[00:02:37] Shawn O’Malley: And they quickly realized that the stock market offered an even bigger channel to profit from using the Kelly formula. The following years, Ed Thorpe would enjoy immaculate returns with his hedge fund, Princeton Newport Partners, while Shannon’s returns would be quite stellar too, outpacing Warren Buffett.

[00:02:54] Shawn O’Malley: Again, this may all sound too good to be true, but there is a dark side to the Kelly formula premised around exploiting an insider’s edge. By the end of the episode, I think you’ll know what I mean. Still, the Kelly formula is a valuable tool for any investor, and the story may lend hope to those aiming to beat the market.

[00:03:08] Shawn O’Malley: But I don’t want to get too far ahead of myself, let’s start back a bit with Claude Shannon. Claude Shannon is, by all accounts, a genius. That really understates things, though. For many who knew him, they considered it an insult to compare Shannon’s intelligence with Einstein’s, not because Einstein was so superior, but because Shannon was.

[00:03:26] Shawn O’Malley: And just because you have not heard of him, does not mean that he hasn’t fundamentally changed the world you live in. To try and understand Shannon’s impact on the world is like trying to measure the impact of inventing the alphabet. Instead of the alphabet for a spoken language, Shannon devised the language to be spoken by computers, that is, binary coding, the world of ones and zeros.

[00:03:45] Shawn O’Malley: He described how this could be captured by electric circuits, where the presence of an electric impulse marked a 1, and no impulse marked a 0. This basic system of coding could be used to convey words, audio, and video, and really all information that we communicate digitally. Shannon is considered among the two or three individuals deserving of the most credit for having invented the electric computer, yet this is not even his greatest achievement.

[00:04:07] Shawn O’Malley: Shannon’s master composition is his information theory, which provides the backbone for the whole of modern technology. From satellite communications to DNA sequencing and virtual reality, much of the modern world owes thanks to Shannon. He accomplished these breakthroughs while working at the avant garde research institution, Bell Labs.

[00:04:25] Shawn O’Malley: The lab traces its roots to Alexander Graham Bell, who used a reward granted to him by the French government for inventing the telephone to fund his own laboratory. Around this time, Bell Telephone Company was formed and, several years later, merged with American Telephone and Telegraph Company, or as we know it today, AT& T.

[00:04:42] Shawn O’Malley: Over the following decades, a network of engineers supporting Bell Labs and its research into telecommunications emerged and would become integral to America’s wartime efforts during World War II to gain technological advantages over the Axis powers. Bell Labs would become one of the most prolifically inventive institutions in human history, building everything from the first lasers and transistors to the first solar batteries and hearing aids, and even helping uncover the origins of the universe.

[00:05:06] Shawn O’Malley: And today, the prestigious group still maintains 10 worldwide locations in places like Shanghai, Finland, Cambridge, Munich, and New Jersey. Over the years, Bell Labs has accumulated an impressive 10 Nobel Prize awards and 5 Turing Awards for contributions to computer science. So Bell Labs has been an exclusive and impactful institution, to say the least.

[00:05:28] Shawn O’Malley: It’s sensitive work has also long been the subject of great intrigue, when Claude Shannon first joined Bell Labs, he began work on what was called Project X, an endeavor with enough scientific pedigree to rival the Manhattan Project of WWII, which created the atom bomb. Project X was a joint undertaking with Britain’s Code and Cypher School, and it included not only Shannon, but other inspiring minds like Alan Turing, namesake of the Turing Award and the Turing Test.

[00:05:53] Shawn O’Malley: They were building what was the first wireless phone, and it was of paramount importance during World War II. Allied leaders sought ways to openly communicate with each other without fear that their messages could be intercepted by the Nazis. Bell Labs built such systems for President Roosevelt in the U.S. and for Winston Churchill to enable safe and seamless transatlantic communications using the only node encryption technology that cannot be cracked, called the one time pad. Shannon’s job was to prove, in fact, that the system truly was impregnable, enabling allied leaders to speak freely. This experience with studying communications technology would be foundational for Shannon, leading him ultimately to his realization that information could be stored in binary code with ones and zeros.

[00:06:33] Shawn O’Malley: In 1948, Shannon published a paper titled A Mathematical Theory of Communication, cementing him as a founding father of information theory and binary language. To understand information theory, we need to understand what computing was like in the early to mid-20th century. According to Bell Labs website, quote, The term computing as it is used today does not reflect the process that existed in the first half of the 20th century.

[00:06:56] Shawn O’Malley: In 1930s, computing meant using mechanical or electrical devices for finding numerical solutions to math problems. At Bell Labs, it also meant designing and building complex telephone switching systems. Automated switching systems were similar to general purpose computers, but without programmability.

[00:07:12] Shawn O’Malley: Human computers, as they were called, were people, often women, who used and operated these machines to find mathematical solutions via carefully crafted procedures, what we call programming today. Building on his understanding of cryptography, Shannon argued in his paper on information theory that human language was largely redundant and much of it could be condensed down into simpler systems. Evidently, he was right.

[00:07:33] Shawn O’Malley: Nothing is simpler than ones and zeros. Reaction to Shannon’s work was immediate and far reaching. Wireline and wireless communication would be revolutionized by information theory, and other industries, such as digital storage, would primarily be based on it. With information theory, we could move from the world of human and mechanical computers tirelessly laboring for days to solve a single problem, to the electronic computers of today that can solve complex problems in a fraction of a second.

[00:08:01] Shawn O’Malley: While information theory made Shannon an A list celebrity in academic circles, that fame hardly translated into daily life where he remained in obscurity. From what little I know of information theory, I know I’m absolutely not doing it any justice here. I’m greatly oversimplifying, but if you do care to go deeper into Shannon’s work, I’ll provide some links in the show notes.

[00:08:20] Shawn O’Malley: But that’s enough on Shannon for now. Let’s turn to one of the story’s other protagonists, and one of my favorite investors to study, Ed Thorpe. As they say, a child’s delayed development of speech can signal a giftedness for mathematical thinking. That proved true for Thorpe, who would not utter his first word until he was three years old.

[00:08:37] Shawn O’Malley: Six months from that first word, Thorpe could reportedly carry a conversation almost like an adult, could count to one million, and had a virtually photographic memory. A few years later, he’d win free ice cream by betting a grocer that he could tally customers’ bills faster than a calculator, and he won.

[00:08:53] Shawn O’Malley: Thorpe, like Shannon, possessed an exceptional mind with an IQ estimated between 170 and 200, and the world is quite fortunate to have Thorpe still walking among us at 92 years old. Thorpe was, for a time, transfixed on beating blackjack. Although it was once considered a woman’s game to distract them while their husbands played craps, Thorpe tried to break down the game’s probabilities, which was something nearly impossible before the rise of modern computers, given the vast amount of possibilities.

[00:09:18] Shawn O’Malley: Thorpe realized that analyses of the odds in blackjack usually made the simple mistake of assuming that the odds of drawing a given card were the same in every deal. Yet, if a dealer pulls three aces and has enough remaining cards to not need to reshuffle the deck, the next deal has only one possible ace.

[00:09:34] Shawn O’Malley: Since aces give players an advantage over the house, then fewer available aces would skew the odds against Thorpe, signaling to him when not to bet aggressively. During a summer break in 1959 at MIT, where he served as a mathematics professor, Thorpe taught himself the early programming language Fortran and programmed MIT’s mainframe computer to assist with his calculations of the odds in blackjack.

[00:09:55] Shawn O’Malley: One of his key findings was that five cards made a bigger difference in favor of the house than any other rank, since they were disadvantageous to players. By simply keeping track of how many five cards have been played, he could dramatically tilt the odds in his favor while playing. Thorpe hoped to publish this finding in academic journals, yet the only member of the National Academy of Sciences who was a mathematician that could submit the paper was none other than Claude Shannon.

[00:10:19] Shawn O’Malley: Thorpe’s goal of beating the odds in blackjack is what would unify these two brilliant minds and Thorpe’s 1962 book, Beat the Dealer, would actually spur casinos to rewrite the rules of blackjack so the house once again had an advantage. But what would really kindle the fire between them was another idea from Thorpe, one premised on predicting how balls might fall in roulette.

[00:10:38] Shawn O’Malley: Thorpe had previously fantasized about building a machine that could predict where a ball might fall on the roulette wheel once it was spun, since casinos continued taking bets for several seconds after a ball had dropped. The two would design something of a wearable computer that could calculate in real time where approximately the ball would land.

[00:10:54] Shawn O’Malley: This was nothing like the wearable computers produced today by Apple and Meta, but it was quite effective at beating the odds in roulette. It would be worn with one wire running into the shoe to track timing of the roulette wheel, while another would run up to an earpiece for receiving the audio based results, with the computer itself simply strapped around the waist.

[00:11:11] Shawn O’Malley: During their tests, Thorpe and Shannon found that the computer gave them a 44 percent edge in roulette, more than enough to make it worth their while. Roughly the size of a pack of cigarettes, the computer itself had 12 transistors that allowed its wearer to time the revolutions of the ball on a roulette wheel and determine where it would end up.

[00:11:29] Shawn O’Malley: Wires led down from the computer to switches in the toes of each shoe, which let the wearer covertly start timing the ball as it passed a reference mark. Another set of wires led up to an earpiece that provided audible output in the form of musical cues. Eight different tones represented octants on the roulette wheel.

[00:11:45] Shawn O’Malley: When everything was in sync, the last tone heard indicated where the person at the table should place their bet. But having an edge doesn’t translate to optimal success. You must know how to balance risk and reward with each bet to achieve the maximum return on your money. Shannon and Thorpe had cracked the code on putting the odds in their favor in both blackjack and roulette, Yet you can still lose money with a modest statistical advantage.

[00:12:08] Shawn O’Malley: In other words, you might be able to theoretically win over time with the odds in your favor, but in practice, if you wipe out your total wealth, you’re done. You have nothing left to bet. Even a great gambler’s lifetime earnings will swing wildly, which is undesirable. The question, then, is knowing when to bet and how much to bet, given the probabilities at hand.

[00:12:26] Shawn O’Malley: That way, catastrophic losses could be avoided while preserving a steadier uptrend in accumulated earnings. Of course, before we can answer that, we need to introduce one other protagonist into today’s story. That is John Kelly, namesake of the Kelly formula, or the formula for earning great fortunes as some say.

[00:12:43] Shawn O’Malley: Born to a petroleum boomtown in 1923, Kelly spent four years flying for the Naval Air Force during World War II, to then do his undergraduate and graduate work at the University of Texas, where he wrote his thesis on the variation of elastic wave velocity with water content in sedimentary rocks, which he hoped would prove useful for the oil industry.

[00:13:02] Shawn O’Malley: This and his later PhD on a similarly esoteric topic Earned Kelly the attention of Bell Labs, and he got a job there. There, colleagues would rate him as the smartest person at Bell Labs, rivaled only by Claude Shannon. Kelly sought an answer to the question that Shannon and Thor bumped up against in Blackjack and Roulette.

[00:13:20] Shawn O’Malley: That is, what is the optimal amount to bet in a given situation? Betting everything, for example, is not a viable strategy. Even with an insider tip on a horse race, there is still some uncertainty, and if you bet everything you have on every inside tip, eventually you’ll be wiped out. But not betting enough can mean letting an opportunity slip out of your hands.

[00:13:38] Shawn O’Malley: A bettor wants to make the most of their insights and compound their returns, in much the same way an investor would think about it. Success should not be measured in dollars, then, but in percentage gains. With Kelly’s formula, it is possible to compound returns at a desirable rate without going broke.

[00:13:52] Shawn O’Malley: It’s truly a have your cake and eat it too sort of thing. To devise this formula for maximizing returns in betting, Kelly studied parimutuel betting at horse races in the US and Japan. Bettors would often set the odds themselves here by adding up a win wager for every race, deducting a track take for expenses and taxes, and distributing the remaining payout.

[00:14:12] Shawn O’Malley: Here’s an example of how the odds and payouts might work, excluding the track’s take. If one sixth of all the money bet in a horse race went to a single horse called Smarty Jones, then all those who bet on Smarty would earn a six times payout from their wagers, as the winnings would only be split among those who bet on Smarty.

[00:14:28] Shawn O’Malley: In odds terms, this would usually be expressed by saying that Smarty is paying out five to one. If you bet ten dollars on Smarty, you’d win fifty dollars, plus the original ten you bet, for a total of sixty dollars. One simple approach Kelly devised was to take the pot of money you plan to bet and split it between every horse in a race.

[00:14:46] Shawn O’Malley: This essentially hedges both your downside and upside because you’re guaranteed to win but are also guaranteed to take losses from the losing horses. In practice, this approach doesn’t work because of the track’s take. That is, the house takes a cut of every bet, a feat of play, that eats away at your cash pile even if you’re winning back some of your bets.

[00:15:04] Shawn O’Malley: On the other extreme, you could imagine betting everything on a single horse if you somehow had a guarantee that that horse would win. If you were 100 percent sure a horse would win, it would be foolish not to bet anything but 100 percent of your money on that horse, which is a strategy with zero diversification.

[00:15:20] Shawn O’Malley: Do you see any resemblance to investing yet? You don’t know what you’re doing, going as diversified as possible and reducing fees to middlemen is your best bet. And as your confidence grows, the more appropriate it becomes for you to take a less diversified approach. There are no guaranteed bets and markets or gambling, but the amount you should bet mirrors your conviction.

[00:15:38] Shawn O’Malley: You’re 90 percent sure a horse will win. Thanks to some insider tip. And you should bet 90 percent of your bankroll on that horse and split the remaining 10 percent on all the others as diversification. We see this all the time with great investors, Warren Buffett, Charlie Munger, and many others are well known for taking concentrated bets on companies.

[00:15:55] Shawn O’Malley: They might only have 5, 10, or 20 companies in their entire portfolio, but that’s not as risky as it seems because their conviction in those bets is much higher. They may not have insider info, but they may have superior analytical skills giving them such conviction. Obviously, nothing in the real world is a sure thing, it is up to you to assess the odds of a bet or an investment and act accordingly.

[00:16:16] Shawn O’Malley: This led Kelly to devise the Kelly formula, or Fortune’s formula, per the title of the book. John Kelly’s formula says to wager X percent of your bankroll on a favorable bet, edge over 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.

[00:16:34] Shawn O’Malley: It’s a fraction because the profit is always in proportion to how much you wager. Odds measures the profits if you win, but odds are set by market forces, what everyone else believes is likely to happen. That’s true in both financial markets and in gambling markets. These beliefs may often be wrong, and in fact, for a better following the Kelly formula, they have to be wrong, otherwise the formula won’t work.

[00:16:58] Shawn O’Malley: That is to say, the Kelly formula is an approach based on the fundamental belief that the odds for something are mispriced. If you’re a subscriber to the Efficient Markets Hypothesis, and the idea that market prices perfectly reflect all available information, then you have no use for the Kelly Formula.

[00:17:13] Shawn O’Malley: The Kelly Formula is for those who know the true odds better than anyone else. In gambling, maybe it’s because you have an inside tip on a horse, or maybe because you have a wearable computer telling you where the roulette ball will fall. In investing, you might have a superior understanding of a company compared to the broader market.

[00:17:28] Shawn O’Malley: Maybe you used to work there, or you have firsthand experience with their products, or maybe you’ve got the pieces put together to understand the stock in a unique way. However you come to it, you have a superior assessment of the true pros and cons of an investment. For example, even something as simple as eating a chipotle every day might reveal to you something not understood by the broader market that changes the odds that the company will continue to hit its earnings targets, like noticing that their portion sizes are shrinking, which is likely to spur a customer revolt.

[00:17:53] Shawn O’Malley: It’s a pretty simple hypothetical, but I actually saw a report from an investment bank recently that compared portion sizes at dozens of different chipotles as part of their research process, though it’s not as silly as it may sound. To be clear, I do not think eating at Chipotle every day will give you an inside scoop that allows you to earn money investing with the Kelly formula, but the point remains that the Kelly formula depends on you having a different and more accurate assessment of the odds than others, and insights for that may come in a variety of ways.

[00:18:19] Shawn O’Malley: Let’s go through an example of the Kelly formula to understand it in action. Imagine that the odds for Secretariat and a horse race are 5 to 1, or just 5 to use betting parlance. Based on your study of Secretariat’s habits and how he performs in certain weather, you determine that with today’s weather, he has closer to a 1 in 3 chance of winning.

[00:18:37] Shawn O’Malley: By betting on Secretariat with 100, you have a 1 third chance to make 600, including the amount you bet. On average, that advantage is worth $200 or one third times a $600 payoff. The average result of $200 gives you an expected profit of $100. That is the expected result of $200 minus the $100 you wager to play.

[00:18:59] Shawn O’Malley: Your edge comes up to one dividing from the expected $100 profit by the $100 wager, and we’d plug in the number one into the numerator of the Kelly formula and the denominator you divide by the odds, which are five for a secretariat. The Cali formula then comes out to 1 5th. That means that you should bet one fifth of your bankroll on Secretariat.

[00:19:19] Shawn O’Malley: I hope that wasn’t too confusing to listen to, and I’ll provide some links to follow along at home for using the Kelly formula. To recap, the Kelly formula is a strategy for maximizing expected returns while minimizing losses over time. It relies on you knowing the odds being offered in the market for something to happen, and your own, more accurate assessment of the real odds.

[00:19:38] Shawn O’Malley: With that, the formula helps you determine the appropriate amount for your bankroll to bet on a given wager. If you do not have any superior insights into the odds that a given horse will win, or the odds that a given stock will compound returns at rates beyond those already expected by the market, then your edge is zero or negative.

[00:19:55] Shawn O’Malley: In that case, the Kelly formula would tell you to bet nothing. Concentrated bets are not appropriate when you don’t know anything that everyone else doesn’t already know. As we’ve discussed a bit, the real limiting factor with the Kelly formula is that bettors rarely have a special advantage. You realize, though, that there’s at least one place you could always make favorable bets.

[00:20:13] Shawn O’Malley: The stock market. In the stock market, risk seeking investors are rewarded for embracing higher uncertainty. On long time periods, government bond investors do not earn the highest returns because they’re not taking on as much risk. Historically, stocks, which are riskier than bonds in aggregate, generate higher average returns over almost every decade plus timeline.

[00:20:33] Shawn O’Malley: The Kelly formula requires that profits are able to be reinvested and that bets can be sized and channeled in different categories at the bettor’s discretion, which are both characteristics that gambling markets and financial markets have in common, enabling the Kelly formula to be used interchangeably between them.

[00:20:48] Shawn O’Malley: Thank you. I’ll just pause for a moment here to clarify that although I’m using the word bet in both the context of gambling and investing, I’m not exactly talking about day trading. I know some investors take offense at the word bet since it implies that they’re reckless gamblers, but I’m really intending to use the word bet in the broadest possible sense to reflect a decision made with imperfect information.

[00:21:07] Shawn O’Malley: And as I outlined with the Chipotle example, an investor’s edge when using the Kelly formula doesn’t necessarily have to stem from illegal insider information. There is a certain ethical ambiguity, though, to the origins of the Kelly formula. Kelly often used the example of rigged horse races where a better exploited an unfair advantage over others to show what an edge may look like.

[00:21:26] Shawn O’Malley: The story of the Kelly formula is a story of secrets, one where bettors and investors alike stop betting when their secret advantage is gone. It’s up to the user’s interpretation to determine what constitutes an advantage, but I’m in no way encouraging anyone to trade on illegal insider information, which would be the most obvious stock market parallel to a rigged horse race.

[00:21:46] Shawn O’Malley: Going back to the story in January 1961, Ed Thorpe went before the American Mathematical Society to present a version of the paper submitted previously by Kelly and Shannon. A reporter at the event wrote a story about the so called fortunes formula and its potential in blackjack, spurring a nationwide frenzy among bettors, hoping to get in touch with Thorpe.

[00:22:04] Shawn O’Malley: Thorpe got more outreach sent to him in response to the paper than every other mathematician at MIT had for their past published papers combined. In all, he received thousands of letters. Thorpe was keen to test out his approach to blackjack in a real casino, and Shannon suggested to him that he try using Kelly’s formula since it would reduce the risk of loss and tell him exactly how much to bet depending on how favorable the deck was.

[00:22:27] Shawn O’Malley: A wealthy New Yorker offered to fund Thorpe’s endeavor, giving him 100, 000 to take on the Vegas casinos in exchange for a 90 percent share of the gambling profits that Thorpe earned. Blackjack is a game of mostly even money bets, and only occasionally do opportunities arise that give the better an edge that would encourage them to bet following the Kelly formula.

[00:22:47] Shawn O’Malley: The problem and practice was that carefully watching the game while only rarely betting would too obviously raise suspicion. Thorpe had to adjust his approach to account for making a number of low stakes losing bets, which would help him avoid allegations of card counting. After about 30 hours of play, Thorpe turned 10, 000 into 21, 000, but he was kicked out of a number of casinos along the way.

[00:23:08] Shawn O’Malley: No one could quite figure out what he was doing, but they all knew he was winning more than he should, and casinos don’t exactly owe you an explanation, they can just kick you out at their discretion. The beauty of using the Kelly formula is that even if Thorpe had an unlucky streak of bets at the casino, the formula minimizes his potential ruin.

[00:23:25] Shawn O’Malley: Since it prescribes bets as a proportion of your current bankroll, the nominal size of your bets shrinks as you incur losses, which are still inevitable even with the Kelly formula. But you can never run out of money following the Kelly formula because every recommended bet scales with the size of your bankroll.

[00:23:40] Shawn O’Malley: The only real world limitation is if your bankroll becomes too small, the dollar amount that the Kelly formula tells you to bet may not meet minimum wager requirements. And assuming you have an edge in the long run, you’ll compound your winnings and grow your bankroll or portfolio size. Whereas the Cali formula minimizes losses over time, you can imagine an alternative bettor who doubles down after winning and likes to make outsized bets with all or most of their bankroll.

[00:24:06] Shawn O’Malley: While this bettor can accumulate wealth faster with large bets, they will inevitably get wiped out. Running a simulation a thousand times, the all or nothing better has considerably worse outcomes on average because even just a few incidents of being wiped out skews the expected results dramatically downward.

[00:24:22] Shawn O’Malley: Making proportional bets as a percentage of net worth is not unique to Kelly’s formula though. So what then makes the Kelly formula so special? The simple answer is that after rigorously testing as many conceivable approaches to betting as possible, the Kelly formula delivers the best results over time by far.

[00:24:38] Shawn O’Malley: You don’t need to really deeply understand the math to recognize that three of the smartest mathematical thinkers of the 20th century landed on the Kelly formula as the optimal approach to sizing bets, depending on how much edge you have working in your favor. In a way, you can think of this formula as an optimal way for allocating capital, uncovered by some of the most brilliant minds to ever live.

[00:24:57] Shawn O’Malley: Kelly formula strikes the statistically proven ideal balance between making undersized and oversized bets. Over time, if you’re not making large enough bets based on the odds and edge at hand, you’re inefficiently allocating capital and will earn worse returns. And as I suggested a minute ago, oversized bets are the fastest way to wipe out your bankroll over time.

[00:25:17] Shawn O’Malley: As Thorpe went around racking up winnings at casinos using the Kelly formula and his approach to card counting, the casino industry actively conspired against him, employing dealers who would try to cheat him, especially after he published his book Beat the Dealer, which they saw as offering anyone in the public a playbook for taking money out of their pocket.

[00:25:33] Shawn O’Malley: Thorpe had become something of an existential threat to casinos, exposing the vulnerabilities of one of their most popular games. Shortly thereafter, the Vegas Resort and Hotel Association announced new rules for Blackjack, making the game harder for card counters adhering to the Kelly formula. Turning to financial markets, I want to outline how stock prices should work in theory.

[00:25:53] Shawn O’Malley: Changes in stock prices should essentially be completely random, based on whether incremental news about the company improves upon current expectations subtracts from them. Fluctuations, in one way or another, are responses to new information relative to what is already priced in, at least if you assume markets are generally efficient.

[00:26:10] Shawn O’Malley: Assuming a stock accurately reflects expectations for a company’s performance, its future stock price performance is inherently unpredictable because only new information deviating from those expectations would materially move prices. Companies that somehow deliver results perfectly in line with market expectations should not swing meaningfully up or down.

[00:26:27] Shawn O’Malley: It is only results above or below expectations that swing the stock. A stock then is subjected to an unknowable and constant stream of new information about both the company and the environment it operates in, which will revise expectations upward or downwards. This implies that someone who buys a stock and sells it immediately is as likely to have a loss as a gain.

[00:26:47] Shawn O’Malley: A speculator’s edge is therefore zero. But the question to next consider is to what extent do stocks actually reflect available information? You might think of the question as being similar to asking whether the earth is round. If you’re asking roughly whether the earth is flat or round, the simple answer is yes, it is.

[00:27:03] Shawn O’Malley: And markets roughly do a good job at accounting for available information and then reflecting that information into prices in real time. Yet, if you’re asking a more nuanced question about whether the Earth is precisely round, the answer is no, because it’s not a perfect sphere. The same is true with markets.

[00:27:19] Shawn O’Malley: It remains hotly debated, but we know there are real limits to how efficiently markets reflect available information. The original academics who hypothesized the Efficient Markets Hypothesis, as it’s known, outlined three different degrees of market efficiency. The weakest form of efficiency does not perfectly reflect available information about a company’s fundamentals, but says that no one can gain an edge by simply evaluating stock charts.

[00:27:40] Shawn O’Malley: That is to say, technical analysis of price data should provide no advantage. The stronger form of efficiency adds in publicly available information about a company’s operations and finances, such that no one can gain an edge by digging through financial statements to predict future results. And then the most extreme form of market deficiency that is far less commonly accepted is the idea that insider information is priced into stocks too.

[00:28:03] Shawn O’Malley: That would mean, for example, that if an insider tried to sell shares because they knew the stock would miss earnings, they would receive no benefit from doing so because that information would have already been leaked to the broader market and priced in. There have been some examples of this where insiders didn’t benefit as expected from inside information, but I don’t think most believe that this is broadly true.

[00:28:22] Shawn O’Malley: In thinking about gaining an edge in gambling as well as in the market, Claude Shannon ironically had a famed indifference to money. He saw himself as an academic devoted to uncovering the truth, who disdained the blinding and corrupting effects that money could have. For years, he kept his money only in a checking account, earning no interest, and he refused to invest in stocks or bonds.

[00:28:41] Shawn O’Malley: But when one of Shannon’s friends started the company Harrison Labs, Shannon invested. The stock would go on a surge and be acquired by Hewlett Packard, leaving Shannon with exceptional gains. In the 1950s, Shannon began an intensive study of the stock market out of intellectual curiosity and a newfound interest in personal profit.

[00:28:59] Shawn O’Malley: Shannon was, of course, not the first great mathematical mind to think that they could extend their talents to success in the stock market. Top of mind for Shannon was determining just how efficient the stock market is and to what extent he could leverage betting formulas like the Kelly formula to help him exploit inefficiencies.

[00:29:14] Shawn O’Malley: While Shannon alongside Thorpe and Kelly had put considerable thought into applying the Kelly formula to gambling, horse races in Blackjack are not quite the same as the stock market. Horse races have defined outcomes, a horse can either win or lose. The range of outcomes for a stock, though, is in a way limitless, it can rise by any amount or fall to zero.

[00:29:34] Shawn O’Malley: Horse races also occur over defined periods, they have a beginning and an end. Stocks trade for 6. 5 hours a day, 5 days a week, and they do so indefinitely. Unless a company delists, its stock could trade forever. Investors can also stay invested for as long or as short as they’d like. There is a time element to investing that doesn’t exist in the same way when gambling.

[00:29:54] Shawn O’Malley: Yet Shannon remained confident in rejecting the efficient market’s hypothesis and in using edges in investing to earn abnormally large returns without taking on additional risk. Meanwhile, Ed Thorpe was also turning his interest to the stock market after deciding that he might risk bodily harm if he pushed his luck any further gambling at casinos.

[00:30:13] Shawn O’Malley: Thorpe was particularly interested in stock options. After just an hour of researching them, he had devised a model for pricing options. Stock options gave the option to buy or sell stock at a certain price. For example, a stock option might enable you to purchase shares in a company at 25 per share. If the stock’s price goes to 29, the option must be worth at least 4 because it enables you to buy shares at a 4 discount to current prices.

[00:30:38] Shawn O’Malley: But what is this option worth if the stock is trading below 25? It must be worth something, because until the option expires, it remains possible that the stock’s price will go above 25 and leave you with a profitable opportunity to buy discounted shares. And options pricing, then, essentially reflects the consensus odds that the option will in fact be profitable at some point, as well as a bet on how profitable it will be.

[00:31:00] Shawn O’Malley: As an option’s price falls The market is implying that it’s less likely that the stock’s price will exceed 25 before the option expires. While he realized that there were too many random variables to try and predict that might fundamentally happen to a company over a short period of time, he could make bets based on probability distributions.

[00:31:18] Shawn O’Malley: Stocks with higher average daily price fluctuations tend to have a wider range of possible future prices. Knowing that, Thorpe found that most options were priced too expensively. Based on probability distributions, they didn’t offer sufficiently high enough odds to be profitable to justify their price.

[00:31:33] Shawn O’Malley: He compared the options market at the time to a carnival game where many participants were simply hoping to win. While he didn’t want to buy those options, he could short them. Shorting expensive options slipped the odds in his favor. He could profitably collect premiums from selling options that did not accurately reflect the risks that he was incurring.

[00:31:50] Shawn O’Malley: Like the house in a casino, the odds would be in his favor and he had an edge to exploit by more accurately pricing options than the rest of the market. Similar to how you might bet on all the horses in a race to win and exploit mispriced odds, Thorpe hedged his options positions by purchasing the company’s stock.

[00:32:06] Shawn O’Malley: That way, if the stock did surge, the gains he earned from owning the stock would offset the losses he took from selling options against them. The practice of long short hedging certainly predates Thorpe and the Kelly formula, but Thorpe’s advantage was in using the Kelly formula and his calculations of the true odds for options pricing to determine the optimal amount of stock he needed to go long to mitigate the risks he took from shorting the stock’s options.

[00:32:28] Shawn O’Malley: This practice is known as Delta Hedging, and you profit from the irrational price of the option coming into line with the stock’s pricing. The beauty of options is that they offer an explicit date for price reconciliation. A company’s stock may be irrationally priced for long periods of time, but the option will either be worthless or profitable by its expiration date.

[00:32:47] Shawn O’Malley: Thorpe continued to refine his system and grew his 40, 000 portfolio into over 100, 000, but his approach wasn’t bulletproof. Many options traded in illiquid markets, and he could move those markets with just his own trading. It also required constant attention to ensure that, as stock and option prices fluctuated, he maintained the proper degree of hedging.

[00:33:07] Shawn O’Malley: With this hedging, his portfolio could be market neutral, meaning he could still profit regardless of what was happening in the broader market. A down year for stocks may still be very profitable for him. While many of his bets weren’t profitable, he was able to use the Kelly formula to ensure he wasn’t betting too big, so his losses were minimized.

[00:33:25] Shawn O’Malley: Thorpe was earning returns of 30 50 percent per year, and he felt he had so many ideas for how to continue generating those returns, he thought he could give away some of his secret sauce, as he did with the book Beat the Dealer. His next book was called Beating the Market, and he hoped that by publishing it, it would enable him to raise more money for his hedge fund.

[00:33:42] Shawn O’Malley: It worked. By November 1969, Thorpe’s hedge fund was in business. Taking his learnings about options, Thorpe fixated on valuing convertible bonds, which are essentially regular corporate bonds with an added option to convert the bonds into shares of stock. While there wasn’t really a universal formula for doing so at the time, Thorpe had cracked the code in 1967.

[00:34:02] Shawn O’Malley: He used a version of an approach that is now mainstream in options pricing, the Plaque Scholes model. Valuing options, whether on their own or connected to a convertible bond, then depended on the strike price to execute the option, the stock’s current price, the time until expiration, and the amount of expected fluctuation in the stock’s price, known as volatility.

[00:34:20] Shawn O’Malley: He gained an edge by finding mispriced convertible bonds, betting on or against them, and using the underlying stock to hedge his bets. In its first full year, even after extracting hefty fees for the fund, Thorpe delivered a return that beat the S& P 500 by 10 percentage points and would double the S& P’s performance over the following year.

[00:34:40] Shawn O’Malley: Most of the time, his bets were well below the prescribed Kelly limit, but that was all he could spend. There are only so many options in convertible bonds trading, so there’s a hard limit on how big those bets can be. On occasion though, he did identify opportunities that were as close to a sure thing as possible, and he bet more than 30 percent of the total portfolio on them, aligning with the Kelly’s formula outline to bet more when you have a greater edge and expected profit.

[00:35:05] Shawn O’Malley: Eventually, as the Black Scholes pricing model for options became more widely understood, options markets and convertible bond markets became more efficiently priced, and Thorpe had to look elsewhere for an edge. By the mid-1970s, Thorpe was generating solidly positive returns while the stock market averages fell, and that success ballooned his assets under management to more than 20 million.

[00:35:24] Shawn O’Malley: In one example, in 1974, Thorpe purchased AMC’s convertible bonds with a 1, 000 face value which had fallen to a price of 600. Convertible bondholders maintained the right to convert their bonds into 100 shares of stock, and with the stock selling at 6 per share, the convertible bonds were effectively selling at exactly the same price as the stock, yet the bonds paid 5 percent interest while the stock paid no dividends.

[00:35:48] Shawn O’Malley: Owning the bond had all the upside of owning the stock, with a cushion from the interest paid and bankruptcy protection since bondholders get repaid in bankruptcy before stockholders. There was seemingly nothing that could go wrong. If the stock went up, he could convert the bond into shares of stock, and if it didn’t, he could collect interest payments and would eventually get repaid.

[00:36:08] Shawn O’Malley: To hedge his position in case the company went bankrupt, he shorted the stock after having bought its convertible bonds. So if the company went under, he would profit from his short bet while still receiving at least a partial payout on his bonds via bankruptcy distributions to creditors. Because the trade was a sure thing with no risk of ruin, the Kelly formula permitted him to leverage his bet with borrowed money, which is exactly what he did.

[00:36:31] Shawn O’Malley: It was a huge winner, and while Thorpe has said these types of situations were few and far between, they made their living off of them. While academics might argue that these inefficiencies in markets should not be able to persist for very long as arbitrageurs leap in to make easy profits, Thorpe found that these too good to be true opportunities were far more persistent.

[00:36:51] Shawn O’Malley: Limits to arbitrage, such as transaction costs, liquidity, and available funds for exploiting an arbitrage contributed to mispricings in markets that could last for days or weeks. What’s interesting to me is that while reading this book, or really any accounting of what great investors have done in the past, it’s really easy to chalk it up to a bygone era when it was simply easier to exploit opportunities.

[00:37:13] Shawn O’Malley: People were not dumber back then though, we can’t just write off opportunities of the past as having only existed because of collective incompetence or because of technological inferiority. Academics in the 1970s were just as confident then that markets were efficient as they are now, yet in hindsight there were clear opportunities to exploit and people were doing that.

[00:37:32] Shawn O’Malley: Although looking for option market mispricings and arbitrage opportunities in convertible bonds are well documented tactics on Wall Street now, they were novel for a time. The point being, I don’t think it’s accurate to conclude that it’s impossible to gain an edge in markets today just because strategies that worked in the past look obvious now.

[00:37:49] Shawn O’Malley: In Thorpe’s era, computers were new, and few people had access to them at scale and knew how to program them. His familiarity with them, investment in them to maximize his competing power on hand, and mathematical background enabled him to do things that were at the forefront of his era. He found a unique way to garner an edge.

[00:38:07] Shawn O’Malley: So no, you can’t look at what he did in the 1970s and try to do the same thing yourself, because that’s no longer a fresh strategy. There is no edge to be had anymore. But that doesn’t mean that there aren’t opportunities being overlooked in markets. It just means that to find them, you need to be like Thorpe.

[00:38:21] Shawn O’Malley: You must dare to venture where no one else has looked before, or dig to uncover realities that are, for some reason, not being properly accounted for. In other words, markets are efficient in ways they weren’t decades ago, thanks to the pioneering accomplishments of people like Ed Thorpe, but that doesn’t mean markets are perfectly efficient now.

[00:38:38] Shawn O’Malley: If you intend to do anything but invest in passive index funds, you must honestly look yourself in the mirror and ask what your edge is. What insight or information do you have that is not understood by the millions of investors who pour over the same economic and financial data? I’ll also add that when it comes to the Kelly formula, it is not without at least some controversy.

[00:38:57] Shawn O’Malley: The drama is a bit academic, yet not everyone agrees that the Kelly formula is the optimal approach to investing. Yes, it generates the best returns over time, but it also endorses arguably aggressive investment strategies and constant reinvestment. While it prevents ruin in the long run, it can still incur significant losses over shorter periods of time, and when we’re talking about people’s life savings on the line, they may not be willing to tolerate those swings.

[00:39:20] Shawn O’Malley: In the long run, we’re all dead, and you might not be able to wait forever for theoretically optimal investment results. It’s sort of funny to say, but in some cases, it’s really not always about maximizing your theoretically possible wealth at all costs. If you don’t want to reinvest all available funds or can’t afford to wait decades, the Kelly formula isn’t necessarily the best model to adhere to.

[00:39:40] Shawn O’Malley: More conservative approaches to match your lifestyle are of course justifiable, but there was some concern when financial advisors first began modeling their guidance around approaches similar to the Kelly formula that people may adopt overly aggressive investment strategies. There was a wave of academic backlash against the Kelly formula in the 1970s for this reason, as they felt it gave the false promise of universally helpful financial advice that didn’t actually cater to people’s lifestyle goals and risk tolerance.

[00:40:06] Shawn O’Malley: As an alternative, the half Kelly formula has become popular in certain circles as a way to tame volatility, where you bet half of the prescribed amount by the Kelly formula each time. It’s appealing because in probability models, this approach cuts volatility drastically while only reducing returns by a quarter.

[00:40:22] Shawn O’Malley: While there is some reason to consider making bets that are a fraction of what the Kelly formula advises to limit volatility in the intermediate term, betting more than what is recommended by the Kelly formula is a fast track to ruin. Because returns swing so wildly when sizing positions in excess of what the Kelly formula would recommend, over the long term, a better betting more than the formula recommends almost always gets wiped out at some point.

[00:40:44] Shawn O’Malley: Turning back to Thorpe’s hedge fund, his success seemed to defy gravity, and he became incredibly rich. At the peak of his wealth, he purchased a 10 bedroom house with a fallout shelter. It was apparently fortified enough to withstand a nuclear bomb that could explode less than a mile away. Thorpe’s success wouldn’t last forever, though.

[00:41:02] Shawn O’Malley: But that wasn’t because of his failures as an investor or shortcomings in the Kelly formula. Instead, it was because of the legally dubious entanglings of his partner. While Thorpe devised the strategies for the hedge fund from his comfortable home in California, his business partner made relationships across Wall Street at a time when the mob and other criminal elements were still quite active in New York City.

[00:41:22] Shawn O’Malley: A crackdown on organized crime and its linkages left Thorpe bewildered when attention turned to his firm thanks to his partner’s shady connections. While Thorpe himself didn’t face charges and is thought to have been telling the truth when saying he wasn’t aware of the sort of illicit deals being made by his partner, Thorpe still chose to dissolve his hedge fund in 1988, disrupting one of the most impressive careers in investment history.

[00:41:44] Shawn O’Malley: Thorpe’s track record from 1969 to 1988 was truly breathtaking. Over that period, a dollar invested with him would have grown to almost 15. That is a 15. 1 percent compound annual growth rate after fees crushing the S and P 500 8. 8 percent return over the same period. Only legends like George Soros and Warren Buffett have topped that sort of outperformance on a multi decade timeline.

[00:42:08] Shawn O’Malley: Yet, Thorpe’s risk adjusted returns, as they say, were unrivaled. While we’ve talked in previous episodes about how price volatility is an inadequate measure of risk, there’s still something to be said about generating excellent returns while also avoiding stress inducing price swings. Thorpe’s returns were remarkably steady, far less volatile than Soros or Buffett, and even less volatile than market averages.

[00:42:30] Shawn O’Malley: Adjusting returns for volatility incurred is the standard for determining truly exceptional returns in the hedge fund world, and by that measure, Thorpe was in a league of his own. Volatility is often said to be the price paid for earning the 8 10 percent returns that the stock market is known to generate over time, but it is certainly better to earn those returns without volatility if possible.

[00:42:50] Shawn O’Malley: It’s just that doing so has not really been possible for many besides Thorpe. Thorpe would go on to launch another hedge fund in 1994, where he improved his track record with even better annual returns. In one note, Thorpe argued that, in case anyone thought his success was due to luck, the sheer frequency of bets he made should be enough to reasonably prove his skill.

[00:43:09] Shawn O’Malley: Over his career, he estimated that he made more than 80 billion worth of trades, broken into 250, 000 different bets. With, on average, hundreds of positions in place at any one time. Like Thorpe, Shannon also relied on the Kelly formula and had a stellar track record. In 1986, Shannon’s returns had outpaced almost every major money manager except for three, despite running an operation of just himself and his wife, while the others sometimes employed hundreds of people.

[00:43:35] Shawn O’Malley: From the late 1950s to 1986, his annual returns were in the neighborhood of 28%, rivaling Warren Buffett’s record over a similar period. Unlike Thorpe, though, Shannon was more of a buy and hold investor, seeking to extract signal from noise. He would evaluate companies management and products, and then derive projections of their earnings performance in the coming years.

[00:43:55] Shawn O’Malley: He recognized that in the long run, stock prices must follow earnings growth and felt most technical analysis of price charts was a very noisy reproduction of the most important underlying data. Shannon embraced the idea that what matters is not how a stock has done in the last few days or weeks, but how its earnings have changed.

[00:44:12] Shawn O’Malley: One of his biggest holdings was actually Teledyne, and Shannon served on the board there, helping to advise Henry Singleton, who was profiled in the book The Outsiders, and who Warren Buffett once said had one of the best track records of capital allocation in US corporate history. At one point, Teledyne made up as much as 80 percent of his portfolio, yet that sort of extreme concentration when he felt he had an edge, since he knew Singleton so well, didn’t faze him.

[00:44:35] Shawn O’Malley: From Teledyne to his earlier picks of leading tech companies, Shannon leveraged his personal relationships, career background at Bell Labs, and other experiences and knowledge to invest where he had an edge. Evidently, he did a phenomenal job, knowing not only what bets to make and where he had an edge, but also how to properly size those bets thanks to the Kelly formula.

[00:44:55] Shawn O’Malley: In conclusion today, I hope you can see the beauty of the Kelly formula and its resistance to wipeouts, which enables it to maximize long term returns. The formula is almost paranoid in a way, because following the Kelly formula, one would be pushed to avoid bets that introduce even microscopic risks of total loss.

[00:45:11] Shawn O’Malley: It is an acknowledgement that infrequent but severe losses are inevitable, and that the only way to win long term is to avoid any possibility of having your wealth reset to zero. In 2003, the legendary investor Bill Miller wrote that the Kelly formula was critical to his decision making process, yet he thought few of his colleagues in the investment management business used it because it didn’t stem from more mainstream academic sources.

[00:45:32] Shawn O’Malley: This Surprisingly, it is still a somewhat fringe topic. I’m not going to say most professional investors aren’t at least familiar with it, but I doubt that most use it in their daily decision making process. As Poundstone writes in the book, this is a story that has gone everywhere, except toward a clear ending.

[00:45:48] Shawn O’Malley: I found it fascinating to dig into the life stories of Shannon, Kelly, and Thorpe, and I couldn’t believe how many different fascinating stories overlapped with the origins of the Kelly formula. From the genesis of AT& T to Time Warner’s connections to the mob, what really caused long term capital management to blow up, and Rudy Giuliani’s crackdown on Wall Street in the 1980s, this book touched on a fascinating array of topics and histories.

[00:46:11] Shawn O’Malley: I’d really encourage you to read it for yourself, since obviously I couldn’t cover every rabbit hole in today’s episode. And as is fundamental to the Kelly formula, I want you to ask what your edge is in investing. I know it’s a question I’ll be reflecting on myself. With that, I’ll leave you with a line from Ed Thorpe himself.

[00:46:28] Shawn O’Malley: Thorpe says, quote, gambling is a tax on ignorance. People often gamble because they think they can win. They’re lucky. They have hunches, that sort of thing, whereas in fact, they’re going to be remorselessly ground down over time. I hope you enjoyed today’s episode and I’ll see you again next time.

[00:46:44] Outro: Thank you for listening to TIP. Make sure to follow Millennial Investing on your favorite podcast app and never miss out on our episodes to access our show notes, transcripts, or courses, go to theinvestorspodcast.com. This show is for entertainment purposes only. Before making any decision, consult a professional. This show is copyrighted by The Investor’s Podcast Network. Written permission must be granted before syndication or rebroadcasting.