all winnings in each round for 52 rounds. On a wheel with a high geometric mean, $1 could grow into millions. On a wheel with a zero-outcome possibility, the geometric mean is zero; ending up with nothing is almost a certainty.
Using the example of betting on a fair coin toss with a highly favorable 6-to-1 payout, the Kelly system prescribes betting 40% of your current bankroll on each toss. After four tosses, this “pinball machine” of possible outcomes produces a wide range of results: while the best-case scenario turns $100 into $8,100, the worst case leaves you with just $12.96. The most common outcome, with two wins and two losses, grows your bankroll to $324, which is both the geometric mean and the median—higher than with any other betting system.
The Pinball Machine Despite its mathematical appeal, the Kelly criterion does not eliminate risk. Even in this favorable setup, 5 out of 16 possible outcomes leave you with less than your starting amount, and there is always a small chance of a catastrophic loss. The system maximizes long-term and median wealth but cannot guarantee against unlucky streaks or large short-term losses. For risk-averse individuals, this small but real chance of substantial loss may outweigh the benefits of optimal growth, highlighting the trade-off between maximizing returns and managing risk.
Strategies that maximize the expected (arithmetic) wealth can have a higher average due to a few extreme outcomes but often result in a median wealth that is much lower than the mean. In contrast, the Kelly criterion maximizes the wealth that a typical investor is most likely to achieve, rather than being skewed by rare, outsized wins.
Excrept from “Fortune’s Formula” The 1/n rule in Kelly betting strategy suggests in an infinite series of bets, the chance that your bankroll will ever drop to a fraction 1/n of its starting value is exactly 1/n.
Half Kelly cuts volatility by half while sacrificing return by only a quarter
Excrept from “Fortune’s Formula” Kelly betting and fat tail events
Suppose you are betting on a simultaneous toss of biased coins believed to have a 55% chance of coming up heads. But on this toss, only 45% of the coins are heads, a fat tail event.
The Kelly bettor still preserves most of their bankroll because the formula never risks everything on a single outcome. This approach allows the bettor to recover from bad luck or adjust their strategy if their probability estimates were wrong, thus avoiding total ruin.
In contrast, a highly leveraged bettor, using something like 30-times leverage, risks far more than their actual capital and can be completely wiped out—and even end up in debt—if a fat tail event occurs.
The core lesson from Kelly’s philosophy is that even unlikely disasters will eventually happen, so always keeping some margin for error is crucial.
Kelly betting protects against catastrophic loss and ensures survival, as the ultimate compound return rate is acutely sensitive to fat tails.
“The gambler using borrowed money must determine how much he can lose without touching off a disastrous chain of misfortunes from which recovery is impossible.”
[…]
“People think that if things are bounded in a certain historical range, there’s necessity or causality here,” Thorp explained. Of course, there’s not. In 1998, when the spread widened suddenly to 6 percent, “they said this was a one-in-a-million-year event. A year or two later, it got wider, and two years after that, it got wider yet.”
[…]
“Estimates about the market’s probabilities are always going to be just that: estimates. It is good practice to have a sense of how far off these estimates may be, and how much likely errors would affect the results. “Margins of error” are themselves estimates. Human nature often skews these estimates optimistically.
[…]
A decade rarely passes without a market event that some respected economist claims, with a straight face, to be a perfect storm, a ten-sigma event, or a catastrophe so fantastically improbable that it should not have been expected to occur in the entire history of the universe from the big bang onward. In a world where financial models can be so incredibly wrong, the extreme downside caution of Kelly betting is hardly out of place. For reasons mathematical, psychological, and sociological, it is a good idea to use a money management system that is relatively forgiving of estimation errors.”
Excrept from Fortune’s Formula, Part Six: Blowing Up