Twenty years after his hedge fund went bust, Victor Haghani conducted a study on how finance people manage risk. Together with a colleague, Richard Dewey, he devised a game in which players start with a bankroll of $25 and spend thirty minutes betting on the outcome of a coin flip. To make it interesting, the coin is weighted: 60% of the time it comes up heads, 40% of the time tails. Participants can stake any share of their bankroll on each flip: if it lands on heads, they double their money, if it lands on tails, they lose. All the rules are transparent. Everyone is told the coin is weighted and players are encouraged to strategise accordingly. You can play the game here. 

Haghani pulled together a sample of 61 participants. They comprised the best and the brightest: college age students in economics and finance and young professionals at finance firms. Around a quarter of the group were analyst and associate level employees from two leading asset management firms. 

Plainly, the game offers a clear edge and Haghani assumed that players would exploit it. He calculated in advance that the most agile-fingered would be able to make around 300 bets in the available time. If they played optimally they would be able to take home over $3.2 million, so he prudently capped the maximum payout at $250 and gave players the thumbs-up when they hit it.

Yet few players got close. Only 21% of participants reached the maximum payout, well below the 95% that should have got there if they’d simply bet a constant 10-20% of their bankroll on each flip. More players than that – 28% of the total – went bust, receiving no payout at all. 

Perhaps players got lazy? Among those in the middle – those that neither hit the max nor crashed out – the average ending bankroll was $75. That’s not a bad return: tripling your money in the space of half an hour is not to be sneezed at. But it is suboptimal given the size of the edge available. (And it doesn’t explain why two thirds of the subjects bet on tails at some point during the game; as Haghani and Dewey write, “Betting on tails once or twice could potentially be attributed to curiosity…but 29 players bet on tails more than five times.”)

Given more time to strategise, players may arrive at a few observations. The first is that one way to the big bucks is to go big. Betting everything on heads each time maximises total expected value. A $25 bet on the first flip has a 60% chance of returning $25 and a 40% chance of losing $25: its expected value is $5. The problem is that such a strategy also maximises the chance of going bust. On any given flip, there’s a 40% chance you’re taken out of the game. 

So going all-in is out of the question. A curious player may then think about what happens if they bet half their bankroll on each flip. It turns out they lose a lot of money in that scenario as well. If you earn 50% on heads and lose 50% on tails, then over 100 flips your expected return is 1.5^60 × 0.5^40 which equals 0.033, or just over 3%. You end up losing 97% of your bankroll before the thirty minutes is even out; not much better than losing everything. If such a harsh outcome doesn’t seem especially intuitive, think about the impact of flipping tails straight after heads: you’ve just lost a quarter of your total pot (1.5 × 0.5 = 0.75). 

What’s becoming apparent is that geometric returns aren’t especially intuitive. In financial markets, there’s a quip: What’s the definition of a stock down 90%? It’s a stock down 80% that then halves.

A curious player could keep iterating to arrive at an optimal bet. Fortunately, there’s a formula. In 1956, John Kelly, a researcher working at Bell Labs, got to thinking about how a gambler might make bets if he had control of a private communications channel running between a baseball stadium and his bookmaker. Control of the wire gives the gambler time to learn the result and delay it getting to the bookmaker long enough to place a winning bet. Obviously, if the wire was perfect you would bet everything on the result but what if the wire had an intermittent fault, how much do you then bet? Kelly calculated that if you bet the size of your edge, your bankroll will grow geometrically. He derived a formula that on the flip of a coin, you would bet 2p-1 of your bankroll each time, where p is the probability of winning.¹ We’ve already seen that 50% is too high in this case; the formula reveals the optimal bet to be 20%.²

Haghani was dismayed at how few participants grasped how to play their edge, even at an intuitive level. As well as betting on tails, they’d over-bet, under-bet, and erratically bet. Yet position sizing is a huge determinant of performance in gambling and in financial markets. Victor Haghani of all people knows this. “One of the great lessons from LTCM is about trade sizing,” he told an interviewer. “If we had had a third of the positions that we had, things would have been different.”

Position sizing is often treated as a secondary consideration to identifying a good investment. But in portfolio management it is critical. On the identification side, “you only need to be right 55% of the time,” according to Walleye Capital CEO, Will England. The trick is to size picks appropriately. In his book, 10½ Lessons From Experience, Paul Marshall, founder of hedge fund firm Marshall Wace, describes the cost of poor position sizing. He identifies a contributor to his fund who gets 64% of his picks right yet fails to make money. His edge is 14% but he insists on betting on tails. 

At the time Haghani was conducting his experiment, Sam Bankman-Fried was fully immersed in his career at Jane Street. Bankman-Fried was no stranger to coin-flipping games. According to author Michael Lewis, whose book, Going Infinite, gives us some background, he was subjected to them as part of the selection process for his job. One such game was a more elaborate version of Haghani’s. Bankman-Fried was given ten coins, each of which had a different weighting not revealed to him, and he could choose which to flip. His takeaway was sound risk advice: “He started the game willing never to find the optimal coin so long as he found a good enough one,” writes Lewis.

Somewhere along the way, though, Bankman-Fried’s risk barometer became corrupted. As an intern at Jane Street, he initiated a bet, offering to pay $1 to any of his fellow interns who would flip a coin with him for $98. Lewis explains: “To the Jane Street way of thinking, Sam was offering free money. A Jane Street intern had what amounted to a professional obligation to take any bet with a positive expected value. The coin toss itself was a 50-50 proposition, and so the expected value to the person who accepted Sam’s bet was a dollar: (0.5 × $98) – (0.5 × $98) + $1 = $1.” Matt Levine at Bloomberg Money Stuff has written extensively about this bet. “Huh, that’s aggressive, what are they teaching those interns?” he asks.

But by then, Bankman-Fried was all-in on the idea of expected value. “Every decision Sam made involved an expected value calculation,” writes Lewis. He would continuously estimate probabilities of outcomes (for example, becoming President of the United States: 5%; going to Texas tomorrow: 60%) and act accordingly based on their expected value. The problem is that the framework blinded him (perhaps wilfully) to the risk of going bust. 

“One of the sort of takeaways that often ends up coming from really thinking hard and critically about expected values is that you should go for it way more than is generally understood,” he told interviewer Jacob Goldstein in May 2022. “You should really go really big, even if you probably will fail and wind up with zero?” asked Goldstein. “That’s absolutely right,” responded Bankman-Fried. 

The exchange with Goldstein came some time after Bankman-Fried had proposed his own coin-flipping game on Twitter (as it was then known). Bankman-Fried proposed flipping a coin more heavily weighted than Haghani’s, landing on heads 10% of the time and tails 90% of the time, with a payout that’s also higher. Instead of doubling your money on heads, you win 10,000x your bet; in the event tails lands, you lose just whatever you stake. Your bankroll is $100,000. How much do you bet?

Bankman-Fried knows Kelly’s formula, which suggests a bet size of around $10,000. “But I, personally, would do more,” he tweets. “I’d probably do more like $50k.” In a one-flip game, he might be right. But in a repeated game, where the bankroll compounds, he will fall flat. Nick Maggiulli simulates the results of 100 flips in his post, Where Michael Lewis Went Wrong. Bankman-Fried would end up bankrupt around 25% of the time and although he would 10,000x his bankroll 60% of the time, the lower Kelly stake would get you there almost 100% of the time. 

Bankman-Fried justified his stance on the basis that his “utility function isn’t really logarithmic. It’s closer to linear.” But the coin doesn’t know that. In a series of repeated flips, bankroll grows geometrically, and Kelly’s formula maximises that, as Nick Maggiulli’s simulation illustrates. Kelly himself recognises that the gambler’s own “utility function” has little bearing on the matter: “At every bet [the gambler] maximizes the expected value of the logarithm of his capital. The reason has nothing to do with the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large numbers applies.”

Sadly for many, Bankman-Fried’s miscalibrated risk engine led him astray. Asked in court what Bankman-Fried told her about risk, Caroline Ellison, former colleague, girlfriend and now witness for the prosecution said, “He said it was OK if positive EV, expected value. He said he was willing to take large coin flips – he talked about being willing to flip a coin and destroy the world, as long as a win would make it twice as good.”

In the one world in which we live that’s not a great strategy. Perhaps Bankman-Fried thought he was running his crypto exchange across multiple parallel universes, in which case fine. As Matt Levine points out, that’s kind of the way things work at Jane Street. Each trader operates in their own world, managing their own bankroll. It doesn’t matter to Jane Street if one of them goes bust, there are lots of other traders making different bets.

Indeed, Michael Lewis recounts a story at Jane Street where Bankman-Fried lost $300 million betting on the outcome of the 2016 US presidential election – “the single worst trade in Jane Street history” – and his bosses shrugged it off; the process was good and it had after all been a positive expected value trade. Levine concludes that the lesson traders are taught at Jane Street works fine as part of an ensemble but not when you’re out on your own. Going Infinite is an odd title for Lewis to choose for his book; perhaps it relates to the number of universes Bankman-Fried thinks he operates in. 

Victor Haghani reckons that Bankman-Fried’s risk preferences made it almost certain that he’d go bust, and quickly. For everyone else, his experiment is a good test. “It ought to become part of the basic education of anyone interested in finance or gambling,” says Ed Thorp. Future investors may insist on it...