A Man for All Markets

(Zen Buddha Silence by Marilyn Barbone)

January 10, 2022

A Man for All Markets: From Las Vegas to Wall Street, How I Beat the Dealer and the Market (Random House, 2017) is the autobiography of Edward O. Thorp, a remarkable person.  Here’s the beginning:

Join me in my odyssey through the worlds of science, gambling, and the securities markets.  You will see how I overcame risks and reaped rewards in Las Vegas, Wall Street, and life.  On the way, you will meet interesting people from blackjack card counters to investment experts, from movie starts to Nobel Prize winners.  And you’ll learn about options and other derivatives, hedge funds, and why a simple investment approach beats most investors in the long run, including experts.

The simple approach to which Thorp refers is investing in ultra-low-cost index funds.  Thorp’s view here is similar to Warren Buffett’s:  https://boolefund.com/warren-buffett-jack-bogle/



Even as a young child, Thorp loved learning.  And he especially loved testing ideas by doing experiments:

A trait that showed up about this time was my tendency not to accept anything I was told until I had checked it out for myself.

From the beginning, I loved learning through experimentation and exploration how my world worked.

Thorp also demonstrated awesome powers of concentration:

When I was reading or just thinking, my concentration was so complete that I lose all awareness of my surroundings.

Thorp was influenced by a few great teachers, including Jack Chasson:

Jack was twenty-seven then, with wavy brown hair and the classic good looks of a Greek god.  He had a ready, warm smile and a way of saying something that boosted the self-esteem of everyone he met… my first great teacher…



Thorp became fascinated by radio and electronics.  The ability to hear voices from the air amazed him:

The mechanical world of wheels, pulleys, pendulums, and gears was ordinary.  I could see, touch, and watch it in action.  But this new world was one of invisible waves that traveled through space.  You had to figure out through experiments that it was actually there and then use logic to grasp how it worked.

Eventually this led Thorp to think things through for himself and also to design experiments:

I was learning to work things out for myself, not limited to prompting from teachers, parents, or the school curriculum.  I relished the power of pure thought, combined with the logic and predictability from science.  I loved visualizing an idea and then making it happen.

Learning, thinking, and experimenting were all great fun for Thorp, leading him to contemplate becoming a scientist at a university:

An academic life was becoming my dream.  I liked all the science experiments I was doing and the knowledge they led to.  If I could have a career continuing this kind of playing, I would be very happy.  And the way to have that kind of life was by joining the academic world where they had the laboratories, the kinds of experiments and projects I enjoyed, and maybe the chance to work with other people like me.

In the summer of 1948, Thorp read through a list of 60 great novels, mostly American literature but also including some foreign authors like Dostoyevski and Stendhal.  Thorp’s teacher Jack Chasson had given him the list and then lent him the books from his personal library.



Thorp was the number one student in his chemistry class, but lost that position when he was cheated.  When the mistake was not corrected, Thorp changed his major to physics:

This rash decision, which led me to change my school and my major subject, would change my whole path in life.  In hindsight, it turned out for the best, as my interests and my future were in physics and mathematics.

In graduate school, after transferring from Berkeley to UCLA, Thorp completed all the course work for the PhD and was halfway through his thesis on the structure of atomic nuclei.  All he had to do was finish the thesis work and pass a final oral exam.  But he would have to learn much more mathematics in order to finish the complex quantum mechanical calculations.  Thorp realized that he could earn a PhD in mathematics much sooner than he would likely be able to earn a PhD in physics.  So he got the PhD in mathematics.

While in graduate school, Thorp had become re-acquainted with Vivian Sinetar.  Thorp says he was lucky she was still single, despite family pressure to marry.  Also, Vivian, whose parents were immigrant Hungarian Jews, would be the first to marry outside the Jewish faith.  Fortunately, Vivian’s parents liked Thorp even though he was an academic rather than a doctor or a lawyer.



Ed Thorp and his wife Vivian spent one Christmas vacation in Las Vegas because the city had turned itself into a bargain vacation spot (to attract gamblers).  The city was different at that time:

Back then the long, straight, uncrowded highway had a dozen or so one-story hotel-casino complexes scattered on either side with hundreds of yards of sand and tumbleweeds separating them.

Just before this trip to Vegas, Thorp had learned from a colleague what is now called basic strategy for blackjack.  This strategy gave the player the smallest statistical disadvantage – 0.62 percent – of any casino game.  Thorp thought he would have fun by risking a few dollars trying out basic strategy.

Before this trip, Thorp had already realized that roulette could be beaten.  Why not blackjack?

The belief that casinos must come out ahead in the long run was supported by conventional wisdom, which argued that if blackjack could be beaten, the casinos would have to either change the rules or drop the game.  Neither had happened.  But, confident from my experiments that I could predict roulette, I wasn’t willing to accept these claims about blackjack.  I decided to check for myself if the player could systematically win.



Thorp explains:

It wasn’t the money that drew me to blackjack.  Though we could certainly use extra dollars, Vivian and I expected to lead the usual low-budget academic life.  What intrigued me was the possibility that merely by sitting in a room and thinking, I could figure out how to win.

Back from vacation, Thorp went to the section of the UCLA library where mathematical and statistical research articles were.

I started with the fact that the strategy I had used in the casino assumed that every card had the same chance of being dealt as any other during play.  This cut the casino’s edge to just 0.62 percent, the best odds of any game being offered.  But I realized that the odds as the game progressed actually depended on which cards were still left in the deck and that the edge would shift as play continued, sometimes favoring the casino and sometimes the player.  The player who kept track could vary his bets accordingly. 

The player would keep his bets small when the casino had the advantage, which was most of the time.  But the player would bet much more when the odds were in his favor.  Over a large number of hands, the casino would win most of the small bets, but the player would win most of the big bets.  As long as the deal was fair—otherwise the player should learn to quit right away—Thorp’s strategy would be profitable over time.

Thorp began to do calculations to see how the player’s advantage changed based on which cards had already been played.  Thorp figured out that what mattered was the proportion of each type of card left as a percentage of the total number of cards left.

When Thorp started teaching mathematics at MIT, he had access to an IBM 704 computer, which he used to test his blackjack approximations.  Next he used the computer to figure out how the odds changed when all four of a specific card were missing from the remaining deck.  The math also showed that if removing a specific group of cards shifted the odds in one direction, adding an equal number of the same cards would move the odds the other way by the same amount.

Eventually Thorp was able to calculate the player advantage based on the specific cards that had been played.  He decided to publish his results in Proceedings of the National Academy of Sciences.  But he needed a member of the academy to approve and forward his work.  The only mathematics member of the academy at MIT was Claude Shannon.  Shannon was famous for the invention of information theory, the foundation of modern computing.

To Thorp’s surprise, Shannon was fascinated by Thorp’s ideas.  A few minutes became an hour and a half of animated dialogue.  Shannon said Thorp had likely made a theoretical breakthrough.  But he suggested the paper be titled “A Favorable Strategy for Twenty-One” instead of “A Winning Strategy for Blackjack.”

Then Shannon asked Thorp if he was working on anything else involving games of chance.  Thorp told him about his idea that roulette was predictable.  This led to several more hours of excited conversation.  Shannon and Thorp decided to work together to build a small, wearable computer that could be used to beat roulette.



Thorp had decided to present his blackjack system at the annual meeting of the American Mathematical Society in Washington, DC.  Thorp made this decision because previous mathematicians (centuries earlier in some cases) seemed to have proven that no casino game could be beat.  Dick Stewart of The Boston Globe had heard about Thorp’s upcoming talk.  Stewart called Thorp to ask about it.  The newspaper also sent a photographer to take Thorp’s picture.  The next morning Stewart’s article and Thorp’s picture were on the front page.

When the day of the meeting arrived, instead of the usual scholarly audience of forty or fifty, there were hundreds of curious people, including many with sunglasses, pinkie rings, or cigars.  Thorp writes:

In the abstract, life is a mixture of chance and choice.  Chance can be thought of as the cards you are dealt in life.  Choice is how you play them.  I chose to investigate blackjack.  As a result, chance offered me a new set of unexpected opportunities.

Thorp was deluged by offers to back a casino test, ranging from a few thousand dollars to $100,000.  Many were curious about whether Thorp’s system would really work.  Thorp felt he owed his readers proof.

The most promising offer came from two New York multimillionaires.  Thorp called them Mr. X and Mr. Y.  Initially, Thorp was concerned about the dangers of a bankroll provided by strangers.  But Mr. X kept calling, so Thorp finally decided to meet him.

Emmanual “Manny” Kimmel (Mr. X) arrived at Thorp’s residence in a Cadillac with two good-looking young blondes.  Kimmel introduced the two women as his “nieces.”  Kimmel dealt blackjack to Thorp for a couple of hours while asking him about his research.  Then they agreed to plan a trip to Nevada.  When Manny was leaving, he grabbed several pearl necklaces from his pocket and offered a strand to Vivian.  The pearls stayed in Thorp’s family and are now warn by his daughter.

Kimmel and his friend (Mr. Y) gave Thorp a bankroll of $100,000.  (This is the equivalent of $800,000 in 2017 dollars.)  But Thorp insisted on starting with only $10,000 in order to prove first that his system worked.

This plan, of betting only at a level at which I was emotionally comfortable and not advancing until I was ready, enabled me to play my system with a calm and disciplined accuracy.  This lesson from the blackjack tables would prove invaluable throughout my investment lifetime as the stakes grew ever larger.

Thorp’s system worked.  But the blackjack player had to understand randomness and odds over a very long series of bets.  Most small bets the casino would win.  And there would also be times when the player was unlucky on bigger bets, despite favorable odds.  But eventually, over time, Thorp’s system worked.

…the Ten-Count System had shown moderately heavy losses mixed with ‘lucky’ streaks of the most dazzling brilliance.  I learned later that this was a characteristic of a random series of favorable bets.  And I would see it again and again in real life in both the gambling and the investment worlds.

Note:  Thorp’s system worked as long as the deal was fair most of the time.  But the player had to learn to spot signs of cheating.  The player also had to quit games where losses were happening fast.  (Fast losses usually meant cheating.)

Cheating was so relentless during those days in Las Vegas that I spent as much time learning about the many ways it was being done as I did playing.  Everywhere we went, we reached a point where we were cheated, barred from play,  or the dealer reshuffled the cards after every hand.



Thorp and Shannon created a wearable computer that would allow the player to win at roulette.

Thorp was now in a position to win a good deal of money—compared to his salary as a mathematics professor—by playing blackjack and roulette.  But introspection revealed to him that he would enjoy life more as an academic than as a gambler:

I was at a point in life where I could choose between two very different futures.  I could roam the world as a professional gambler winning millions per year.  Switching between blackjack and roulette, I could spend some of the winnings as perfect camouflage by also betting on other games offering a small casino edge, like craps or baccarat.

My other choice was to continue my academic life.  The path I would take was determined by my character, namely, What makes me tick?  As the Greek philosopher Heraclitus said, ‘Character is destiny.’



Thorp writes:

Gambling is investing simplified.  The striking similarities between the two suggested to me that, just as some gambling games could be beaten, it might also be possible to do better than the market averages.  Both can be analyzed using mathematics, statistics, and computers.  Each requires money management, choosing the proper balance between risk and return.  Betting too much, even though each individual bet is in your favor, can be ruinous… On the other hand, playing safe and betting too little means you leave money on the table.  The psychological makeup to succeed at investing also has similarities to that for gambling.  Great investors are often good at both.

Thorp made several mistakes when he started investing.  The first stock he bought dropped 50%.  Thorp decided to wait until he could get even.  This happened after four years.  One thing Thorp learned from this experience is to avoid anchoring.

Learn about the anchoring effect here:  https://boolefund.com/why-simple-quant-models-beat-experts-in-a-wide-variety-of-areas/

Thorp’s second mistake was investing based on momentum.  It didn’t work.  Thorp learned not to expect momentum to continue unless you have good reasons to think it will.

Thorp’s third mistake was to buy silver on margin.  Initially silver rose and Thorp used the profits to buy even more silver on margin.  Then the silver price dropped, which wiped Thorp out because he was on margin. After that, silver started going up again, but Thorp had already lost his whole investment due to his use of margin.  This experience taught Thorp about proper risk management.

Thorp learned how to invest in undervalued warrants while hedging the position:

To form a hedge, take two securities whose prices tend to move together, such as a warrant and the common stock it can be used to purchase, but which are comparatively mispriced.  Buy the relatively underpriced security and sell short the relatively overpriced security.  If the proportions in the position are chosen well, then even though prices fluctuate, the gains and losses on the two sides will approximately offset or hedge each other.  If the relative mispricing between the two securities disappears as expected, close the position in both and collect a profit.

Thorp figured out a formula for pricing warrants and options.  An option to buy a stock is like a warrant except that usually the company issues warrants.  Thorp began investing portfolios for friends and acquaintances.



Ralph Waldo Gerard was an early investor with Thorp.  Previously, Gerard had invested in the Buffett Partnership.  Gerard was related to the father of value investing, Benjamin Graham (Buffett’s teacher and mentor).

Gerard invited Thorp and his wife to his home for dinner with Susie and Warren Buffett.  Buffett is arguably the most successful investor of all time.  But Thorp learned that Buffett had to work extremely hard in order to find a few excellent long-term investments.

By contrast, Thorp’s quantitative, statistical investment strategy seemed much easier than analyzing in detail thousands of companies.  Thorp’s approach would give him more free time to enjoy family and to pursue his academic career.

Later, Buffett invited Gerard and Thorp to his home in Emerald Bay, California for an afternoon of bridge.  Thorp:

Bridge is what mathematicians call a game of imperfect information.  The bidding, which precedes the play of the cards, gives some information about the four concealed hands held by two pairs of players who are opposing each other.  As the cards are played, players use the bidding and the cards they have seen so far to make inferences about who has the remaining unplayed cards.  The stock market also is a game of imperfect information and even resembles bridge in that both have their deceptions.  As in bridge, you do better in the market if you get more information sooner and put it to better use.  It’s no surprise that Buffett, arguably the greatest investor in history, is a bridge addict.

Thorp was impressed by Buffett and made a prediction:

Impressed by Warren’s mind and his methods, as well as his record as an investor, I told Vivian that I believed he would eventually become the richest man in America.  Buffett was an extraordinarily smart evaluator of underpriced companies, so he could compound money much faster than the average investor.  He also could continue to rely mainly on his own talent even as his capital grew to an enormous amount.  Warren furthermore understood the power of compound interest and, clearly, planned to apply it over a long time.

Thorp partnered with a New York stockbroker, Jay Regan, who had studied philosophy at Dartmouth.  Together, they launched Convertible Hedge Associates—later renamed Princeton Newport Partners.  They aimed to raise $5 million, but only reached $1.4 million.  They went ahead anyways.



Princeton Newport Partners (PNP) specialized in the hedging of convertible securities—warrants, options, convertible bonds and preferreds, and other types of derivative securities.  PNP not only hedged each individual position.  But it also hedged the portfolio against changes in interest rates and changes in the overall market level.  PNP’s near total reliance on quantitative methods—using mathematical formulas, economic models, and computers—made them the earliest “quants.”

Thorp was motivated to reduce risk:

Influenced by having been born during the Great Depression and by my early investment experiences, I made reducing risk a central feature of my investing approach.

The hedges protected us against losses but at the expense of giving up some of the gains in the big up-markets.

In 1973-1974, each $1,000 invested in the S&P 500 would have shrunk to $618, whereas each $1,000 invested in PNP grew to $1,160.

Thorp’s wife, Vivian, not only raised their three children.  She was also active in local politics, helping reelect a decent congressman.  And Vivian organized and ran a large phone bank that helped elect the first black man to a California statewide office.  Moreover, she influenced many people one on one.

One time, a woman complained to Vivian about “those Jews.”  Vivian was Jewish and had lost several relatives in Nazi World War II prison camps.  Ed Thorp:

When she told us about meeting the woman, we expected to hear how she tore her to shreds.  Explaining why she did not, Vivian pointed out that the woman would have learned nothing and simply would have become an enemy.  Vivian patiently educated this basically good person and they became friends for the rest of their lives.

Thorp’s PhD thesis had been in pure mathematics and this continued to be his focus for fifteen years.  Although Thorp loved teaching and research, eventually he resigned his full professorship at the University of California, Irvine.  He felt a sense of loss, but it turned out to be for the best.  Thorp continued his friendships and research collaborations.  He continued to present his work at meetings and publish it in the mathematical, financial, and gambling literature.



Thorp and his colleagues continued to solve problems for valuing derivatives before academics did.  This gave PNP a large edge from 1967 to 1988, when PNP closed.

Hedging with derivatives was a key source of profits for PNP during its entire nineteen years.  Such hedging also became a core strategy for many later hedge funds like Citadel, Stark, and Elliott, which each went on to manage billions.

Some risks cannot be hedged:

There is another kind of risk on Wall Street from which computers and formulas can’t protect you.  That’s the danger of being swindled or defrauded.  Being cheated at cards in the casinos in the 1960s was valuable preparation for the far greater scale of dishonesty I would encounter in the investment world.  The financial press reveals new skulduggery on a daily basis.



PNP’s dream for the 1980s was to expand their expertise into new areas.

Of the scores of indicators we systematically analyzed, several correlated strongly with past performance.  Among them were earnings yield (annual earnings divided by price), dividend yield, book value divided by price, momentum, short interest…, earnings surprise…, purchases and sales by company officers, directors, and large shareholders, and the ratio of total company sales to the market price of the company.  We studied each of these separately, then worked out how to combine them.  When the historical patterns persisted as prices unfolded into the future, we created a trading system called MIDAS (multiple indicator diversified asset system) and used it to run a separate long/short hedge fund (long the “good” stocks, short the “bad” ones).  The power of MIDAS was that it applied to the entire multitrillion-dollar stock market, with the possibility of investing very large sums.

From November 1, 1979 through January 1, 1988, PNP’s capital expanded from $28.6 million to $273 million.  The partnership earned 22.8 percent per year before fees, which meant 18.2 percent per year for limited partners.

Furthermore, PNP invented excellent new products that could allow the fund to manage billions.  They included:

  • State-of-the-art convertible, warrant, and option computerized analytic models and trading systems
  • Statistical arbitrage
  • Expert investments based on interest rates
  • OSM Partners, a “fund of hedge funds”



In the 1970s, less established companies had to scramble for funding.  A young financial innovator named Michael Milken had an idea:

Milken’s group underwrote issues of low-rated, high-yielding bonds—the so-called junk bonds—some of which were convertible or came with warrants to purchase stock… Filling a gaping need and hungry demand in the business community, Milken’s group became the greatest financing engine in Wall Street history.

Such innovation outraged the old line establishment of corporate America, who were initially transfixed like deer in the headlights as a horde of entrepreneurs, funded with seemingly unlimited Drexel-generated cash, began a wave of unfriendly takeovers.  Many old firms were vulnerable because the officers and directors had done a poor job of investing the shareholders’ equity.  With subpar returns on capital, the stocks were cheap…

The officers and directors of America’s big corporations were happy with the way things had been.  They enjoyed their hunting lodges and private jets, made charitable donations for their personal aggrandizement and objectives, and granted themselves generous salaries, retirement plans, bonuses of cash, stock, and stock options, and golden parachutes.  All these things were designed by and for themselves and paid for with corporate dollars, the expenses routinely ratified by a scattered and fragmented shareholder base.  Economists call this conflict of interest between management, or agents, and the shareholders, who are the real owners, the agency problem.  It continues today, one example being the massive continuing grants of stock options by management to itself…

Rudolph Giuliani, U.S. Attorney for the Southern District of New York, was on a campaign to prosecute real and alleged Wall Street criminals.  As a part of his effort to prosecute Michael Milken at Drexel Burnham and Robert Freeman at Goldman Sachs, Giuliani went after Thorp’s partner Jay Regan, who knew both Milken and Freeman well.

Giuliani went after the Princeton office of PNP.  The Newport office, where Thorp and forty others worked, did not have any knowledge of the alleged acts in the Princeton office.  No one at the Newport office was implicated in, or charged with, any wrongdoing in this (or any other) matter.

To apply more pressure, the U.S. Attorney began contacted the limited partners of PNP.  They subpoened them to come to New York and testify before the grand jury.  Thorp explains that the limited partners were passive participants in PNP.  The subpoenas thus had no real value for Giuliani’s case.  It seems Giuliani wanted to disturb and upset these limited partners so that they might withdraw from PNP.

In the end, convictions for racketeering and tax fraud against a few PNP defendants were thrown out by the Second Court of Appeals.  Thorp writes:

In January 1992, having achieved their real goal, which was to convict Milken and Freeman, the prosecutors dropped the remaining charges against four of the five PNP defendants and a relate charge against the Drexel trader.  Princeton’s head trader and the Drexel defendant were still facing fines and three-month prison terms for their remaining counts.  In September 1992, a federal judge vacated these sentences as well.

Thorp later explains:

The old establishment financiers were lucky in that prosecutors would find numerous violations of securities laws within the Milken group and among its allies, associates, and clients.  However, it is difficult to judge how relatively bad these were, compared with the incessant violations that have always been, and continue to be, endemic in business and finance, because only a few of the many violators  are caught, and when they are prosecuted it may be for only a tiny fraction of their offenses.  This contrasts with the case of Drexel, where the searchlight of government was focused to reveal as many violations as possible.  It’s like the case of the man who was cited three times in a single year for driving while intoxicated.  His neighbor would also drink and drive, but was never pulled over.  Who is the greater criminal?  Now suppose I tell you that the caught man did it only three times and was apprehended every time, whereas his neighbor did it a hundred times and was never caught.  How could this happen?  What if I tell you that the two men are bitter business rivals and that the traffic cop’s boss, the police chief, gets large campaign contributions from the man who got no traffic citations.  Now who is the greater criminal?

Thorp considered launching a partnership that would be similar to PNP.  But he loved the quantitative analysis part of the business, not operations and marketing.  So he decided to wind down the Newport office.



Although the closing of PNP erased billions in future wealth for Thorp and his colleagues, Thorp and his wife had more than enough money to be free to spend their time exactly as they wanted.

Around this time, Thorp discovered the greatest financial fraud.  He had been hired to examine some hedge fund investments.  Thorp approved them with one exception: Bernard Madoff Investment.

Madoff claimed to use a split-strike price strategy:  He would buy a stock, sell a call option at a higher price, and use the proceeds to pay for a put option at a lower price.

I explained that, according to financial theory, the long-run impact on portfolio returns from many properly priced options with zero net proceeds should also be zero.  So we expect, over time, that the client’s portfolio return should be roughly the same as the return on equities.  The returns Madoff reported were too large to be believed.  Moreover, in months when stocks are down, the strategy should produce a loss—but Madoff wasn’t reporting any losses.  After checking the client’s account statements I found that losing months for the strategy were magically converted to winners by short sales of S&P Index futures.  In the same way, months that should have produced very large wins were ‘smoothed out.’


…At my suggestion, the client then hired my firm to conduct a detailed analysis of their individual transactions to prove or disprove my suspicions that they were fake.  After analyzing about 160 individual option trades, we found that for half of them no trades occurred on the exchange where Madoff said that they supposedly took place.  For many of the remaining half that did trade, the quantity reported by Madoff just for my client’s two accounts exceeded the entire volume reported for everyone.  To check the minority of remaining trades, those that did not conflict with the prices and volumes reported by the exchanges, I asked an official at Bear Stearns to find out in confidence who all the buyers and sellers of the options were.  We could not connect any of them to Madoff’s firm.

Thorp had proved Madoff’s investment operation was a fraud.  Madoff was running a Ponzi scheme.

In 1991, Thorp was seeking a partner to whom to sell their statistical arbitrage software.  This led him to meet with Bruce Kovner, a successful commodities trader.

About this time he realized large oil tankers were in such oversupply that the older ones were selling for little more than scrap value.  Kovner formed a partnership to buy one.  I was one of the limited partners.  Here was an interesting option.  We were largely protected against loss because we could always sell the tanker for scrap, recovering most of our investment;  but we had a substantial upside:  Historically, the demand for tankers had fluctuated widely and so had their price.  Within a few years, our refurbished 475,000-ton monster, the Empress Des Mers, was profitably plying the world’s sea-lanes stuffed with oil.  I liked to think of my ownership as a twenty-foot section just forward of the bridge… The Empress Des Mers operated profitably into the twenty-first century, when the saga finally ended.  Having generated a return on investment of 30 percent annualized, she was sold for scrap in 2004, fetching almost $23 million, far more than her purchase price of $6 million.

Thorp discusses traders who always try to save a tiny amount on each trade.  The problem is that the trader may do this successfully twenty times in a row, but then miss a trade that goes up so much that it wipes out the savings on the previous twenty trades.

What the hagglers and the traders do reminds me of the behavioral psychology distinction between two extremes on a continuum of types:  satisficers and maximizers.  When a maximizer goes shopping, looks for a handyman, buys gas, or plans a trip, he searches for the best (maximum) possible deal.  Time and effort don’t matter much.  Missing the very best deal leads to regret and stress.  On the other hand, the satisficer, so-called because he is satisfied with a result that is close to the best, factors in the costs of searching and decision making, as well as the risk of losing a near-optimal opportunity and perhaps never finding anything as good again.

This is reminiscent of the so-called secretary or marriage problem in mathematics.  Assume that you will interview a series of people, from which you will choose one.  Further, you must consider them one at a time, and having once rejected someone, you cannot reconsider.  The optimal strategy is to wait until you have seen about 37 percent of the prospects, then choose the next one you see who is better than anybody among this first 37 percent that you passed over.  If no one is better you are stuck with the last person on the list.




…Some exchanges, such as NASDAQ, let HF [High Frequency] traders peek at customer orders ahead of everyone else for thirty milliseconds before the order goes to the exchange.  Seeing an order to buy, for instance, the HF traders can buy first, pushing the stock price up, then resell to the customer at a profit.  Seeing someone’s order to sell, the HF trader sells first, causing the stock to fall, and then buys it back at the lower price.  How is this different from the crime of front-running, described in Wikipedia as ‘the illegal practice of a stock broker executing orders on a security for its own account while taking advantage of advance knowledge of pending orders from its customers’?

Some securities industry spokesmen argue that harvesting this wealth from investors somehow makes the markets more efficient and that ‘markets need liquidity.’  Nobel Prize-winning economist Paul Krugman disagrees sharply, arguing that high-frequency trading is simply a way of taking wealth from ordinary investors, serves no useful purpose, and wastes national wealth because the resources consumed create no social good.

Since the more the rest of us trade the more we as a group lose to the computers, here’s one more reason to buy and hold rather than trade, unless you have a big enough edge.



Thorp discusses a statistical arbitrage investment project:

The idea of the project was to study how the historical returns of securities were related to various characteristics, or indicators.  Among the scores of fundamental and technical measures we considered were the ratio of earnings per share to price per share, known as the earnings yield, the liquidation or “book” value of the company compared with its market price, and the total market value of the company (its “size”).  Today our approach is well known and widely explored but back in 1979 it was denounced by massed legions of academics who believed market prices already had fully adjusted to such information.  Many practitioners disagreed.  The time was right for our project because the necessary high-quality databases and the powerful new computers with which to explore them were just becoming affordable.

The idea for statistical arbitrage was based on the discovery (by one of Thorp’s researchers) that the stocks that had gone up the most over the previous two weeks did the worst as a group over the ensuing few weeks, while the stocks that had gone down the most over the previous two weeks did the best.

In 1994, Thorp launched a new investment partnership, Ridgeline Partners.  Limited partners gained 18 percent per year over eight and a quarter years.

We charged Ridgeline Partners 1 percent per year plus 20 percent of net new profits.  We voluntarily reduced fees during a period when we felt disappointed in our performance.  We gave back more than $1 million to the limited partners.  Some of today’s greedy hedge fund managers might say our return of fees was economically irrational, but our investors were happy and we nearly always had a waiting list.  Ridgeline was closed a large part of the time to new investors, and current partners were often restricted from adding capital.  To maintain higher returns, we sometimes even reduced our size by returning capital to partners.

Instead of charging more fees, Thorp says he sought to treat limited partners as he would wish to be treated if he were in their place.  Thorp closed the fund down in the fall of 2002 because returns had declined due to more hedge funds using statistical arbitrage programs.  More importantly, Ed and Vivian wanted time to travel, read, and learn, and to be with their family.




The consensus of industry studies of hedge fund returns to investors seems to be that, considering the level of risk, hedge funds on average once gave their investors extra return, but this has faded as the industry expanded.  Later analyses say average results are worse than portrayed.  Funds voluntarily report their results to the industry databases.  Winners tend to participate much more than losers.  One study showed that this doubled the reported average annual return for funds as a group from an actual 6.3 percent during 1996-2014 to a supposed 12.6 percent.

The study goes one to point out that if returns over the years are given weights that correspond to the dollars invested, then the returns are ‘only marginally higher than risk-free [U.S. Treasury Bonds] rates of return.’  Another reason that reports by the industry look better than what investors experienced is that they combined higher-percentage returns from the earlier years, when the total invested in hedge funds was smaller, with the lower-percentage returns later, when they managed much more money.

It’s difficult to get an edge picking stocks.  Hedge funds are little businesses just like companies that trade on the exchanges.  Should one be any better at picking hedge funds than we are at picking stocks?

Thorp points out that you will rarely find an investment that is better than an ultra-low-cost index fund over time.  Also, some hedge funds and mutual funds create spectacular records early on but mediocre results when assets under management have grown:

One method that leads to this has also been used to launch new mutual funds.  Fund managers sometimes start a new fund with a small amount of capital.  They then stuff it with hot IPOs (initial public offerings) that brokers give them as a reward for the large volume of business they have been doing through their established funds.  During this process of ‘salting the mine,’ the fund is closed to the public.  When it establishes a stellar track record, the public rushes in, giving the fund managers a huge capital base from which they reap large fees.  The brokers who supplied the hot IPOs are rewarded by a flood of additional business from the triumphant managers of the new fund.  The available volume of hot IPOs is too small to help returns much once the fund gets big, so the track record declines to mediocrity.  However, the fund promoters can use more hot IPOs to incubate yet another spectacularly performing new fund;  and so it goes on.

Like Buffett, Thorp predicts the gradual disappearance of any excess returns produced by hedge funds as a group.  Here is Buffett’s view:  https://boolefund.com/warren-buffett-jack-bogle/




Call any investment that mimics the whole market of listed U.S. securities ‘passive’ and notice that since each of these passive investments acts just like the market, so does a pool of all of them.  If these passive investors together own, say, 15 percent of every stock, then ‘everybody else’ owns 85 percent and, taken as a group, their investments also are like one giant index fund.  But ‘everybody else’ means all the active investors, each of whom has his own recipe for how much to own of each stock and none of whom has indexed.  As Nobel Prize winner Bill Sharpe says, it follows from the laws of arithmetic that the combined holdings of all the active investors also replicates the index.

Reducing risk through diversification is a reason to own an index fund.  An even more important reason to own an index fund is to reduce your costs.  Ultra-low costs are why index funds outperform, necessarily, the vast majority of investors, especially over the course of several decades.  Thorp explains:

Investors who don’t index pay on average an extra 1 percent a year in trading costs and another 1 percent to what Warren Buffett calls ‘helpers’—the money managers, salespeople, advisers, and fiduciaries that permeate all areas of investing.  As a result of these costs, active investors as a group trail the index by 2 percent or so, whereas the passive investor who selects a no-load (no sales fee), low-expense-ratio (low overhead and low management fee) index fund can pay less than 0.25 percent in fees and trading costs.  From the gambling perspective, the return to an active investor is that of a passive investor plus the extra gain or loss from paying (on average) 2 percent a year to toss a fair coin in some (imaginary) casino.  Taxable active investors do even worse, because a high portfolio turnover means short-term capital gains, which currently are taxed at a higher rate than gains from securities, the sales of which have been deferred for a year.

Furthermore, notes Thorp, one way an investor could mimic an index fund is simply to buy a portfolio of at least twenty stocks.  If the choices are randomized, then the returns from this portfolio should track the index over time.  Consider, for instance, that the Dow Jones Industrial Average—comprised of thirty stocks—has closely tracked the S&P 500 Index over time.

Moreover, the portfolio of twenty stocks could be even lower cost than an ultra-low-cost index fund because the 20-stock portfolio likely would not require any trading at all, whereas a broad market ultra-low-cost index fund would have to make minor adjustments over time in order to keep tracking the index.



Thorp observes that for a perfectly efficient market, one you can’t beat, we expect:

  • All information to be instantly available to many participants.
  • Many participants to be financially rational.
  • Many participants to be able instantly to evaluate all available relevant information and determine the current fair price of every security.
  • New information to cause prices immediately to jump to the new fair price, preventing anyone from gaining an excess market return by trading at intermediate prices during the transition.

Supporters of the EMH (Efficient Market Hypothesis) typically argue that these conditions hold as an approximation.

In the real world of investing, Thorp writes that the market is somewhat inefficient.  In particular:

  • Some information is instantly available to the minority that happen to be listening at the right time and place.  Much information starts out known only to a limited number of people, then spreads to a wider group in stages.  This spreading could take from minutes to months, depending on the situation.  The first people to act on the information capture the gains.  The others get nothing or lose.  (Note:  The use of early information by insiders can be either legal or illegal, depending on the type of information, how it is obtained, and how it’s used.)
  • Each of us is financially rational only in a limited way.  We vary from those who are almost totally irrational to some who strive to be financially rational in nearly all their actions.  In real markets the rationality of the participants is limited.
  • Participants typically have only some of the relevant information for determining the fair price of a security.  For each situation, both the time to process the information and the ability to analyze it generally very widely.
  • The buy and sell orders that come in response to an item of information sometimes arrive in a flood within a few seconds, causing the price to gap or nearly gap to the new level.  More often, however, the reaction to news is spread out over minutes, hours, days, or months, as the academic literature documents.

These realities tell us how to beat the market, says Thorp:

  • Get good information early.  How do you know if your information is good enough or early enough?  If you are not sure, then it probably isn’t.
  • Be a disciplined rational investor.  Follow logic and analysis rather than sales pitches, whims, or emotion.  Assume you may have an edge only when you can make a rational affirmative case that withstands your attempts to tear it down.  Don’t gamble unless you are highly confident you have the edge.  As Buffett says, ‘Only swing at the fat pitches.’
  • Find a superior method of analysis.  Ones that you have seen pay off for me include statistical arbitrage, convertible hedging, the Black-Scholes formula, and card counting at blackjack.  Other winning strategies include superior security analysis by the gifted few and the methods of the better hedge funds.
  • When securities are known to be mispriced and people take advantage of this, their trading tends to eliminate the mispricing.  This means the earliest traders gain the most and their continued trading tends to reduce or eliminate the mispricing.  When you have identified an opportunity, invest ahead of the crowd.

Thorp sums it up:

Note that market inefficiency depends on the observer’s knowledge.  Most market participants have no demonstrable advantage.  For them, just as the cards in blackjack or the numbers at roulette seem to appear at random, the market appears to be completely efficient.

To beat the market, focus on investments well within your knowledge and ability to evaluate, your ‘circle of competence.’  Be sure your information is current, accurate, and essentially complete.  Be aware that information flows down a ‘food chain,’ with those who get it first ‘eating’ and those who get it late being eaten.  Finally, don’t bet on an investment unless you can demonstrate by logic, and if appropriate by track record, that you have an edge.



Thorp wraps up his book by sharing some of what he learned on his odyssey through science, mathematics, gambling, hedge funds, finance, and investing:

Education has made all the difference for me.  Mathematics taught me to reason logically and to understand numbers, tables, charts, and calculations as second nature.  Physics, chemistry, astronomy, and biology revealed wonders of the world, and showed me how to build models and theories to describe and to predict.  This paid off for me in both gambling and investing.

Education builds software for your brain.  When you’re born, think of yourself as a computer with a basic operating system and not much else.  Learning is like adding programs, big and small, to this computer, from drawing a face to riding a bicycle to reading to mastering calculus.  You will use these programs to make your way in the world.  Much of what I’ve learned came from schools and teachers.  Even more valuable, I learned at an early age to teach myself.  This paid off later on because there weren’t any courses in how to beat blackjack, build a computer for roulette, or launch a market-neutral hedge fund.

I found that most people don’t understand the probability calculations needed to figure out gambling games or to solve problems in everyday life.  We didn’t need that skill to survive as a species in the forests and jungles.  When a lion roared, you instinctively climbed the nearest tree and thought later about what to do next.  Today we often have the time to think, calculate, and plan ahead, and here’s where math can help us make decisions…

Thorp later writes that economists have found one factor that explains a nation’s future economic growth more than any other:  its output of scientists and engineers.  Therefore it’s crucial to have the best education system we can.  It’s essential that we strive to keep talented American-born scientists and engineers in the United States, and that we also seek to keep gifted foreign-born scientists and engineers after they have received advanced degrees in the United States.  Thorp:

To starve education is to eat our seed corn.  No tax today, no technology tomorrow.

Thorp concludes:

Life is like reading a novel or running a marathon.  It’s not so much about reaching a goal but rather about the journey itself and the experiences along the way.  As Benjamin Franklin famously said, ‘Time is the stuff life is made of,’ and how you spend it makes all the difference.

…Whatever you do, enjoy your life and the people who share it with you, and leave something good of yourself for the generations to follow.



An equal weighted group of micro caps generally far outperforms an equal weighted (or cap-weighted) group of larger stocks over time.  See the historical chart here:  https://boolefund.com/best-performers-microcap-stocks/

This outperformance increases significantly by focusing on cheap micro caps.  Performance can be further boosted by isolating cheap microcap companies that show improving fundamentals.  We rank microcap stocks based on these and similar criteria.

There are roughly 10-20 positions in the portfolio.  The size of each position is determined by its rank.  Typically the largest position is 15-20% (at cost), while the average position is 8-10% (at cost).  Positions are held for 3 to 5 years unless a stock approaches intrinsic value sooner or an error has been discovered.

The mission of the Boole Fund is to outperform the S&P 500 Index by at least 5% per year (net of fees) over 5-year periods.  We also aim to outpace the Russell Microcap Index by at least 2% per year (net).  The Boole Fund has low fees.


If you are interested in finding out more, please e-mail me or leave a comment.

My e-mail: jb@boolefund.com



Disclosures: Past performance is not a guarantee or a reliable indicator of future results. All investments contain risk and may lose value. This material is distributed for informational purposes only. Forecasts, estimates, and certain information contained herein should not be considered as investment advice or a recommendation of any particular security, strategy or investment product. Information contained herein has been obtained from sources believed to be reliable, but not guaranteed. No part of this article may be reproduced in any form, or referred to in any other publication, without express written permission of Boole Capital, LLC.

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