Valaris (VAL)—The Cheapest Stock I’ve Ever Seen

(Image: Zen Buddha Silence, by Marilyn Barbone)

March 8, 2020

We continue with examples of Boole’s quantitative investment process in action.

A few weeks ago, we looked at Ranger Energy Services (RNGR):

Before that, we looked at Macro Enterprises (Canada: MCR.V):

This week, we are going to look at Valaris (VAL)—the largest offshore oil driller in the world and sample research paper thesis writing encouragement can i buy viagra in europe peut on se procurer du viagra nursing dissertation topics stroke source site source url mahatma gandhi research paper source link ovulet clomiphene online homework help uk poem essay example go go site prodigy essay creative writing assignment go to link go here help starting compare contrast essay follow writing essay website thesis topics in guidance and counselling content of executive summary of business plan popular dissertation conclusion writer website us the cheapest stock I’ve ever seen—which comes out near the top of the quantitative screen employed by the Boole Microcap Fund.  This results from four steps.

Step One

First we screen for cheapness based on five metrics.  Here are the numbers for Valaris:

    • EV/EBITDA = 4.01
    • P/E = 0.14
    • P/B = 0.01
    • P/CF = 0.07
    • P/S = 0.03

These figures—especially P/E and P/B—make Valaris one of the top ten cheapest companies out of over two thousand that we ranked.  (Note: This assumes a medium-case recovery with EBITDA at $1,510 million and net income at $700 million.  The current market capitalization is $99 million.)

Step Two

Next we calculate the Piotroski F-Score, which is a measure of the fundamental strength of the company.  For more on the Piostroski F-Score, see my blog post here:

Valaris has a Piotroski F-Score of 7.  (The best score possible is 9, while the worst score is 0.)  This is a very good score.

Step Three

Then we rank the company based on low debt, high insider ownership, and shareholder yield.

We measure debt levels by looking at total liabilities (TL) to total assets (TA).  Valaris has TL/TA of 45.0%, which is reasonable.

Insider ownership is important because that means that the people running the company have interests that are aligned with the interests of other shareholders.  At Valaris, insider ownership is approximately 5%.  This isn’t a high percentage, but it does represent a total insider ownership of $5 million.

Shareholder yield is the dividend yield plus the buyback yield.  The company has no dividend and is not buying back shares.  Thus, shareholder yield is practically zero.

Each component of the ranking has a different weight.  The overall combined ranking of Valaris places it in the top 5 stocks on our screen, or the top 0.2% of the more than two thousand companies we ranked.

Step Four

The final step is to study the company’s financial statements, presentations, and quarterly conference calls to (i) check for non-recurring items,  hidden liabilities, and bad accounting; (ii) estimate intrinsic value—how much the business is worth—using scenarios for low, mid, and high cases.

See the company presentation (dated February, 2020):

Valaris is the largest offshore oil driller in the world, with presence in six continents and nearly all major offshore markets.  The company has a large and diverse customer base including major, national, and independent E&P companies.

Valaris has 16 drillships, 10 semisubmersibles, and 50 jackups.  Valaris has one of the highest-quality fleets: 11 of its 16 drillships are the highest-specification.  13 of its 50 jackups are heavy duty ultra harsh and harsh environment jackups.  High-spec assets are preferred by customers.

Valaris is also one of the best capitalized drillers.  Valaris has a market capitalization of $99 million.  The company has $2.5 billion in contracted revenue backlog (excluding bonus opportunities).  It has $1.7 billion in liquidity, including $100 million in cash and $1.6 billion in credit available.  And it has only $858 million in debt maturities to 2024.  Valaris is one of two public offshore drillers with no guaranteed or secured debt in the capital structure.  With the asset value of its fleet at $9.1 billion (according to third party estimates), Valaris has ample flexibility to raise additional capital if needed.

In April 2019, Ensco plc (ESV) and Rowan Companies plc (RDC) merged in an all-stock transaction.  The combination (renamed Valaris) has brought together two world-class operators with common cultures.  Both companies have strong track records of safety and operational excellence.  And both companies have a strategic focus on innovative technologies that increase efficiencies and lower costs.  Ensco was rated #1 in customer satisfaction for nine straight years according to a leading independent survey.

As a result of the merger, Valaris has already achieved cost savings of $135 million pre-tax per year.  The company expects to achieve additional savings of $130 million a year.  The full savings will be $265 million a year, which is $100 million more a year than the company initially projected.

Intrinsic value scenarios:

    • Low case: If oil prices languish below $55 (WTI) for the next 3 to 5 years, Valaris will be a survivor, due to its large fleet, globally diverse customer base, industry leading performance, low cost structure, and well-capitalized position.  In this scenario, Valaris is likely worth at least 10 percent of current book value (which is depressed) of $48.15.  That’s $4.82, about 860% higher than today’s $0.50.
    • Mid case: If oil prices are in a range of $55 to $75 over the next 3 to 5 years—which is likely based on long-term supply and demand—then Valaris is probably worth at least current book value (which is depressed) of $48.15 a share, roughly 9,530% higher than today’s $0.50.
    • High case: EBITDA under a full recovery is approximately $4 billion.  Fair value can be conservatively estimated at 6x EV/EBITDA.  That would be EV (enterprise value) of $24 billion, which implies a market cap of $17.6 billion.  That works out to $88.94 a share, over 17,685% higher than today’s $0.50.

Bottom Line

Valaris (VAL) is the cheapest stock I’ve ever encountered.  Assuming the return of normal circumstances within the next 3 to 5 years, the potential upside is roughly 9,530%.  We are “trembling with greed” to buy VAL for the Boole Microcap Fund.



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:

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:




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.

Preparing to Buy Beaten-Down Stocks: Covid-19 Is Going Global

(Image: Zen Buddha Silence, by Marilyn Barbone)

March 1, 2020

No one can predict what the stock market will do, even if a recession appears to be unfolding.  That said, 2009 to 2019 was a bull market for U.S. stocks.  It wouldn’t be surprising if the S&P 500 Index experienced a bear market.  Covid-19,  the disease associated with the novel virus SARS-COV-2, may cause a U.S. recession—even if mild—which could in turn cause a bear market for U.S. stocks.  In fact, a bear market for stocks may already have begun.

Periodically there are bear markets. The important thing for a long-term value investor is to be prepared to buy the most undervalued stocks in the event of a bear market.  It’s important to be able to buy when everybody else is selling.

Let’s now look at an article by The Economist, “Covid-19 is now in 50 countries, and things will get worse.”  Link:

The article states:

Studies suggest that the number of people who have left China carrying the disease is significantly higher than would be inferred from the cases so far reported to have cropped up elsewhere, strongly suggesting that the virus’s spread has been underestimated.  Some public-health officials still talk in terms of the window for containment coming closer and closer to closing.  In reality, it seems to have slammed shut.

The article continues:

As of the morning of February 27th, stock markets had fallen by 8% in America, 7.4% in Europe and 6.2% in Asia over the past seven days.  The industries, commodities and securities that are most sensitive to global growth, cross-border commerce and densely packed public spaces got whacked particularly hard, with the prices of oil and shares in airlines, cruise-ship owners, casinos and hotel companies all tumbling.  Investors have taken refuge in assets that are perceived to be safe: yields on ten-year Treasury bonds reached an all-time low of 1.3%.  The place least hit was China, where a huge sell-off took place some time ago.  Investors, like some public-health officials, are starting to think that the epidemic there is, for now, under control… But if economic models developed for other diseases hold good, the rich world stands a distinct chance of slipping into recession as the epidemic continues.  That will bring China, and everyone else, a fresh set of problems.

The article adds:

How the virus will spread in the weeks and months to come is impossible to tell. Diseases can take peculiar routes, and dally in unlikely reservoirs, as they hitchhike around the world.  Two cases in Lebanon lead to worries about the camps in which millions of people displaced from Syria are now crowded together and exposed to the winter weather.  But regardless of exactly how the virus spreads, spread it will.  The World Health Organisation (WHO) has not yet pronounced covid-19 a pandemic—which is to say, a large outbreak of disease affecting the whole world.  But that is what it now is.

Part of the WHO’s reticence is that the P-word frightens people, paralyses decision making and suggests that there is no further possibility of containment.  It is indeed scary—not least because, ever since news of the disease first emerged from Wuhan, the overwhelming focus of attention outside China has been the need for a pandemic to be avoided.  That many thousands of deaths now seem likely, and millions possible, is a terrible thing.  But covid-19 is the kind of disease with which, in principle, the world knows how to deal.

The course of an epidemic is shaped by a variable called the reproductive rate, or R.  It represents, in effect, the number of further cases each new case will give rise to.  If R is high, the number of newly infected people climbs quickly to a peak before, for want of new people to infect, starting to fall back again… If R is low the curve rises and falls more slowly, never reaching the same heights.  With SARS-COV-2 now spread around the world, the aim of public-health policy, whether at the city, national or global scale, is to flatten the curve, spreading the infections out over time.

If R is low and fewer people are infected, that gives health-care systems more time to develop better treatments, which would mean a lower death rate.

The virus seems to be transmitted mainly through droplets that infected people cough or sneeze into the air.  So transmission can be reduced through good hygiene, physical barriers, and reducing the various ways that people mingle.  These measures are routinely used to lessen the spread of the influenza virus, which kills hundreds of thousands of people a year.  The article continues:

Influenza, like many other respiratory diseases, thrives in cold and humid air.  If covid-19 behaves the same way, spreading less as the weather gets warmer and drier, flattening the curve will bring an extra benefit.  As winter turns to spring then summer, the reproductive rate will drop of its own accord.  Dragging out the early stage of the pandemic means fewer deaths before the summer hiatus and provides time to stockpile treatments and develop new drugs and vaccines—efforts towards both of which are already under way.

Ben Cowling, an epidemiologist at the University of Hong Kong, says that the intensity of the measures countries employ to flatten the curve will depend on how deadly SARS-COV-2 turns out to be.  It is already clear that, for the majority of people who get sick, covid-19 is not too bad, especially among the young: a cough and a fever.  In older people and those with chronic health problems such as heart disease or diabetes, the infection risks becoming severe and sometimes fatal.  How often it will do so, though, is not known.

Epidemiologists estimate the fatality rate of covid-19 in the range of 0.5-1%.  This is higher than the fatality rate of 0.1% of the seasonal flu in America.  But it’s lower than the 10% fatality rate for SARS, a disease caused by another coronavirus that broke out in 2003.

However, the fatality rate depends not only on the disease itself, but also on the quality of care received.  This means that poorer countries are at more risk than richer countries.  Moroever, the economic effects of covid-19 will be worse in poorer countries.  The article says:

As the pandemic unfolds, the reproductive rate in different parts of the world will differ according both to the policies put in place and the public’s willingness to follow them.  Few countries will be able to impose controls as strict as China’s.  In South Korea the government has invoked the power to forcibly stop any public activities, such as mass protests; schools, airports and military bases are closed.  Japan is urging companies to introduce staggered working hours and virtual meetings, limiting both crowding on public transport and mingling at work.  Other developed countries are mostly not going that far, as yet.  Something that is acceptable in one country might result in barely any compliance, or even mass protests in another.

The article again:

Some hints of what may be to come can be gleaned from an economic model of an influenza pandemic created by Warwick McKibbin and Alexandra Sidorenko, both then at Australian National University, in 2006.  Covid-19 is not flu: it seems to hit people in the prime of their working life less often, which is good, but to take longer to recover from, which isn’t.  But the calculations in their model—which were being updated for covid-19 as The Economist went to press—give some sense of what may be to come…

Mr McKibbin says the moderate scenario in that paper looks closest to covid-19, which suggests a 2% hit to global growth.  That corresponds to calculations by Oxford Economics, a consultancy, which put the possible costs of covid-19 at 1.3% of GDP.  Such a burden would not be evenly spread.  Oxford Economics sees America and Europe both being tipped into recession—particularly worrying for Europe, which has little room to cut interest rates in response, and where the country currently most exposed, Italy, is already a cause for economic concern.  But poor countries would bear the biggest losses from a pandemic, relative to their economies’ size.

The article concludes:

As the world climbs the epidemic curve, biomedical researchers and public-health experts will rush to understand covid-19 better.  Their achievements are already impressive; there is realistic talk of evidence on new drugs within months and some sort of vaccine within a year.  Techniques of social distancing are already being applied.  But they will need help from populations that neither dismiss the risks nor panic.



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:

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:




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.

Saudi America

(Image: Zen Buddha Silence, by Marilyn Barbone)

February 16, 2020

Journalist Bethany McLean has written an excellent book, Saudi America: The Truth About Fracking and How It’s Changing the World.  (McLean is the co-author of The Smartest Guys in the Room: The Amazing Rise and Scandalous Fall of Enron.)

Just recently, in January 2020, U.S. crude oil production has approached 13 million barrels a day, outproducing both Saudi Arabia and Russia.  The EIA (U.S. Energy Information Administration) predicts that U.S. crude oil production will average 13.2 million barrels a day in 2020 and 13.6 million barrels a day in 2021.   McLean writes:

Few saw this coming.  This remarkable transformation in the U.S. was brought about by American entrepreneurs who figured out how to literally force open rocks often more than a mile below the surface of the earth, to produce gas, and then oil.  Those rocks—called shale, or source rock, or tight rock, and once thought to be impermeable—were opened by combining two technologies: horizontal drilling, in which the drill bit can travel well over two miles horizontally, and hydraulic fracturing, in which fluid is pumped into the earth at a high enough pressure to crack open hydrocarbon bearing rocks, while a so-called proppant, usually sand, holds the rock open a sliver of an inch so the hydrocarbons can flow.

McLean later adds:

The country’s newest hot spot, Texas’s Permian Basin, now ranks second only to Saudi Arabia’s legendary Ghawar oilfield in production per day, according to oil company ConocoPhillips.  Stretching through northern Appalachia, the Marcellus shale could be the second largest natural gas field in the world, according to geologists at Penn State.  Shale gas now accounts for over half of total U.S. production, according to the EIA, up from almost nothing a decade ago.

McLean again:

“It [shale] is monstrous,” says Will Fleckenstein, who drilled his first horizontal well in 1990 and is now a professor of petroleum engineering at the Colorado School of Mines.  In part due to ongoing improvements in technology, he says, “It is impossible to overstate the hydrocarbons that it is technically and economically feasible to produce.”

McLean continues:

CME Group executive director and senior economist Erik Norland calls fracking “one of the top five things reshaping geopolitics.”  Ever since President Franklin D. Roosevelt met the first Saudi king, Abdul-Aziz al Saud, aboard the USS Quincy in the Suez Canal in 1945, we’ve had a devil’s bargain: our protection in exchange for their oil.  The superficial analysis boils down to a simple question: If America doesn’t need Saudi oil, does America need Saudi Arabia?

Still, there are reasons to doubt the sustainability of U.S. fracking, as McLean explains:

The fracking of oil, in particular, rests on a financial foundation that is far less secure than most people realize.

The most vital ingredient in fracking isn’t chemicals, but capital, with companies relying on Wall Street’s willingness to fund them.  If it weren’t for historically low interest rates, it’s not clear there would even have been a fracking boom.

“You can make an argument that the Federal Reserve is entirely responsible for the fracking boom,” one private equity titan told me.  That view is echoed by Amir Azar, a fellow at Columbia University’s Center on Global Energy Policy.  “The real catalyst of the shale revolution was… the 2008 financial crisis and the era of unprecedentedly low interest rates it ushered in,” he wrote in a recent report.  Another investor puts it this way: “If companies were forced to live within the cash flow they produce, U.S. oil would not be a factor in the rest of the world, and would have grown at a quarter to half the rate that it has.”

Here’s the outline:


    • America’s Most Reckless Billionaire
    • The Brain Trust
    • Debt
    • Skeptics
    • Bust
    • It Changes the World, But It Ends in Tears


    • America First
    • Permania
    • Game of Thrones
    • A New Era?
    • Make America Great Again
    • Losing the Race
    • Epilogue


Part One: Shale Revolution


McLean writes about Aubrey McClendon, co-founder of Chesapeake Energy Corporation and worth over $2 billion in early 2008:

If McClendon did die broke, it wouldn’t have been out of character.  During his years as an oil and gas tycoon, he fed on risk, and was as fearless as he was reckless.  He built an empire that at one point produced more gas than any American company except ExxonMobil.  Once, when an investor asked on a conference call, “When is enough?”, McClendon answered bluntly: “I can’t get enough.”

Many think that without McClendon’s salesmanship and his astonishing ability to woo investors, the world would be a far different place today.  Stories abound about how at industry conferences, executives from oil majors like Exxon would find themselves speaking to mostly empty seats, while people literally fought for space in the room where McClendon was holding forth.

“In retrospect, it was kind of like Camelot,” says Henry Hood, Chesapeake’s former general counsel, who worked at Chesapeake initially as a consultant from 1993 until the spring of 2013.  “There was a period of time that will never be duplicated with a company that will never be duplicated.”

Forbes once called McClendon “America’s Most Reckless Billionaire,” which for many in the industry defined the man, notes McLean.  McLean adds:

You might think of McClendon as a bit of J.R. Ewing, the fictional character in the television series Dallas, mixed with Michael Milken, the junk bond king who pioneered an industry and arguably changed the world, but spent several years in prison after pleading guilty to securities fraud.

McLean continues:

McClendon, who was always immensely popular, was the president and co-valedictorian of his senior class.  He headed to Duke, where he was the rush chairman of his fraternity… “He was super competitive and aggressive,” recalls someone who knew him at Duke.  “If he had a few drinks, he’d want to wrestle.  He was big and strong and a little bit out of control.”


In college, he also met Hood, who recalls a driven if rambunctious young man.  “Aubrey was thoughtful, tall, and handsome, but incredibly clumsy,” Hood recalls.  “He always had an ink stain on his shirt from the pen in his front pocket.  We called him ‘Aubspill,’ because he was always spilling.  In basketball, he was always throwing elbows like a bull in a China shop.”  Another variation of the nickname, Hood recalls, was “‘Aubkill,’ because McClendon’s outsized competitive instinct made him dangerous in physical activities.”

Initially, McClendon was going to be an accountant.  But then he read an article in the Wall Street Journal.

“It was about two guys who had drilled a big well in the Anadarko Basin that had blown out, and it was alleged to be the biggest blowout in the history of the country,” McClendon told Rolling Stone.  “They sold their stake to Washington Gas and Light and got a $100 million check.  I thought, ‘These are two dudes who just drilled a well and it happened to hit.’  So that really piqued my interest.”

McLean observes that the geologist M. King Hubbert predicted that U.S. oil production would peak in some time between 1965 and 1970.  American oil production did peak at 9.6 million barrels a day in 1970.  Interestingly, before then, the Texas Railroad Commission controlled the international price of oil by setting production at a certain level and maintaining spare capacity.  But after U.S. production peaked, OPEC—created in 1960 by Iran, Iraq, Saudi Arabia, and Venezuela—started gaining the power to control the price of oil.

When McClendon graduated Duke in 1981, energy prices began declining.  McLean writes:

But McClendon was never one to be deterred.  He thought there was opportunity in assembling packages of drilling rights—for gas, not oil—either to be sold to bigger companies or to be drilled… In order to drill, you just have to persuade someone to give you a lease.  McClendon became what’s known in the oil and gas business as a land man, the person who negotiates the leases that allow for drilling.

That, it turned out, would make him the perfect person for the new world of fracking, which is not so much about finding the single gusher as it is about assembling the rights to drill multiple wells.  “Landmen were always the stepchild of the industry,” he later told Rolling Stone.  “Geologists and engineers were the important guys—but it dawned on me pretty early that all their fancy ideas aren’t worth very much if we don’t have a lease.  If you’ve got the lease and I don’t, you win.”

In 1983, McClendon partnered with another Oklahoman: Tom Ward.  Six years later, they formed Chesapeake Energy with a $50,000 investment.  McLean:

In some ways, they were an odd couple.  Bespectacled and balding, Ward came across as more of a typical businessman, whereas McClendon, with his flowing hair and Hollywood good looks, was the dynamo of the duo.  They divided the responsibilities, with McClendon happily playing the front man, raising money, and talking to the markets, while Ward stayed in the background running the business.  They operated out of separate buildings, with separate staffs…

Neither Ward nor McClendon were technological pioneers.  That distinction, most people agree, goes to a man named George Mitchell, who drew on research done by the government to experiment on the Barnett Shale, an area of tight rock in the Fort Worth basin of North Texas.  Using a combination of horizontal drilling and hydraulic fracturing, Mitchell’s team cracked the code for getting gas out of rock that was thought to be impermeable.

At the time, however, most companies—including ExxonMobil—and most observers ignored what Mitchell was doing, believing that it would be too expensive.  McLean continues:

McClendon, however, was the pioneer in the other essential part of the business: raising money.  “As oxygen is to life, capital is to the oil and gas business,” says Andrew Wilmot, a Dallas-based mergers and acquisitions advisor to the oil and gas industry at Purposed Ventures.  “This industry needs capital to fire on all cylinders, and the founder and father of raising capital for shale in the U.S. is Aubrey McClendon.”… “I never let Aubrey McClendon in the door for a meeting,” says an analyst who works for a big investment firm.  “Because we would have bought a ton of stock and it would not have ended well.  He was that good.”

In the early 1990s, Bear Stearns helped Chesapeake sell high-yield debt in a first-of-its-kind sort of deal.  This was no small achievement.  After all, Chesapeake didn’t have much of a track record, and there was less than zero interest in the oil and gas business from the investment community.  “I watched him convince people in these meetings,” says a banker who was there.  “He was so good, so sharp, with such an ability to draw people in.”

On February 12, 1993—a day McClendon would later describe as the best one of his career—he and Ward took Chesapeake public.  They did so despite the fact that their accounting firm, Arthur Andersen, had issued a “going concern” warning, meaning its bean-counters worried that Chesapeake might go out of business.  So McClendon and Ward simply switched accounting firms.  “Tom and I were thirty-three-year-old land men at the time, and most people didn’t think we had a clue of what we were doing, and probably in hindsight they were at least partially right,” McClendon told one interviewer in 2006.  Their IPO reduced their ownership stake to 60 percent, but both men kept for themselves as important perk, one that would play a key role in the Chesapeake story: They got the right to take a personal 2.5 percent stake in every well Chesapeake drilled.

McLean adds:

In the years following its IPO, Chesapeake was one of the best-performing stocks on Wall Street, climbing from $1.33 per share (split adjusted) to almost $27 per share.

There was an area called Austin Chalk where some highly successful wells had been drilled.  McClendon “went all in”: Chesapeake leased more than a million acres.  Unfortunately, much of this land turned out not to be productive.  Chesapeake took a $200 million charge against earnings, wiping out the previous three years of profits.  By 1998, Chesapeake stock was selling for $0.75.  McLean:

As he would do again and again, McClendon survived by borrowing yet more money to acquire more properties.  “Simply put, low prices cure low prices as consumers are motivated to consume more and producers are compelled to produce less,” he wrote in Chesapeake’s 1998 annual report.  McClendon essentially made a giant bet that gas prices would rise on their own, and he billed the properties he acquired “low risk.”  Luckily, he was right.  Within a few years, prices were soaring again, and McClendon had gotten out of a jam, for now.

Wall Street loved Chesapeake because of the fees it paid.

From 2001 to 2012, Chesapeake sold $16.4 billion in stock and $15.5 billion of debt, and paid Wall Street more than $1.1 billion in fees, according to Thomson Reuters Deals Intelligence.  McClendon was like no other client.

McClendon developed a reputation of overpaying.  There were stories of him offering ten times the amount of other offers.  But he was extremely hard-working, and would frequently send an email at 4:00 a.m.

McLean writes:

But with the sort of price increase that the market was experiencing at the time, it didn’t seem to matter what McClendon had paid.  Gas prices steadily marched upward, and by their peak in June 2008, they had more than doubled in just a few years.  Chesapeake’s stock moved in lockstep, recovering its losses from the 1990s, and more.  It hit over $65 a share in the summer of 2008, giving the company a market value of more than $35 billion.  That made McClendon’s shares worth some $2 billion.

McLean again:

To Wall Street investors, McClendon was delivering on what they wanted most: consistency and growth.  His pitch was that fracking had transformed the production of gas from a hit or miss proposition to one that operated with an on and off switch.  It was manufacturing, not wildcatting.  He became a flag waver for natural gas…

For a man steeped in the industry’s history of booms and busts, McClendon had by now convinced himself that gas prices would never fall.  In August 2008, he predicted that gas would stay in the $8 to $9 range for the foreseeable future.

In the spring and summer of 2008, Chesapeake raised $2.5 billion by selling stock, and $2 billion selling debt.  Moreover, with Chesapeake stock near an all-time high, McClendon continued to buy shares on margin.  Someone who knew him said McClendon always had to be “all in” because it was “so important to him to win.”

McLean notes that McClendon forgot that just as low prices cure low prices, so high prices cure high prices.



McLean writes about EOG Resources, which others called “the Brain Trust.”  EOG was a natural gas company.  But the CEO Mark Papa realized that natural gas prices would be low for decades.  Papa had EOG change its focus to oil production.

First, EOG began drilling for oil in the Bakken, a formation of about 200,000 square miles below parts of Montana, North Dakota, and Saskatchewan.  Then EOG began drilling for oil in the Eagle Ford shale under Texas.  McLean:

In the spring of 2010, EOG announced to a crowd of Wall Street investors at the Houston Four Seasons that the Eagle Ford contained over nine hundred million barrels of oil, enough to rival the Bakken.

McLean reports that money was pouring into the American energy business, which was a rare area of growth after the Great Recession.  Oil prices kept increasing, and no one thought they would fall again.  Soon the boom changed the U.S. economy.



McLean writes:

As gas prices began to fall in 2008, so did Chesapeake’s stock, from its peak of $70 in the summer of 2008 to $16 by October.  With that steep slide, the value of the shares he’d pledged to banks in exchange for loans also fell—and the banks called his margin loans.  Rowland, whose office was about fifty feet away from McClendon’s, recalls him walking in and saying, “Marc, they’re selling me out.”  “It was a one minute conversation,” says Rowland.  “He went from a $2 billion net worth to a negative $500 million.  There wasn’t any sweat in his eye or anything like that.  It was just the way he was.”

Indeed, from October 8 to October 10, McClendon had to sell 94 percent of his Chesapeake stock.  “I would not have wished the past month on my worst enemy,” he said in a meeting.

The Chesapeake board of directors, which was one of the highest paid in the industry, gave McClendon a $75 million bonus.  This brought McClendon’s total pay for the year to $112 million, which made him the highest paid CEO in corporate America that year.  McLean:

Underlying all of McClendon’s enterprises was a vast and tangled web of debt.  That 2.5 percent stake in the profits from Chesapeake’s wells that McClendon and Ward had kept for themselves at the IPO had come with a hitch: McClendon had to pay his share of the costs to drill the wells.  Over time, according to a series of investigative pieces done by Reuters, he quietly borrowed over $1.5  billion from various banks and private equity firms, using the well interests as collateral.  Reuters, which entitled one piece “The Lavish and Leveraged Life of Aubrey McClendon,” also reported that much of what McClendon owned, from his stake in the Oklahoma City Thunder to his wine collection to his venture capital and hedge fund investments, was also mortgaged.

As for Chesapeake, the company continued to bleed cash.  McLean:

From 2002 to the end of 2012, there was never a year in which Chesapeake reported positive free cash flow (meaning the cash it generated from operations less its capital expenditures).  Over the decade ending in 2012, Chesapeake burned through almost $30 billion.

McLean says that Chesapeake loaded up on debt and sold stock to investors in order to cover its costs.  The company also sold some of its land.  Chesapeake was going to need higher natural gas prices in order to survive.

In January 2013, Chesapeake announced that McClendon would retire.  Right after leaving, McClendon started American Energy Partners, an umbrella company for a smorgasbord of new companies.  Within only a few years, McClendon had raised $15 billion in capital and his companies employed eight hundred people.  Meanwhile, the price of gas rose, hitting $6.



Short-sellers weren’t just skeptical of McClendon.  They were skeptical of the entire industry, which seemed to consume more capital than it produced.  The famous short-seller David Einhorn made a detailed case for shorting the shale industry at the 2015 Ira W. Sohn Investment Research Conference.  McLean:

Einhorn found that from 2006 to 2014, the fracking firms had spent $80 billion more than they had received from selling oil and gas.  Even when oil was at $100 a barrel, none of them generated excess cash flow—in fact, in 2014, when oil was at $100 for part of the year, the group burned through $20 billion.

McLean continues:

A key reason for the terrible financial results is that fracked oil wells in particular show an incredibly steep decline rate.  According to an analysis by the Kansas City Federal Reserve, the average well in the Bakken declines 69 percent in its first year and more than 85 percent in its first three years, while a conventional well might decline by 10 percent a year.  One energy analyst calculated that to maintain production of 1 million barrels per day, shale requires up to 2,500 wells, while production in Iraq can do it with fewer than 100.  For a fracking operation to show growth requires huge investment each year to offset the decline from the previous years’ wells.  To Einhorn, this was clearly a vicious circle.

Furthermore, the shale revolution wouldn’t have been possible were it not for the ultra-low interest rate policy by the Federal Reserve.  McLean:

Amir Azar, a fellow at Columbia’s Center on Global Energy Policy, wrote than by 2014, the industry’s net debt exceeded $175 billion, a 250 percent increase from its 2005 level.  But interest expense increased at less than half the rate debt did, because interest rates kept falling.



In late November, 2014, at a meeting in Vienna, Saudi Arabia—led by its oil minister Ali Al-Naimi—and OPEC decided to leave production where it was rather than cut it.  The marginal cost to produce a barrel of oil in Saudi Arabia is “at most” $10, according to Al-Naimi, whereas it was as much as five times that figure for U.S. frackers to produce a barrel of oil.  Many people interpreted OPEC’s decision not to cut as an attempt to drive U.S. frackers out of business by causing the oil price to collapse.

By February 2016, a barrel of oil sold for just $26.  Soon came the reckoning:

One after another, debt-laden companies began to declare bankruptcy, with some two hundred of them eventually going bust.

Acquisitions had not worked out well:

As one investor put it: “All of the acquisitions of shale assets done by the majors and by international companies have been disasters.  The wildcatters made a lot of money, but the companies haven’t.”




Even in those dark days, McClendon remained a true believer.  Rather than back down he doubled down.  He announced deal after deal… Even those who were skeptical of him were amazed.  “Look at this guy who mortgaged it all to start a new company in the teeth of a terrible decline,” says one financier.  “If he went out, he was going to go out in a blaze of glory.”

McLean continues:

Those who know McClendon and who backed him believe he might have survived the financial hell, maybe even raised the capital for a third go round.  But he could not escape the legal hell he also found himself in.

…In the spring of 2014, a year after McClendon had left Chesapeake, the state of Michigan brought criminal charges against the firm for conspiring with other companies to rig the bids in a 2010 state auction for oil and gas rights… McClendon and the CEO of a Canadian company named Encana had divvied up the state, agreeing not to bid on leases in each other’s allocated counties, according to the charges.  “Should we throw in 50/50 together here rather than trying to bash each other’s brains out on lease buying?” McClendon once asked an Encana executive, according to evidence filed in the case.

A raft of civil lawsuits were filed alleging that Chesapeake had used a similar strategy in other hot plays.


Part Two: Saudi America


McLean writes:

Citigroup chief economist Ed Morse said that the U.S. had the potential to become the “new Middle East,” and Leonardo Maugeri, a former director at Italian energy firm Eni who became a fellow at the Harvard Kennedy School’s Belfer Center for Science and International Affairs, coined the phrase “Saudi America” when his 2012 report predicted that the U.S. could one day rival Saudi Arabia’s fabled oil production.

McLean adds:

During the Obama Administration, U.S. and European politicians began pushing for America to accelerate the granting of permits for new LNG facilities so that the U.S. could export natural gas to Europe, weakening Russia’s ability to use its energy supplies as a political weapon… Ambassadors from Hungary, Poland, Slovakia, and the Czech Republic sent a letter asking Congress to allow the faster sale of more natural gas to Europe.

McLean describes how the U.S. ban on oil exports was eventually repealed.

Studies came out from various industry-friendly organizations arguing that free trade in oil would create nearly a million jobs, add billions to the economy, and would lower the trade deficit.


…Repeal of the export ban was ultimately tucked into the sprawling $1.1 trillion year-end 2015 spending bill.




While on the campaign trail, then-candidate Donald Trump began to talk about energy independence.  Upon election, he installed one of the most energy heavy cabinets in modern history, from ExxonMobil CEO Rex Tillerson as Secretary of State; to former Oklahoma Attorney General Scott Pruitt as head of the Environmental Protection Agency, which he had sued more than a dozen times to protect the interests of energy companies; to former oil and gas consultant Ryan Zinke as Secretary of the Interior.

Einhorn and other skeptics of fracking ended up being wrong.  McLean writes:

As it turns out, the demise of fracking that seemed so inevitable wasn’t inevitable after all.  There are a few reasons why that was the case, but it all begins with the Permian—Permania, some are now calling it.

Fly into Midland, and all you see across the flat dry land are windmills and drilling rigs.  “Outside of Saudi Arabia, the whole oil story today is West Texas,” one investor tells me.

McLean explains:

What set the Permian apart from other plays is geological luck.  Its oil- and gas-bearing rocks are laid down in horizontal bands… That means that one lease can give you multiple layers of hydrocarbons, and also that you can drill more efficiently, because you only have to use one expensive rig to access multiple layers.

Furthermore, unlike the Bakken and the Marcellus, the Permian already had infrastructure in place including pipelines.

In 2010, the Permian was producing nearly a million barrels a day.  In 2017, it was producing 2.5 million barrels a day.  Today (February 2020), the Permian is producing over 4.8 million barrels a day, making it the largest oil field in the world.  See:

As for overall oil production, the EIA predicts that the U.S. will produce 13.2 million barrels a day in 2020 and 13.6 million barrels a day in 2021.  See:

This makes the U.S. the world’s biggest energy power, since Saudi Arabia produces a maximum of 11 million barrels a day and is currently producing just under 10 million barrels a day in order to move oil prices higher.  Russia produces just under 11 million barrels a day.

McLean writes:

Until recently, the history of shale drilling was that operators would watch what others did, and if someone’s new technique got more oil and gas out of the ground, then everyone else would start doing that.

But now drillers are more data-driven, looking for repeatable ways to get oil out of the ground at the lowest cost.  Horizontal wells used to be about a mile long, but now one company’s well extended almost four miles.  Moreover, while the average fracked well used 4 million pounds of sand in 2011, now the average fracked well uses 12 million pounds of sand.

McLean adds:

At the same time, other things are getting more minute and efficient.  Drillers are also executing smaller, more complex, and more frequent fractures.  These more precise fracks reduce the risk that the wells “communicate”—that one leaks into another, rendering them inoperable—so the wells can be drilled more closely together…

According to a 2016 paper by researchers at the Federal Reserve, not only are rigs drilling more wells, but each well is producing far more.  The extraction from the new wells in their first month of production has roughly tripled since 2008.  Break-even cost—the estimate of what it costs to get a barrel of oil out of the ground—has plunged.  Before the bust, it was supposedly around $70; analysts say it’s less than $50 now…


The dramatic rebound make skeptics look spectacularly wrong—no one more so than David Einhorn.

Then McLean makes an important point:

And yet, even today, it is unclear if we will look back and see fracking as the beginning of a huge and lasting shift—or if we will look back wistfully, realizing that what we thought was transformative was merely a moment in time.  Because in its current financial form, the industry is still unsustainable, still haunted by McClendon’s twin ghosts of heavy debt and lack of cash flow.  “The industry has burned up cash whether the oil price was at $100, as in 2014, or at about $50, as it was during the past three months,” one analyst calculated in mid-2017.  According to his analysis, the biggest sixty firms in aggregate had used up an average of $9 billion per quarter from mid-2012 to mid-2017.

The availability of capital, made possible by low interest rates, has been central to the success of the shale industry, notes McLean:

Wall Street’s willingness to fund money-losing shale operators is, in turn, a reflection of ultra-low interest rates.  That poses a twofold risk to shale companies.  In his paper for Columbia’s Center on Global Energy Policy, Amir Azar noted that if interest rates rose, it would wipe out a significant portion of the improvement in break-even costs.

But low interest rates haven’t just meant lower borrowing costs for debt-laden companies.  The lack of return elsewhere also led pension funds, which need to be able to pay retirees, to invest massive amounts of money with hedge funds that invest in high yield debt, like that of energy firms, and with private equity firms—which, in turn, shoveled money into shale companies, because in a world devoid of growth, shale at least was growing.  Which explains why Lambert, the portfolio manager of Nassau Re, says “Pension funds were the enablers of the U.S. energy revolution.”

McLean points out that private equity firms raised over $100 billion—almost five times the usual amount—in 2016 to invest in natural resources.  Many private equity investors have done well because other investors have been willing to invest in energy companies based upon acreage owned rather than based upon a multiple of profits.  McLean:

“I view it as a greater fool business model,” one private equity executive tells me.  “But it’s one that has worked for a long time.”

Some believe that the U.S. is squandering its shale reserves by focusing on growth-at-all-costs.  McLean:

“Our view is that there’s only five years of drilling inventory left in the core,” one prominent investor tells me.  “If I’m OPEC, I would be laughing at shale.  In five years, who cares?  It’s a crazy system, where we’re taking what is a huge gift and what should be real for many years, not five years, and wasting it…”

That said, McLean notes that some longtime skeptics believe the industry is moving in the right direction by focusing on generating positive free cash flow.



McLean writes:

OPEC wasn’t as impervious to lower prices as Ali Al-Naimi, the Bedouin shepherd-turned-oil-minister, seemed to have suggested in the fall of 2014.  True, Saudi Arabia and other OPEC members spend a lot less money to get a barrel of oil out of the ground than do U.S. frackers.  But it turns out that Al-Naimi’s measurement was flawed.  That’s because the wealthy (and not-so-wealthy) oil-rich states have long relied on their countries’ natural resources to support patronage systems, in which revenue from selling natural resources underwrites generous social programs, subsidies, and infrastructure spending.  That, in turn, has helped subdue potential political upheavals…

This is an expensive way to run a country, and so the patronage system gave rise to the notion of the fiscal break-even price, which essentially is the average oil price that an oil state needs to balance its budget each year.  And it really does all come down to oil: McKinsey noted in a December 2015 analysis that Saudi Arabia gets about 90 percent of its government revenue from oil.

McLean adds:

At the same time, a Game of Thrones was starting in Saudi Arabia.  In late 2014, King Abdullah bin Abdulaziz Al Saud, who had led Saudia Arabia for a decade, passed away, leading to the elevation of his brother, King Salman bin Abdulaziz, to the throne.  With that came the accession of the new king’s son, Mohammed bin Salman.  The new deputy crown prince, who has been nicknamed MBS, was then just thirty years old.  His father gave him unprecedented control over a huge portion of the economy, including oil.

McLean continues:

Transforming the system is exactly what Saudi Arabia is trying to do.  MBS is viewed as the architect of a breathtakingly audacious plan called Saudi Vision 2030.

Part of the plan is to increase the number of Saudis in private employment.  The funding for the plan is going to come from the recent IPO of Saudi Aramco.  The IPO valued the company at about $1.7 trillion, making Saudi Aramco the largest public company in the world.  However, the IPO only raised about $25.6 billion, roughly a third of what MBS hoped for.  That’s because the company is only listed on the Tadawul, Saudi’s stock exchange, instead of being listed in New York or London, as originally planned.  Due in part to weak oil prices, there wasn’t sufficient interest in Saudi Aramco’s IPO from global investors.  See:



McLean writes about the U.S. lowering oil imports significantly:

Limiting America’s dependence on unstable regions of the world may seem like an unalloyed positive, but the larger effects are quite murky.  “It’s a very, very difficult question as to whether it makes for a safer world or a less safe world,” says Norland… He worries that economic hardship can lead to increased terrorism and even civil war—a horror anywhere, but one that is especially horrible in countries like Angola, where a civil war than began in 1975 just ended.

McLean continues:

The shale revolution also figures into America’s often vexing relationship with China, the world’s second-largest economy.  China has overtaken the U.S. as the world’s biggest oil importer, but what’s even more astounding is that shipments of U.S. crude oil to China, which were nothing before the lifting of the export ban, hit almost $10 billion in 2017.  China is also on track to become the biggest importer of U.S. LNG… This could help reduce our trade deficit, and could be a healthy change from a world where the U.S. and China compete for scarce energy supplies.

Meanwhile, Saudi Arabia is worried that the oil for security bargain between Saudi Arabia and the U.S. is breaking down, as U.S. oil imports from Saudi Arabia continue to decline.  However, notes McLean:

There are a lot of reasons it’s in America’s interest to have a stable Middle East, whether it’s fighting terrorism, resisting the spread of nuclear weapons, protecting Israel—or keeping the global economy functioning.  Even if America doesn’t need Middle Eastern oil, its allies in Europe do, and China certainly does.  This isn’t just altruism.  In a world where over 40 percent of the S&P 500’s revenues come from outside the U.S., the American economy is dependent on the global economy.



McLean says:

What is obvious, even today, is the enormous impact of shale gas on the domestic economy.  “The U.S. has the lowest cost energy prices of any OECD nation,” noted the Energy Center’s Stephen Arbogast.  Prices are hovering at less than half of prices in much of the rest of the world.  That means the energy intensive manufacturing in the U.S. “enjoys a significant advantage versus Europe, Japan, or China, all of whom depend upon imported oil and LNG for marginal supplies.”

McLean adds:

Since 2010, over three hundred chemical industry projects worth $181 billion have been announced in the U.S., according to the American Chemistry Council, a trade group representing chemical companies.

Furthermore, cheap natural gas is gradually replacing coal:

Cheap natural gas is what’s destroying coal, not tree-hugging liberals.  It’s now cheaper to build a power plant that uses natural gas than to build one fueled by coal.  As a result, the market share for coal in power generation has fallen from 50 percent in 2005 to around 30 percent today.

McLean points out that the replacement of coal by natural gas is probably positive for the environment.  In 2017, U.S. carbon dioxide emissions hit a twenty-five years low.  However, there has been a huge rise in emissions of methane due to the natural gas supply chain.  Some scientists believe that methane is a more potent greenhouse gas than carbon dioxide.

Meanwhile, instead of trying to improve the natural gas supply chain, the Trump Administration has been busy trying to bail out the coal industry.  McLean:

In the fall of 2017, the Department of Energy, under Rick Perry, announced a plan that would in effect force regional electricity grids to purchase large amounts of coal.  The ostensible reason is to ensure a supply of fuel that can be stored and called upon in the event of a disruption—but in addition to distorting markets and likely causing an increase in energy prices, the plan ran counter to the Department’s own study, which reported that increased reliance on natural gas and renewables was not reducing the reliability of the grid.

That said, the Trump Administration is taking other actions that would increase the supply of natural gas:

Trump’s initial proposal was to unlock our “$50 trillion in untapped shale, oil, and natural gas reserves,” and to further that goal, he signed an executive order to ease regulations on offshore drilling and eventually allow more to occur, particularly in the Arctic Ocean.  His administration has also proposed allowing drilling in the Arctic National Wildlife Refuge for the first time in forty years.  The effect of all this would be to make it harder to control methane leaks, and it would also likely crater prices, thereby making the economics of drilling even less attractive than they already are.  If this comes about at a time of rising interest rates and the end of the era of cheap capital, we may soon begin talking about how the Trump Administration killed the shale revolution.



No one can predict how long it will take for the world to transition to a place where we no longer use oil and gas.  McLean writes:

Most scholars think the transition will take decades.  But there are those who say it might come much more quickly.  Most prominently, in a 2015 study, Stanford University engineering professor Mark Jacobson and colleagues have argued that it was technically feasible for all fifty states to run on clean renewable energy by 2050, with an 80 percent conversion possible by 2030.  Of course, there’s also the opposite argument.  Fatih Birol, the executive director of the International Energy Agency, a Paris-based nongovernmental organization, doesn’t think we’ll see peak demand for oil any time soon, mainly because very few countries have any sort of fuel economy standards for trucks, and the use of renewables in freight transportation lags passenger vehicles enormously.  And while passenger cars make up about 25 percent of oil demand, other modes of transportation, like shipping, aviation, and freight account for almost 30 percent… OPEC, too, projects that demand will keep increasing through 2040.

McLean says that not knowing the precise timing doesn’t mean the U.S. government, led by Trump, should pretend that the transition isn’t coming.

President Trump has proposed slashing the budget for a division of the Department of Energy called the Office of Energy Efficiency and Renewable Energy… The proposed spending cuts caused the last seven heads of the office, including three who served under Republican presidents, to write a letter to Congress.  “We are unified that cuts of this magnitude… will do serious harm to this office’s critical work and America’s energy future,” they wrote.  Trump has also imposed tariffs on foreign-made solar panels, which could decrease installation volumes in coming years; Bloomberg called the tariffs “the biggest blow to renewables yet.”

Meanwhile, China and other countries, even including Saudi Arabia, are far ahead of the U.S. when it comes to the transition to clean energy.



McLean writes:

I found that it’s a fool’s errand to make bold predictions about what’s to come, but the most honest answer I found about the future came from research firm IHS Markit.

The firm has three scenarios.  The first, called Rivalry, is the base case.  Rivalry, IHS says, means “intense competition among energy sources plus evolutionary social and technology change.  Gas loosens oil’s grip on transport demand.  Renewables become increasingly competitive with gas, coal, and nuclear in power generation.”

The second scenario, called Autonomy, is a much faster-than-expected transition away from fossil fuels.  “Revolutionary changes in market, technology, and social forces decentralize the global energy supply and demand system.”

The last scenario is called Vertigo.  Vertigo means “economic and geopolitical uncertainty drive volatility and boom-bust cycles with economic concerns slowing the transition to a less carbon-intensive economy.”

McLean adds:

The potential for vertigo helps explain why Charlie Munger, the famous investor and thinker and longtime Warren Buffett sidekick, believes that we should conserve what we have, instead of drilling frenetically.  Munger argues that for all the eventual certainty of renewables, there is still no substitute for hydrocarbons in several essential aspects of modern life, namely transportation and agriculture.

Part of why the U.S. can feed its population is because yield per acre has increased dramatically in the modern era—but that’s in large part due to pesticides, nitrogenous fertilizers and other agricultural products that are made through use of hydrocarbons.

McLean continues:

As history shows, even oil and gas executives don’t have a clue what’s going to happen next.  Charlie Munger might be right.  Or shale oil and gas might do what shale oil and gas have done since the revolution began, and surprise to the upside.  EOG might discover ways to get oil economically out of other places we never thought we could get oil economically.  Or there could be a battery breakthrough tomorrow that renders oil obsolete more quickly than anyone ever dreamed.

McLean observes that we should recognize that America’s oil and gas resources are very different.  America’s natural gas is ultra-low cost and will last at least a century.  As for oil, it’s not clear what the ultimate supply is, nor is it clear at what price oil is recoverable.

Furthermore, notes McLean, we should keep in mind the extent to which both oil and (to a lesser extent) gas drillers have been dependent upon low-cost capital, which may be less available when interest rates eventually rise.  McLean:

For the first time in perhaps forever, at least some long-term investors are aligned with conservationists, and they are trying to send a message that isn’t drill, baby, drill—but rather drill thoughtfully and profitably, so that more people benefit from America’s resources for longer, and it isn’t only executives getting a payday.

McLean adds:

In a recent letter to the firm’s clients, JP Morgan chief strategist Michael Cembalest wrote that one thing he considered critical for our future was “the ability to develop natural gas-powered vehicles and trains with lower fuel costs than gasoline- or diesel-powered counterparts, and with greater geopolitical fuel security.”

…The energy grid of the future will likely consist of mostly renewables but with the ability to rapidly add backup power from natural gas when wind, solar, and hydropower generation is low.



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:

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:




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.

Cheap, Solid Microcaps Far Outperform the S&P 500

(Image: Zen Buddha Silence, by Marilyn Barbone)

February 2, 2020

The wisest long-term investment for most investors is an S&P 500 index fund.  It’s just simple arithmetic, as Warren Buffett and Jack Bogle frequently observe:

But you can do significantly better—roughly 7% per year (on average)—by systematically investing in cheap, solid microcap stocks.  The mission of the Boole Microcap Fund is to help you do just that.

Most professional investors never consider microcaps because their assets under management are too large.  Microcaps aren’t as profitable for them.  That’s why there continues to be a compelling opportunity for savvy investors.  Because microcaps are largely ignored, many get quite cheap on occasion.

Warren Buffett earned the highest returns of his career when he could invest in microcap stocks.  Buffett says he’d do the same today if he were managing small sums:

Look at this summary of the CRSP Decile-Based Size and Return Data from 1927 to 2015:

Decile Market Cap-Weighted Returns Equal Weighted Returns Number of Firms (year-end 2015) Mean Firm Size (in millions)
1 9.29% 9.20% 173 84,864
2 10.46% 10.42% 178 16,806
3 11.08% 10.87% 180 8,661
4 11.32% 11.10% 221 4,969
5 12.00% 11.92% 205 3,151
6 11.58% 11.40% 224 2,176
7 11.92% 11.87% 300 1,427
8 12.00% 12.27% 367 868
9 11.40% 12.39% 464 429
10 12.50% 17.48% 1,298 107
9+10 11.85% 16.14% 1,762 192

(CRSP is the Center for Research in Security Prices at the University of Chicago.  You can find the data for various deciles here:

The smallest two deciles—9+10—comprise microcap stocks, which typically are stocks with market caps below $500 million.  What stands out is the equal weighted returns of the 9th and 10th size deciles from 1927 to 2015:

Microcap equal weighted returns = 16.14% per year

Large-cap equal weighted returns = ~11% per year

In practice, the annual returns from microcap stocks will be 1-2% lower because of the difficulty (due to illiquidity) of entering and exiting positions.  So we should say that an equal weighted microcap approach has returned 14% per year from 1927 to 2015, versus 11% per year for an equal weighted large-cap approach.

Still, if you can do 3% better per year than the S&P 500 index (on average)—even with only a part of your total portfolio—that really adds up after a couple of decades.



By systematically implementing a value screen—e.g., low EV/EBITDA or low P/E—to a microcap strategy, you can add 2-3% per year.



You can further boost performance by screening for improving fundamentals.  One excellent way to do this is using the Piotroski F_Score, which works best for cheap micro caps.  See:



In sum, over time, a quantitative value strategy—applied to cheap microcap stocks with improving fundamentals—has high odds of returning at least 7% (+/- 3%) more per year than an S&P 500 index fund.

If you’d like to learn more about how the Boole Fund can help you do roughly 7% better per year than the S&P 500, please call or e-mail me any time.

E-mail:  (Jason Bond)



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:

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:




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.

Ranger Energy Services (RNGR)

(Image:  Zen Buddha Silence by Marilyn Barbone.)

January 26, 2020

We continue with examples of Boole’s quantitative investment process in action.

Last week, we looked at Macro Enterprises (Canada: MCR.V):

This week, we are going to look at Ranger Energy Services (RNGR), which may be an even better opportunity.

Ranger Energy Services comes out near the top of the quantitative screen employed by the Boole Microcap Fund.  This results from four steps.

Step One

First we screen for cheapness based on five metrics.  Here are the numbers for Ranger:

    • EV/EBITDA = 2.96
    • P/E = 17.51
    • P/B = 0.53
    • P/CF = 2.20
    • P/S = 0.31

These figures—especially EV/EBITDA, P/B, and P/CF—make Ranger Energy Services one of the top ten cheapest companies out of over two thousand that we ranked.

Step Two

Next we calculate the Piotroski F-Score, which is a measure of the fundamental strength of the company.  For more on the Piostroski F-Score, see my blog post here:

Ranger has a Piotroski F-Score of 8.  (The best score possible is 9, while the worst score is 0.)  This is an excellent score.

Step Three

Then we rank the company based on low debt, high insider ownership, and shareholder yield.

We measure debt levels by looking at total liabilities (TL) to total assets (TA).  Ranger has TL/TA of 34.3%, which is fairly low.

Insider ownership is important because that means that the people running the company have interests that are aligned with the interests of other shareholders.  At Ranger, insiders own 60% of the shares.  This puts Ranger Energy Services in the top 2.2% of the more than two thousand companies we ranked according to insider ownership.

Shareholder yield is the dividend yield plus the buyback yield.  The company has no dividend.  Also, while it has bought back some shares—which is good because the shares appear quite undervalued (see Step Four)—this has been offset by the issuance and exercise of stock options.  Thus overall, the shareholder yield is zero.

Each component of the ranking has a different weight.  The overall combined ranking of Ranger Energy Services places it in the top 5 stocks on our screen, or the top 0.2% of the more than two thousand companies we ranked.

Step Four

The final step is to study the company’s financial statements, presentations, and quarterly conference calls to (i) check for non-recurring items,  hidden liabilities, and bad accounting; (ii) estimate intrinsic value—how much the business is worth—using scenarios for low, mid, and high cases.

See the company presentation:

Ranger operations are reported in three segments:

Completion & Other Services

    • Primary operations include wireline (plug & perf and pump down)
    • Well Testing, Snubbing, Fluid Hauling and Tank Rental
    • Locations in Permian and DJ Basins

Well Service Rigs & Related Services

    • Primary operations include well completion support, workovers, well maintenance and P&A
    • Related equipment rentals include power swivels, well control packages, hydraulic catwalks, pipe racks and pipe handling tools
    • Locations in the Permian, DJ, Bakken, Eagle Ford, Haynesville, Gulf Coast, SCOOP/STACK

Processing Solutions

    • Primary operations include the rental of Modular Mechanical Refrigeration Units (“MRUs”) and other natural gas processing equipment
    • Locations in the Permian, Bakken, Utica, San Joaquin and Piceance

Ranger Energy Services has a market cap of $107 million and an enterprise value of $148 million.  The company recently signed new multi-year contracts with oil majors including Chevron and Conoco.  Ranger continues to gain market share due to its high spec rigs and relatively low debt levels.  Moreover, the company continues to pay down its debt—its target debt level is $0.

Intrinsic value scenarios:

    • Low case: Ranger is probably not worth less than book value, which is $12.98 per share.  That’s about 90% higher than today’s share price of $6.83.
    • Mid case: The company can achieve EBITDA of $75 million, and is likely worth at least EV/EBITDA of 5.0.  That translates into a share price of $21.41, which is 214% higher than today’s $6.83.
    • High case: Ranger has said that in a recovery, it could do EBITDA of $100 million.  The company may easily be worth at least EV/EBITDA of 6.0.  That translates into a share price of $35.83, which is about 425% higher than today’s $6.83.

Bottom Line

Ranger Energy Services is one of the top 5 most attractive stocks out of more than two thousand microcap stocks that we ranked using our quantitative screen.  Moreover, all the intrinsic value estimates—including the low case—are far above the current stock price.  As a result, we are “trembling with greed” to accumulate this stock for the Boole Microcap Fund.


In addition to company financial statements and presentations, I used information from the following two analyses of Ranger Energy Services:

(Note: If you have trouble accessing the analysis, you can create a guest account, which is free.)



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:

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:




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.

Macro Enterprises (MCR.V)

(Image:  Zen Buddha Silence by Marilyn Barbone.)

January 19, 2020

The Boole Microcap Fund follows a quantitative investment process.  This year (2020), I am going to give examples of Boole’s investment process in action.

The first example is Macro Enterprises Inc. (Canada: MCR.V).  Macro Enterprises builds oil and natural gas pipelines, constructs energy-related infrastructure facilities, and performs maintenance and integrity work on existing pipelines.  The company operates primarily in western Canada and is headquartered in Fort St. John, British Columbia.

Macro Enterprises comes out near the top of the quantitative screen employed by the Boole Microcap Fund.  This results from four steps.

Note: All values in Canadian dollars unless otherwise noted.

Step One

First we screen for cheapness based on five metrics.  Here are the numbers for Macro Enterprises:

    • EV/EBITDA = 1.37
    • P/E = 3.72
    • P/B = 1.08
    • P/CF = 3.51
    • P/S = 0.26

These figures—especially EV/EBITDA, P/E, and P/S—make Macro Enterprises one of the top ten cheapest companies out of over two thousand that we ranked.

Step Two

Next we calculate the Piotroski F-Score, which is a measure of the fundamental strength of the company.  For more on the Piostroski F-Score, see my blog post here:

Macro Enterprises has a Piotroski F-Score of 7.  (The best score possible is 9, while the worst score is 0.)  This is a very good score.

Step Three

Then we rank the company based on low debt, high insider ownership, and shareholder yield.

Warren Buffett, arguably the greatest investor of all time, explains why low debt is important:

At rare and unpredictable intervals… credit vanishes and debt becomes financially fatal.

We measure debt levels by looking at total liabilities (TL) to total assets (TA).  Macro Enterprises has TL/TA of 27.17%, which is fairly low.

Insider ownership is important because that means that the people running the company have interests that are aligned with the interests of other shareholders.  Macro’s founder and CEO, Frank Miles, owns approximately 30%+ of the shares outstanding.  Other insiders own about 3%.  This puts Macro Enterprises in the top 7% of the more than two thousand companies we ranked according to insider ownership.

Shareholder yield is the dividend yield plus the buyback yield.  The company has no dividend.  Also, while it has bought back a modest number of shares, this has been offset by the issuance and exercise of stock options.  Thus overall, the shareholder yield is zero.

Each component of the ranking has a different weight.  The overall combined ranking of Macro Enterprises places it in the top 5 stocks on our screen, or the top 0.2% of the more than two thousand companies we ranked.

Step Four

The final step is to study the company’s financial statements, presentations, and quarterly conference calls to (i) check for non-recurring items,  hidden liabilities, and bad accounting; (ii) estimate intrinsic value—how much the business is worth—using scenarios for low, mid, and high cases.

Macro Enterprises has been in operation for 25 years.  Over that time, it has earned a reputation for safety and reliability while becoming one of the largest pipeline construction companies in western Canada.  The company has a market cap of $121 million and an enterprise value of $98 million.

Macro has built a record backlog of $870+ million in net revenue over the next few years.  That is more than 7x the company’s current market cap.  Presently the company has at least a 16% EBITDA margin.  This translates into a net profit margin of at least 11%.  That means the company will earn at least 80% of its current market cap over the next few years.  (Peak net profit margins were around 15%—at these levels, the company would earn more than 100% of its market cap over the next few years.)

If you look at the $870+ million backlog, there are two large projects.  There’s the $375 million Trans Mountain Project, of which Macro’s interest is 50%.  And there’s the $900 million Coastal GasLink Project, of which Macro’s interest is 40%.  Importantly, both of these projects are largely cost-plus—as opposed to fixed price—which greatly reduces the company’s execution risk.  Macro can be expected to add new profitable projects to its backlog.

Furthermore, Macro performs maintenance and integrity work on existing pipelines.  The company has four master service agreements with large pipeline operators to conduct such work, which is a source of recurring, higher-margin revenue.

Intrinsic value scenarios:

    • Low case: Macro is probably not worth less than book value, which is $3.61 per share.  That’s about 7% lower than today’s share price of $3.89.
    • Mid case: The company is probably worth at least EV/EBITDA of 5.0.  That translates into a share price of $10.39, which is 167% higher than today’s $3.89.
    • High case: Macro may easily be worth at least EV/EBITDA of 8.0.  That translates into a share price of $16.17, which is about 316% higher than today’s $3.89.

Bottom Line

Macro Enterprises is one of the top 5 most attractive stocks out of more than two thousand microcap stocks that we ranked using our quantitative screen.  Moreover, the mid case and high case intrinsic value estimates are far above the current stock price.  As a result, we are “trembling with greed” to buy this stock for the Boole Microcap Fund.


In addition to company financial statements and presentations, I used information from the following three analyses of Macro Enterprises:

(Note: If you have trouble accessing the analyses, you can create a guest account, which is free.)



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:

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:




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.

The Man Who Solved the Market

(Image:  Zen Buddha Silence by Marilyn Barbone.)

January 12, 2020

Gregory Zuckerman’s new book, The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution, is a terrific book.

When Zuckerman set out to write a book about Jim Simons’s Renaissance Technologies, Simons and others made it clear to Zuckerman not to expect any help from them.  Zuckerman wasn’t surprised.  He knew that Simons and team were among the most secretive traders in Wall Street history.  Zuckerman writes:

There were compelling reasons I was determined to tell Simons’s story.  A former math professor, Simons is arguably the most successful trader in the history of modern finance.  Since 1988, Renaissance’s flagship Medallion hedge fund has generated average annual returns of 66 percent… No one in the investment world comes close.  Warren Buffett, George Soros, Peter Lynch, Steve Cohen, and Ray Dalio all far short…

Zuckerman adds:

Simons’s pioneering methods have been embraced in almost every industry, and reach nearly every corner of everyday life.  He and his team were crunching statistics, turning tasks over to machines, and relying on algorithms more than three decades ago—long before these tactics were embraced in Silicon Valley, the halls of government, sports stadiums, doctors’ offices, military command centers, and pretty much everywhere else forecasting is required.

With persistence, Zuckerman ended up doing over four hundred interviews with more than thirty current and former Renaissance employees.  And he did interviews with a larger number of friends, family members, and others.  Zuckerman:

I owe deep gratitude to each individual who spent time sharing memories, observations, and insights.  Some accepted substantial personal risk to help me tell this story.  I hope I rewarded their faith.

(Jim Simons, by Gert-Martin Greuel, via Wikimedia Commons)


Part One: Money Isn’t Everything


In the winter of 1952, Jimmy Simons was fourteen years old.  He was trying to earn spending money at Breck’s garden supply near Newton, Massachusetts, where his home was.  Jimmy was absent-minded and misplaced almost everything.  So the owners asked him to sweep the floor.  A few weeks later, having finished the Christmas-time job, the owners asked Jimmy what he wanted to do.  He replied:

I want to study mathematics at MIT.

Breck’s owners burst out laughing.  How could someone so absent-minded study mathematics?  And at MIT?  Zuckerman writes:

The skepticism didn’t bother Jimmy, not even the giggles.  The teenager was filled with preternatural confidence and an unusual determination to accomplish something special, the result of supportive parents who had experienced both high hopes and deep regrets in their own lives.

Jimmy remained an only child of Marcia and Matthew Simons after Marcia endured a series of miscarriages.

A sharp intellect with an outgoing personality and subtle wit, Marcia volunteered in Jimmy’s school but never had the opportunity to work outside the home.  She funneled her dreams and passions into Jimmy, pushing him academically and assuring him that success was ahead.

Matty Simons was a sales manager for 20th Century Fox.  He loved the job.  But then his father-in-law, Peter Kantor, asked him to work at his shoe factory.  Peter promised Matty an ownership stake.  Matty felt obliged to join the family business.  Zuckerman:

Matty Simons spent years as the general manager of the shoe factory, but he never received the ownership share Peter had promised.  Later in life, Matty told his son he wished he hadn’t forgone a promising and exciting career to do what was expected of him.

Simons says the lesson he learned was to do what you like in life, not what you think you ‘should’ do.  Simons never forgot this lesson.

Even as a young kid, Simons loved to think, often about mathematics.  Zuckerman:

Unlike his parents, Jimmy was determined to focus on his own passions.  When he was eight, Dr. Kaplan, the Simons family doctor, suggested a career in medicine, saying it was the ideal profession “for a bright Jewish boy.”

Jimmy bristled.

“I want to be a mathematician or a scientist,” he replied.

The doctor tried to reason with the boy.  “Listen, you can’t make any money in mathematics.”

Jimmy said he wanted to try.

Zuckerman continues:

He loved books, frequently visiting a local library to take out four a week, many well above his grade level.  Mathematical concepts captivated him most, however.

After finishing high school in three years, the seventeen-year-old took a trip across the country with a friend.  One thing they encountered was severe poverty, which made them more sensitive to the predicaments of society’s disadvantaged.

Soon Simons enrolled at MIT.

(MIT logo, via Wikimedia Commmons)

He skipped the first year of mathematics because he had taken advanced-placement courses in high school.  Zuckerman:

Overconfident during the second semester of his freshman year, Simons registered for a graduate course in abstract algebra.  It was an outright disaster.  Simons was unable to keep up with the classmates and couldn’t understand the point of the assignments and course topics.

Simons bought a book on the subject and took it home for the summer, reading and thinking for hours at a time.  Finally, it clicked.  Simons aced subsequent algrebra classes.  Though he received a D in an upper-level calculus course in his sophomore year, the professor allowed him to enroll in the next level’s class, which discussed Stokes’ theorem, a generalization of Isaac Newton’s fundamental theorem of calculus that relates line integrals to surface integrals in three dimensions.  The young man was fascinated—a theorem involving calculus, algebra, and geometry seemed to produce simple, unexpected harmony.  Simons did so well in the class that students came to him seeking help.

“I just blossomed,” Simons says.  “It was a glorious feeling.”

Simons loved the beauty of mathematics.  Although Simons realized he wasn’t quite the best, he had an imaginative approach to problems and an instinct to focus on the kinds of problems that might lead to breakthroughs.

When Simons returned to MIT to begin his graduate studies, his advisor suggested he finish his PhD at the University of California, Berkeley, so he could work with a professor named Shiing-Shen Chern, a former math prodigy from China and a leading differential geometer and topologist.

Meanwhile, he had met an eighteen-year-old named Barbara Bluestein.  They talked a great deal and eventually decided to get engaged.  Over strong objections from her parents, Barbara decided to go with Simons to Berkeley.  The pair got married in Reno, Nevada.  Simons used the money they had left after the marriage to play poker.  He won enough to buy Barbara a black bathing suit.

Back at Berkeley:

…Simons made progress on a PhD dissertation focused on differential geometry—the study of curved, multidimensional spaces using methods from calculus, topology, and linear algebra.  Simons also spent time on a new passion: trading.  The couple had received $5,000 as a wedding gift, and Simons was eager to multiply the cash.

Simons bought a couple of stocks.  But they didn’t move and he asked a broker if they had anything “more exciting.”  The broker suggested soybeans.

Simons knew nothing about commodities or how to trade futures… but he became an eager student.  At the time, soybeans sold for $2.50 per bushel.  When the broker said Merrill Lynch’s analysts expected prices to go to three dollars or even higher, Simons’s eyes widened.  He bought two futures contracts, watched soybeans soar, and scored several thousand dollars of profits in a matter of days.

Simons was hooked.

“I was fascinated by the action and the possibility I could make money short-term,” he says.

After a respected journal published Simons’s dissertation, he won a prestigious, three-year teaching position at MIT.  However, Simons began to worry that his whole life would be research and teaching.

“Is this it?  Am I going to do this my whole life?” he asked Barbara one day at home.  “There has to be more.”

In 1963, Simons accepted a research position at Harvard.



Zuckerman writes:

In 1964, Simons quit Harvard University to join an intelligence group helping to fight the ongoing Cold War with the Soviet Union.  The group told Simons he could continue his mathematics research as he worked on government assignments.  Just as important, he doubled his previous salary and began paying off his debts.

Simons’s offer came from the Princeton, New Jersey, division of the Institute for Defense Analyses, an elite research organization that hired mathematicians from top universities to assist the National Security Agency—the United States’ largest and most secretive intelligence agency—in detecting and attacking Russian codes and ciphers.

…The IDA taught Simons how to develop mathematical models to discern and interpret patterns in seemingly meaningless data.  He began using statistical analysis and probability theory, mathematical tools that would influence his work.

Simons learned that he liked making algorithms and testing things out on a computer.  Simons became a sleuthing star.

Illustration by Stuart Miles

Also, Simons learned by seeing how the group recruited new researchers.  The recruits were identified by brainpower, creativity, and ambition, rather than for any particular expertise or background.  Simons met Lenny Baum, one of the most-accomplished code breakers.  Baum developed a saying that became the group’s credo:

“Bad ideas is good, good ideas is terrific, no ideas is terrible.”

Zuckerman notes:

The team routinely shared credit and met for champagne toasts after discovering solutions to particularly thorny problems.  Most days, researchers wandered into one another’s offices to offer assistance or lend an ear.  When staffers met each day for afternoon tea, they discussed the news, played chess, worked on puzzles, or competed at Go, the complicated Chinese board game.

Simons and his wife threw regular dinner parties at which IDA staffers became inebriated on Barbara’s rum-heavy Fish House Punch.  The group played high-stakes poker matches that lasted until the next morning, with Simons often walking away with fistfuls of his colleagues’ cash.

Photo by Gadosp

Meanwhile, Simons was making progress in his research on minimal varieties, a subfield of differential geometry in which he long had an interest.  Zuckerman:

He was hoping to discover and codify universal principles, rules, and truths, with the goal of furthering the understanding of these mathematical objects.  Albert Einstein argued that there is a natural order in the world; mathematicians like Simons can be seen as searching for evidence of that structure.  There is true beauty to their work, especially when it succeeds in revealing something about the universe’s natural order.  Often, such theories find practical applications, even many years later, while advancing our knowledge of the universe.

Eventually, a series of conversations with Frederick Almgren Jr., a professor at nearby Princeton University who had solved the problem in three dimensions, helped Simons achieve a breakthrough.  Simons created a partial differential equation of his own, which became known as the Simons equation, and used it to develop a uniform solution through six dimensions…

In 1968, Simons published “Minimal Varieties in Riemannian Manifolds,” which became a foundational paper for geometers, proved crucial in related fields, and continues to garner citations, underscoring its enduring significance.  These achievements helped establish Simons as one of the world’s preeminent geometers.

At the same time, Simons was studying the stock market in an effort to figure out how to make money.  Collaborating with Baum and two other colleagues, Simons developed a stock-trading system.  Simons and his colleagues ignored fundamental information such as earnings, dividends, and corporate news.  Instead, they searched for “macroscopic variables” that could predict the short-term behavior of the stock market.

Here’s what was really unique: [They] didn’t try to identify or predict these states using economic theory or other conventional methods, nor did the researchers seek to address why the market entered certain states.  Simons and his colleagues used mathematics to determine the set of states best fitting the observed pricing data; their model then made its bets accordingly.  The whys didn’t matter, Simons and his colleagues seemed to suggest, just the strategies to take advantage of the inferred states.

Simons and his colleagues used a mathematical tool called a hidden Markov model.

(Hidden Markov models, via Wikimedia Commons)

Also, they weren’t alone.  For instance, mathematician Ed Thorp developed an early form of computerized trading.


Simons was really good at identifying the most promising ideas of his colleagues.  Soon, he was in line to become deputy director of IDA.  Then came the Vietnam war.  Simons wrote a six-paragraph letter to The New York Times, arguing that there were better uses of the nation’s resources than the Vietnam war.  As a result, IDA fired Simons.

A friend asked Simons what his ideal job was.  Simons said he’d like to be the chair of a large math department, but he was too young and didn’t know the right people.  The friend told Simons that he had an idea.  Soon thereafter, Simons received a letter from John Toll, president of SUNY Stony Brook, a public university on Long Island.  The school had already spent five years looking for someone to chair its math department.

Toll was leading a $100 million, government-funded push to make SUNY Stony Brook the “Berkeley of the East.”

In 1968, Simons moved his family to Long Island and he began recruiting.  One person he targeted was a Cornell University mathematician named James Ax, who had won the prestigious Cole Prize in number theory.  Simons and Ax had been friendly as graduate students at Berkeley.  Simons charmed Ax into moving to Stony Brook.  Zuckerman:

Ax’s decision sent a message that Simons meant business.  As he raided other schools, Simons refined his pitch, focusing on what it might take to lure specific mathematicians.  Those who valued money got raises; those focused on personal research got lighter class loads, extra leave, generous research support, and help evading irritating administrative requirements.

Meanwhile, Simons’s marriage to Barbara was struggling.  They eventually divorced.  Barbara went on to earn a PhD in computer science at Berkeley in 1981.  Subsequently, she had a distinguished career.  Asked about her marriage to Simons, she reflected that they had gotten married when they were too young.

Now alone on Long Island, Simons sought a nanny to help when his three children were at his house.  That’s how he met Marilyn Hawrys, a twenty-two-year-old blond who later became a graduate student in economics at Stony Brook.  Simons asked her on a date, and they started seeing each other.

Around this time, Simons made a breakthrough with Shiing-Shen Chern.

On his own, Simons made a discovery related to quantifying shapes in curved, three-dimensional spaces.  He showed his work to Chern, who realized the insight could be extended to all dimensions.  In 1974, Chern and Simons published “Characteristic Forms and Geometric Invariants,” a paper that introduced Chern-Simons invariants… which proved useful in various aspects of mathematics.

In 1976, at the age of thirty-seven, Simons was awarded the American Mathematical Society’s Oswald Veblen Prize in Geometry, the highest honor in the field, for his work with Chern and his earlier research in minimal varieties.  A decade later, theoretical physicist Edward Witten and others would discover that Chern-Simons theory had applications to a range of areas in physics, including condensed matter, string theory, and supergravity.  It even became crucial to methods used by Microsoft and others in their attempts to develop quantum computers capable of solving problems vexing modern computers, such as drug development and artificial intelligence.  By 2019, tens of thousands of citations in academic papers—approximately three a day—referenced Chern-Simons theory, cementing Simons’s position in the upper echelon of mathematics and physics.

Simons was ready for a new challenge.  He had recently invested money with Charlie Freifeld, who had taken a course from Simons at Harvard.  Freifeld used econometric models to predict the prices of commodities.  Soon Simons’s investment with Freifeld had increased tenfold.  This got Simons’s excited again about the challenge of investing.

In 1978, Simons left academia to start an investment firm focused on currency trading.  (World currencies had recently been allowed to float.)  Some academics thought that Simons was squandering a rare talent.



Early summer of 1978, a few miles down the road from Stony Brook University:

Simons sat in a storefront office in the back of a dreary strip mall.  He was next to a woman’s clothing boutique, two doors down from a pizza joint, and across from the tiny, one-story Stony Brook train station.  His space, built for a retail establishment, had beige wallpaper, a single computer terminal, and spotty phone service.  From his window, Simons could barely see the aptly named Sheep Pasture Road, an indication of how quickly he had gone from broadly admired to entirely obscure.

The odds weren’t in favor of a forty-year-old mathematician embarking on his fourth career, hoping to revolutionize the centuries-old world of investing.

Simons hadn’t shown any real talent in investing.  He acknowledged that his investment with Freifeld had been “completely lucky.”

Photo by Ulrich Willmunder

They only held on to the profits because they had agreed to cash out if they ever made a large amount.  Weeks after they sold their position, sugar prices plummeted, which neither Freifeld nor Simons had predicted.  They had barely avoided disaster.  Zuckerman:

Somehow, Simons was bursting with self-confidence.  He had conquered mathematics, figured out code-breaking, and built a world-class university department.  Now he was sure he could master financial speculation, partly because he had developed a special insight into how financial markets operated.


It looks like there’s some structure here, Simons thought.

He just had to find it.

Simons decided to treat financial markets like any other chaotic system.  Just as physicists pore over vast quantities of data and build elegant models to identify laws in nature, Simons would built mathematical models to identify order in financial markets.  His approach bore similarities to the strategy he had developed years earlier at the Institute for Defense Analyses, when he and his colleagues wrote the research paper that determined that markets existed in various hidden states that could be identified with mathematical models.  Now Simons would test the approach in real life.

Simons named his company Monemetrics, combining “money” and “econometrics.”  He then began a process he knew well: hiring a team of big brains.  Zuckerman:

He did have an ideal partner in mind for his fledgling firm:  Leonard Baum, one of the coauthors of the IDA research paper and a mathematician who had spent time discerning hidden states and making short-term predictions in chaotic environments.  Simons just had to convince Baum to risk his career on Simons’s radical, unproven approach.

Baum’s parents had fled Russia for Brooklyn to escape poverty and anti-Semitism.  In high school, Baum was six feet tall and his school’s top sprinter.  He also played tennis.  Baum graduated Harvard University in 1953 and then earned a PhD in mathematics.  Zuckerman:

After joining the IDA in Princeton, Baum was even more successful breaking code than Simons, receiving credit for some of the unit’s most important, and still classified, achievements.


Balding and bearded, Baum pursued math research while juggling government assignments, just like Simons.  Over the course of several summers in the late 1960s, Baum and Lloyd Welch, an information theorist working down the hall, developed an algorithm to analyze Markov chains, which are sequences of events in which the probability of what happens next depends only on the current state, not past events.  In a Markov chain, it is impossible to predict future steps with certainty, yet one can observe the chain to make educated guesses about possible outcomes…

A hidden Markov process is one in which the chain of events is governed by unknown, underlying parameters or variables.  One sees the results of the chain but not the “states” that help explain the progression of the chain… Some investors liken financial markets, speech recognition patterns, and other complex chains of events to hidden Markov models.

The Baum-Welch algorithm provided a way to estimate probabilities and parameters within these complex sequences with little more information than the output of the processes…

Baum usually minimized the importance of his accomplishment.  Today, though, Baum’s algorithm, which allows a computer to teach itself states and probabilities, is seen as one of the twentieth century’s notable advances in machine learning, paving the way for breakthroughs affecting the lives of millions in fields from genomics to weather prediction.  Baum-Welch enabled the first effective speech recognition system and even Google’s search engine.

Zuckerman again:

Baum began working with Simons once a week.  By 1979, Baum, then forty-eight years old, was immersed in trading, just as Simons had hoped.  A top chess player in college, Baum felt he had discovered a new game to test his mental faculties.  He received a one-year leave of absence from the IDA and moved his family to Long Island and a rented, three-bedroom Victorian house lined with tall bookcases…

It didn’t take Baum long to develop an algorithm directing Monemetrics to buy currencies if they moved a certain level below their recent trend line and sell if they veered too far above it.  It was a simple piece of work, but Baum seemed on the right path, instilling confidence in Simons.

Zuckerman adds:

Baum became so certain their approach would work, and so hooked on investing, that he quit the IDA to work full-time with Simons.

Baum would receive 25 percent of the company’s profits.  Simons would test strategies in Monemetrics.  If they worked, he would implement them in a limited investment partnership he launched called Limroy, which included money from outside investors.  Simons had tried to raise $4 million and had gotten close enough that he felt ready to launch.  Zuckerman:

To make sure he and Baum were on the right track, Simons asked James Ax, his prized recruit at Stony Brook, to come by and check out their strategies.  Like Baum a year or so earlier, Ax knew little about investing and cared even less.  He immediately understood what his former colleagues were trying to accomplish, though, and became convinced they were onto something special.  Not only could Baum’s algorithm succeed in currencies, Ax argued, but similar predictive models could be developed to trade commodities, such as wheat, soybeans, and crude oil.  Hearing that, Simons persuaded Ax to leave academia, setting him up with his own trading account.  Now Simons was really excited.  He had two of the most acclaimed mathematicians working with him to unlock the secrets of the markets and enough cash to support their efforts.

One day Baum realized that Margaret Thatcher was keeping the British pound at an unsustainably low level.

Illustration by Tatiana Muslimova

Overcome with excitement, he rushed to the office to tell Simons.  They started buying as much as they could of the British pound, which shot up in value.  They then made accurate predictions for the Japanese Yen, West German deutsche mark, and Swiss franc.  The fund made tens of millions of dollars.  Zuckerman:

After racking up early currency winnings, Simons amended Limroy’s charter to allow it to trade US Treasury bond futures contracts as well as commodities.  He and Baum—who now had their own, separate investment accounts—assembled a small team to build sophisticated models that might identify profitable trades in currency, commodity, and bond markets.

Simons was having a great time, but the fun wouldn’t last.


Simons needed someone to program their computers.  He discovered Greg Hullender, a nineteen-year-old student at the California Institute of Technology.  Simons offered Hullender $9,000 a year plus a share of the firm’s profits.

Limroy proceeded to lose money for the next six months.  Simons was despondent.  At one point he told Hullender, “Sometimes I look at this and feel I’m just some guy who doesn’t really know what he’s doing.”  Zuckerman:

In the following days, Simons emerged from his funk, more determined than ever to build a high-tech trading system guided by algorithms, or step-by-step computer instructions rather than human judgment.


The technology for a fully automated system wasn’t there yet, Simons realized, but he wanted to try some more sophisticated methods.

Simons proceeded to launch a project of gathering as much past data as possible.  He got commodity, bond, and currency prices going back decades, even before World War II in some cases.  Zuckerman:

Eventually, the group developed a system that could dictate trades for various commodity, bond, and currency markets.

The system produced automated trade recommendations, a step short of automated trades, but the best they could do then.  But soon Simons and Baum lost confidence in their system.  They couldn’t understand why the system was making certain recommendations, and also they had another losing streak.

Simons and Baum drifted towards a more traditional investing approach.  They looked for undervalued investments and also tried to get the news faster than others in order to react to it before others.  They were investing about $30 million at that point.  Zuckerman writes:

Their traditional trading approach was going so well that, when the boutique next door closed, Simons rented the space and punched through the adjoining wall.  The new space was filled with offices for new hires, including an economist and others who provided expert intelligence and made their own trades, helping to boost returns.  At the same time, Simons was developing a new passion: backing promising technology companies, including an electronic dictionary company called Franklin Electronic Publishers, which developed the first hand-held computer.

In 1982, Simons changed Monemetrics’ name to Renaissance Technologies Corporation, reflecting his developing interest in these upstart companies.  Simons came to see himself as a venture capitalist as much as a trader.

Meanwhile, Baum excelled by relying on his own research and intuition:

He was making so much money trading various currencies using intuition and instinct that pursuing a systematic, “quantitative” style of trading seemed a waste of time.  Building formulas was difficult and time-consuming, and the gains figured to be steady but never spectacular.  By contrast, quickly digesting the office’s news ticker, studying newspaper articles, and analyzing geopolitical events seemed exciting and far more profitable.

Between July 1979 and March 1982, Baum made $43 million in profits, almost double his original stake from Simons.

Baum tended to hold on to his investments when he thought that a given trend would continue.  Eventually this caused a rift between Baum and Simons.  For example, in the fall of 1979, both Baum and Simons bought gold around $250 an ounce.

(Photo by Daniel Schreurs)

By January 1980, gold had soared past $700.  Simons sold his position, but Baum held on thinking the trend would continue.  Around this time, Simons learned that people were lining up to sell their physical gold—like jewelry—to take advantage of the high prices.  Simons grew concerned that the increase in supply could crush gold prices.

When he got back in the office, Simons ordered Baum to sell.  Baum refused.  Baum was sitting on more than $10 million in profits and gold had shot past $800 an ounce.  Baum was driving Simons crazy.  Finally Simons called the firm’s broker and put the phone to Baum’s ear.  Simons ordered Baum: “Tell him you’re selling.”  Baum finally capitulated.  Within months, gold shot past $865 an ounce and Baum was complaining that Simons had cost him serious money.  But just a few months later, gold was under $500.

In 1983, Federal Reserve Chair Paul Volcker predicted a decline in interest rates, and inflation appeared to be under control.  Baum purchased tens of millions of dollars of US bonds.  However, panic selling overcame the bond market in the late spring of 1984 amid a large increase in bond issuance by the Reagan administration.

Once again, Simons told Baum to lighten up, but Baum refused.  Baum also had a huge bet that the Japanese Yen would continue to appreciate.  That bet was also backfiring.  Zuckerman:

When the value of Baum’s investment positions had plummeted 40 percent, it triggered an automatic clause in his agreement with Simons, forcing Simons to sell all of Baum’s holdings and unwind their trading affiliation, a sad denouement to a decades-long relationship between the esteemed mathematicians.

Ultimately, Baum was very right about US bonds.  By then, Baum was only trading for himself.  Baum also returned to Princeton.  He was now sleeping better and he had time for mathematics.  Baum focused on prime numbers and the Riemann hypothesis.  Also, for fun, he traveled the United States and competed in Go tournaments.

Meanwhile, Simons was upset about the losses.  He considered just focusing on technology investing.  He gave clients an opportunity to withdraw, but most kept faith that Simons would figure out a way to improve results.



Simons came to the conclusion that Baum’s approach, using intellect and instinct, was not a reliable way to make money.  Simons commented: “If you make money, you feel like a genius.  If you lose, you’re a dope.”  Zuckerman:

Simons wondered if the technology was yet available to trade using mathematical models and preset algorithms, to avoid the emotional ups and downs that come with betting on markets with only intelligence and intuition.  Simons still had James Ax working for him, a mathematician who seemed perfectly suited to build a pioneering computer trading system.  Simons resolved to back Ax with ample support and resources, hoping something special would emerge.

Zuckerman continues:

In 1961, Ax earned a PhD in mathematics from the University of California, Berkeley, where he became friends with Simons, a fellow graduate student.  Ax was the first to greet Simons and his wife in the hospital after Barbara gave birth to their first child.  As a mathematics professor at Cornell University, Ax helped develop a branch of pure mathematics called number theory.  In the process, he forged a close bond with a senior, tenured academic named Simon Kochen, a mathematical logician.  Together, the professors tried to prove a famous fifty-year-old conjecture made by the famed Austrian mathematician Emil Artin, meeting immediate and enduring frustration.  To blow off steam, Ax and Kochen initiated a weekly poker game with colleagues and others in the Ithaca, New York, area.  What started as friendly get-togethers, with winning pots that rarely topped fifteen dollars, grew in intensity until the men fought over stakes reaching hundreds of dollars.


Ax spent the 1970s searching for new rivals and ways to best them.  In addition to poker, he took up golf and bowling, while emerging as one of the nation’s top backgammon players.

In 1979, Ax joined Simons.  First Ax looked at fundamentals.  Zuckerman:

Ax’s returns weren’t remarkable, so he began developing a trading system to take advantage of his math background.  Ax mined the assorted data Simons and his team had collected, crafting algorithms to predict where various currencies and commodities were headed.


Ax’s predictive models had potential, but they were quite crude.  The trove of data Simons and others had collected proved of little use, mostly because it was riddled with errors and faulty prices.  Also, Ax’s trading system wasn’t in any way automated—his trades were made by phone, twice a day, in the morning and at the end of the trading day.

Ax soon began to rely on a former professor:  Sandor Straus earned a PhD in mathematics from Berkeley in 1972 and went to teach at Stony Brook.  Straus thrived.  And he wore a long ponytail with John Lennon-style glasses.  In 1976, Straus joined Stony Brook’s computer center.  He helped Ax and others develop computer simulations.  But Straus wasn’t happy and was worried about money.  Simons offered to double Straus’s salary if he joined Monemetrics as a computer specialist.

Against the consistent advice of his father and friends, Straus eventually decided to join Monemetrics.  Straus began collecting a wide range of data.  Zuckerman:

No one had asked Straus to track down so much information.  Opening and closing prices seemed sufficient to Simons and Ax.  They didn’t even have a way to use all the data Straus was gathering, and with computer-processing power still limited, that didn’t seem likely to change.  But Straus figured he’d continue collecting the information in case it came in handy down the road.

Straus became somewhat obsessive in his quest to locate pricing data before others realized its potential value.  Straus even collected information on stock trades, just in case Simons’s team wanted it at some point in the future.  For Straus, gathering data became a matter of personal pride.

Zuckerman adds:

…No one had told Straus to worry so much about the prices, but he had transformed into a data purist, foraging and cleaning data the rest of the world cared little about.

…Some other traders were gathering and cleaning data, but no one collected as much as Strauss, who was becoming something of a data guru.

Straus’s data helped Ax improve his trading results…

Simons asked Henry Laufer, another Stony Brook mathematician, to join Monemetrics.  Laufer agreed.  Zuckerman:

Laufer created computer simulations to test whether certain strategies should be added to their trading model.  The strategies were often based on the idea that prices tend to revert after an initial move higher or lower.  Laufer would buy futures contracts if they opened at unusually low prices compared with their previous closing price, and sell if prices began the day much higher than their previous close.  Simons made his own improvements to the evolving system, while insisting that the team work together and share credit.

In 1985, Ax moved to Huntington Beach, California.  Ax and Straus established a new company: Axcom Limited.  Simons would get 25 percent of the profits, while Ax and Straus split the remaining 75 percent.  Laufer didn’t want to move west, so he returned to Stony Brook.  Zuckerman writes:

By 1986, Axcom was trading twenty-one different futures contracts, including the British pound, Swiss franc, deutsche mark, Eurodollars, and commodities including wheat, corn, and sugar.  Mathematical formulas developed by Ax and Straus generated most of the firm’s moves, though a few decisions were based on Ax’s judgment calls.  Before the beginning of trading each day, and just before the end of trading in the late afternoon, a computer program would send an electronic message to Greg Olsen, their broker at an outside firm, with an order and some simple conditions.  One example: “If wheat opens above $4.25, sell 36 contracts.”

However, Simons and the team were not finding new ways to make money, nor were they improving on their existing methods, which allowed rivals to catch up.  Zuckerman:

Eventually, Ax decided they needed to trade in a more sophisticated way.  They hadn’t tried using more-complex math to build trading formulas, partly because the computing power didn’t seem sufficient.  Now Ax thought it might be time to give it a shot.

Ax had long believed financial markets shared characteristics with Markov chains, those sequences of events in which the next event is only dependent on the current state.  In a Markov chain, each step along the way is impossible to predict with certainty, but future steps can be predicted with some degree of accuracy if one relies on a capable model…

To improve their predictive models, Ax concluded it was time to bring in someone with experience developing stochastic equations, the broader family of equations to which Markov chains belong.  Stochastic equations model dynamic processes that evolve over time and can involve a high level of uncertainty.

Soon Rene Carmona, a professor at University of California, Irvine, got a call from a friend who told him a group of mathematicians was looking for someone with his specialty—stochastic differential equations.  Simons, Ax, and Straus were interested.  Zuckerman:

Perhaps by hiring Carmona, they could develop a model that would produce a range of likely outcomes for their investments, helping to improve their performance.

Carmona was eager to lend a hand—he was consulting for a local aerospace company at the time and liked the idea of picking up extra cash working for Axcom a few days a week.  The challenge of improving the firm’s trading results also intrigued him.

“The goal was to invent a mathematical model and use it as a framework to infer some consequences and conclusions,” Carmona says.  “The name of the game is not to always be right, but to be right often enough.”

In 1987, after having made little progress, Carmona decided to spend the summer working full-time for Axcom.  Yet Carmona still couldn’t generate useful results.  Carmona soon realized that they needed regressions that could capture nonlinear relationships in market data.  Zuckerman explains:

He suggested a different approach.  Carmona’s idea was to have computers search for relationships in the data Strauss had amassed.  Perhaps they could find instances in the remote past of similar trading environments, then they could examine how prices reacted.  By identifying comparable trading situations and tracking what subsequently happened to prices, they could develop a sophisticated and accurate forecasting model capable of detecting hidden patterns.

For this approach to work, Axcom needed a lot of data, even more than Strauss and the others had collected.  To solve the problem, Strauss began to model data rather than just collect it.  In other words, to deal with gaps in historical data, he used computer models to make educated guesses as to what was missing.  They didn’t have extensive cotton pricing data from the 1940s, for example, but maybe creating the data would suffice…

Carmona suggested letting the model run the show by digesting all the various pieces of data and spitting out buy-and-sell decisions.  In a sense, he was proposing an early machine-learning system.  The model would generate predictions for various commodity prices based on complex patterns, clusters, and correlations that Carmona and the others didn’t understand themselves and couldn’t detect with the naked eye.

Elsewhere, statisticians were using similar approaches—called kernel methods—to analyze patterns in data sets.

At first, Simons couldn’t get comfortable because he couldn’t understand why the model was reaching certain conclusions.  Carmona told Simons to follow the data.  Ax urged Simons to let the computers do it.

(Illustration by Dmitry Gorelkin)


When the Axcom team started testing the approach, they quickly began to see improved results.  The firm began incorporating higher dimensional kernel regression approaches, which seemed to work best for trending models…

Simons was convinced they could do even better.  Carmona’s ideas helped, but they weren’t enough.  Simons called and visited, hoping to improve Axcom’s performance, but he mostly served as the pool operator, finding wealthy investors for the fund and keeping them happy, while attending to the various technology investments that made up about half of the $100 million assets now held by the firm.

Seeking even more mathematical firepower, Simons arranged for a well-respected academic to consult with the firm.  That move would lay the groundwork for a historic breakthrough.



Elwyn Berlekamp attended MIT.  In his senior year, Berlekamp won a prestigious math competition to become a Putnam Fellow.  While pursuing a PhD at MIT, he focused on electrical engineering, studying with Peter Elias and Claude Shannon.  One day Shannon pulled Berlekamp aside and told him it wasn’t a good time to invest in the stock market.  Berlekamp had no money, so he laughed.  Also, he thought investing was a game where rich people play and that it doesn’t do much to improve the world.

During the summers of 1960 and 1962, Berlekamp worked as a research assistant at Bell Laboratories research center in Murray Hill, New Jersey.  While there, he worked for John Larry Kelly, Jr.  Kelly was a brilliant physicist who had been a pilot in the US Navy in World War II.  Kelly smoked six packs of cigarettes a day and invented a betting system to bet on college and professional football.  Kelly invented the Kelly criterion.

The Kelly criterion can be written as follows (Zuckerman doesn’t mention this, but it’s worth noting):

    • F = p – [q/o]


    • F = Kelly criterion fraction of current capital to bet
    • o = Net odds, or dollars won per $1 bet if the bet wins (e.g., the bet may pay 5 to 1, meaning you win $5 per each $1 bet if the bet wins)
    • p = probability of winning
    • q = probability of losing = 1 – p

The Kelly criterion has a unique mathematical property: if you know the probability of winning and the net odds (payoff), then betting exactly the percentage determined by the Kelly criterion leads to the maximum long-term compounding of capital, assuming that you’re going to make a long series of bets.  Betting any percentage that is not equal to that given by the Kelly criterion will inevitably lead to lower compound growth over a long period of time.

Berlekamp finished his PhD at the University of California, Berkeley.  He became an assistant professor of electrical engineering.  Zuckerman:

Berlekamp became an expert in decoding digital information, helping NASA decipher images coming back from satellites exploring Mars, Venus, and other parts of the solar system.  Employing principles he had developed studying puzzles and games… Berlekamp cofounded a branch of mathematics called combinatorial game theory and wrote a book called Algebraic Coding Theory, a classic in the field.

By the late 1960s, the Institute for Defense Analyses (IDA) hired Berlekamp.  He met Simons, but the two didn’t hit it off, despite having both spent time at MIT, Berkeley, and IDA.

“His mathematics were different from mine,” Berlekamp says.  “And Jim had an insatiable urge to do finance and make money.  He likes action… He was always playing poker and fussing around with the markets.”

In 1973, Berlekamp became the part owner of a cryptography company.  In 1985, Eastman Kodak acquired the company, giving Berlekamp an unexpected windfall.  Berlekamp complained that the money caused challenges in his marriage.  His wife wanted a bigger house, while he wanted to travel.

While trying to figure out how to invest the money, a friend told him to look at commodities.  This led Berlekamp to contact Simons, who told him, “I have just the opportunity for you.”  Berlekamp started flying to Huntington Beach a couple of times a month to learn to trade and to see if his expertise in statistical information theory might be useful.  Zuckerman says:

For all the brainpower the team was employing, and the help they were receiving from Carmona and others, Axcom’s model usually focused on two simple and commonplace trading strategies.  Sometimes, it chased prices, or bought various commodities that were moving higher or lower on the assumption that the trend would continue.

Photo by Sergio Delle Vedove

Other times, the model wagered that a price move was petering out and would reverse: a reversion strategy.

Ax had access to more extensive pricing information than his rivals, thanks to Straus’s growing collection of clean, historic data.  Since price movements often resembled those of the past, that data enabled the firm to more accurately determine when trends were likely to continue and when they were ebbing.  Computing power had improved and become cheaper, allowing the team to produce more sophisticated trading models, including Carmona’s kernel methods—the early, machine-learning strategy that had made Simons so uncomfortable.  With those advantages, Axcom averaged annual gains of about 20 percent, topping most rivals.

Yet Simons kept asking why returns weren’t better.  Adding to the tension, their rivals were multiplying.

With the Kelly criterion in mind, Berlekamp told Ax that when the odds were higher, they should make bigger bets.  Ax said they would, but seemed noncommittal.  Zuckerman:

Berlekamp discovered other problems with Axcom’s operations.  The firm traded gold, silver, copper, and other metals, as well as hogs and other meats, and grains and other commodities.  But their buy-and-sell orders were still placed through emailed instructions to their broker, Greg Olsen, at the open and close of each trading day, and Axcom often held on to investments for weeks or even months at a time.

That’s a dangerous approach, Berlekamp argued, because markets can be volatile.  Infrequent trading precluded the firm from jumping on new opportunities as they arose and led to losses during extended downturns.  Berlekamp urged Ax to look for smaller, short-term opportunities—get in and get out.  Ax brushed him off again, this time citing the cost of doing rapid trading.  Besides, Straus’s intraday price data was riddled with inaccuracies—he hadn’t fully “cleaned” it yet—so they couldn’t create a reliable model for short-term trades.

Meanwhile, some investors didn’t have faith in Simons’s venture-capital investments.  So Simons closed it down in March 1988 and opened, with Ax, an offshore hedge fund focused solely on trading.  They named the hedge fund Medallion because each had gotten a prestigious math award.

Within six months, Medallion was struggling.  Part of it seemed to be due to the fact that Ax had gotten less focused.  He moved to an isolated spot near Malibu and he rarely came in to the office even though he was managing nearly a dozen employees.  Soon Ax purchased a spectacular home on a cliff in Pacific Palisades.

At one point in 1989, Medallion was down nearly 30 percent from the middle of the previous year.  Simons ordered Axcom to halt all trading based on the firm’s struggling, longer-term predictive signals until Ax and his team developed a plan to improve results.  In the meantime, Ax was only allowed to do short-term trading, which was just 10 percent of the fund’s activity.

Ax thought Simons’s order violated their partnership agreement.  Ax was going to sue Simons.  Technically, however, Axcom was trading for a general partnership controlled by Simons.

Then Berlekamp offered to buy Ax’s stake.  Ax agreed.  The deal left Berlekamp with a 40 percent stake in the firm, while Straus and Simons had 25 percent each.  Ax still had 10 percent.  But he effectively retired from trading.  Ax moved to San Diego.  He wrote poetry, took screenwriting classes, and wrote a science-fiction thriller.



Berlekamp moved the firm close to his home in Berkeley.

The team forged ahead, with Berlekamp focused on implementing some of the most promising recommendations Ax had ignored.  Simons, exhausted from months of bickering with Ax, supported the idea.

“Let’s bank some sure things,” Berlekamp told Simons.

In addition to the issue of cost, short-term trading interested few investors because the gains were tiny.  Zuckerman:

Berlekamp hadn’t worked on Wall Street and was inherently skeptical of long-held dogmas developed by those he suspected weren’t especially sophisticated in their analysis.  He advocated for more short-term trades.  Too many of the firm’s long-term moves had been duds, while Medallion’s short-term trades had proved its biggest winners, thanks to the work of Ax, Carmona, and others.  It made sense to try to build on that success.  Berlekamp also enjoyed some good timing—by then, most of Straus’s intraday data had been cleaned up, making it easier to develop fresh ideas for shorter-term trades.

Their goal remained the same: scrutinize historic price information to discover sequences that might repeat, under the assumption that investors will exhibit similar behavior in the future.

Berlekamp also observed that doing a higher number of shorter-term trades meant that no individual trade could hurt results.  This reduces the portfolio’s risk.  Berlekamp and his colleagues view Medallion as being like a casino.

“If you trade a lot, you only need to be right 51 percent of the time,” Berlekamp argued to a colleague.  “We need a smaller edge on each trade.”

Zuckerman writes:

Simons and his researchers didn’t believe in spending much time proposing and testing their own intuitive trade ideas.  They let the data point them to the anomalies signaling opportunity.  They also didn’t think it made sense to worry about why these phenomena existed.  All that mattered was that they happened frequently enough to include in their updated trading system, and that they could be tested to ensure they weren’t statistical flukes.

Zuckerman again:

Beyond the repeating sequences that seemed to make sense, the system Berlekamp, Laufer, and Straus developed spotted barely perceptible patterns in various markets that had no apparent explanation.  These trends and oddities sometimes happened so quickly that they were unnoticeable to most investors.  They were so faint, the team took to calling them ghosts, yet they kept reappearing with enough frequency to be worth additions to their mix of trade ideas.  Simons had come around to the view that the whys didn’t matter, just that the trades worked.


By late 1989, after about six months of work, Berlekamp and his colleagues were reasonably sure their rebuilt trading system—focused on commodity, currency, and bond markets—could prosper.


The firm implemented its new approach in late 1989 with the $27 million Simons still managed.  The results were almost immediate, startling nearly everyone in the office.  They did more trading than ever, cutting Medallion’s average holding time to just a day and a half from a week and a half, scoring profits almost every day.

At one point, Medallion had every one of its positions with the Stotler Group, a commodity-trading firm run by Karsten Mahlmann.  There were rumors Stotler was in trouble.  Berlekamp wasn’t sure what to do.  But Simons was.  He ordered Berlekamp to move all their positions to a different broker.  So he did.  Soon Stotler filed for bankruptcy.  Zuckerman:

Simons and his firm had narrowly escaped a likely death blow.


Zuckerman writes:

For much of 1990, Simons’s team could do little wrong, as if they had discovered a magic formula after a decade of fumbling around in the lab.  Rather than transact only at the open and close of trading each day, Berlekamp, Laufer, and Strauss traded at noon, as well.  Their system became mostly short-term moves, with long-term trades representing about 10 percent of activity.

One day, Axcom made more than $1 million, a first for the firm.  Simons rewarded the team with champagne, much as the IDA’s staff had passed around flutes of bubbly after discovering solutions to thorny problems.

(Photo by William MacGregor)


Medallion scored a gain of 55.9 percent in 1990, a dramatic improvement on its 4 percent loss the previous year.  The profits were especially impressive because they were over and above the hefty fees charged by the fund, which amounted to 5 percent of all assets managed and 20 percent of all gains generated by the fund.

Simons was convinced that the team had discovered a highly profitable strategy.  Nonetheless, he kept messing around with trade ideas like gold.  Zuckerman:

Berlekamp was baffled.  It was Simons who had pushed to develop a computerized trading system free of human involvement, and it was Simons who wanted to rely on the scientific method, testing overlooked anomalies rather than using crude charts or gut instinct.  Berlekamp, Laufer, and the rest of the team had worked diligently to remove humans from the trading loop as much as possible.  Now Simons was saying he had a good feeling about gold prices and wanted to tweak the system?


“We must still rely on human judgment and manual intervention to cope with a drastic, sudden change,” [Simons] explained in a letter [to clients] that month.

Simons told Berlekamp how much better the fund should be doing.  Simons was convinced the $40 million fund could achieve remarkable success.  Berlekamp didn’t think the fund could do much better than it already had.  Eventually Berlekamp—who was enjoying teaching at Berkeley more than ever—suggested to Simons that Simons buy him out.  Zuckerman:

Which is exactly what Simons did.  In December 1990, Axcom was disbanded; Simons purchased Berlekamp’s ownership interest for cash, while Strauss and Ax traded their Axcom stakes for shares in Renaissance, which began to manage the Medallion fund.  Berlekamp returned to Berkeley to teach and do full-time math research, selling his Axcom shares at a price that amounted to six times what he had paid just sixteen months earlier…

“It never occurred to me that we’d go through the roof,” Berlekamp says.



Zuckerman writes:

Like the technical traders before him, Simons practiced a form of pattern analysis and searched for telltale sequences and correlations in market data.  He hoped to have a bit more luck than investors before him by doing his trading in a more scientific manner, however.  Simons agreed with Berlekamp that technical indicators were better at guiding short-term trades than long-term investments.  But Simons hoped rigorous testing and sophisticated predictive models, based on statistical analysis rather than eyeballing price charts, might help him escape the fate of the chart adherents who had crashed and burned.

Photo by Maxkabakov

But Simons didn’t realize that others were busy crafting similar strategies, some using their own high-powered computers and mathematical algorithms.  Several of these traders already had made enormous progress, suggesting that Simons was playing catch-up.


Edward Thorp became the first modern mathematician to use quantitative strategies to invest sizable sums of money.  Thorp was an academic who had worked with Claude Shannon, the father of information theory, and embraced the proportional betting system of John Kelly, the Texas scientist who had influenced Elwyn Berlekamp.

By the late 1980s, Thorp’s fund had $300 million under management, while Simons’s Medallion fund had only $25 million.  Thorp’s fund traded warrants, options, convertible bonds, and other derivative securities.

I wrote about Thorp’s autobiography, A Man for All Markets, here:


Gerry Bamburger, a computer-science graduate of Columbia University, gave technical support to Morgan Stanley’s stock traders.  Zuckerman:

Bamburger sensed opportunity.  If the bank created a database tracking the historic prices of various paired stocks, it could profit simply by betting on the return of these price-spreads to their historic levels after block trades or other unusual activity.  Bamburger’s bosses were swayed, setting him up with half a million dollars and a small staff.  Bamburger began developing computer programs to take advantage of “temporary blips” of paired shares… By 1985, he was implementing his strategy with six or seven stocks at a time, while managing $30 million, scoring profits for Morgan Stanley.

Morgan Stanley gave Bamburger a new boss: Nunzio Tartaglia, an astrophysicist.  This prompted Bamburger to quit Morgan Stanley and join Ed Thorp’s hedge fund.

Meanwhile, Tartaglia renamed his group Automated Proprietary Trading (APT).  Zuckerman:

New hires, including a former Columbia University computer-science professor named David Shaw and mathematician Robert Frey, improved profits.  The Morgan Stanley traders became some of the first to embrace the strategy of statistical arbitrage, or stat arb…. The team’s software ranked stocks by their gains or losses over the previous weeks, for example.  APT would then sell short, or bet against, the top 10 percent of the winners within an industry while buying the bottom 10 percent of the losers on the expectation that these trading patterns would revert.

Eventually, APT suffered losses.  Morgan Stanley ended up shutting the group down.  Zuckerman comments:

It wouldn’t be clear for many years, but Morgan Stanley had squandered some of the most lucrative trading strategies in the history of finance.


One of the Medallion fund’s chief competitors was a fund run by David Shaw, who had been a part of Morgan Stanley’s APT group.  Zuckerman:

Shaw, a supercomputing expert, hired math and science PhDs who embraced his scientific approach to trading.  He also brought on whip-smart employees from different backgrounds.  English and philosophy majors were among Shaw’s favorite hires, but he also hired a chess master, stand-up comedians, published writers, an Olympic-level fencer, a trombone player, and a demolitions specialist.

Soon Shaw’s fund became successful and was managing several hundred million dollars.  Jim Simons wasn’t sure what methods Shaw was using, but he did realize that he needed to hire more help in order to catch up.  Simons contacted Donald Sussman, a hedge-fund manager who had helped Shaw launch his fund.  Perhaps Sussman would give Simons a similar boost.



Zuckerman writes about Simons meeting with Sussman:

Simons had discarded a thriving academic career to do something special in the investing world.  But, after a full decade in the business, he was managing barely more than $45 million, a mere quarter the assets of Shaw’s firm.  The meeting had import—backing from Sussman could help Renaissance hire employees, upgrade technology, and become a force on Wall Street.

Simons’s presentation to Sussman went well.  But in the end, because Sussman was the sole source of capital for D. E. Shaw—which was generating 40 percent returns per year—Sussman decided not to invest in Renaissance.  Simons approached other potential backers, but no one would invest.  Most thought it was crazy to rely on trading models generated by computers.  Many also thought Simons’s fees were too high.  Simons charged 5 percent of assets plus 20 percent of profits, whereas most hedge funds at the time charged 2 percent of assets plus 20 percent of profits.  Perhaps most importantly, Renaissance had fewer than two years of impressive performance.

Simons knew he needed to hire more help.  He turned to the mathematician Henry Laufer.  Zuckerman:

Joining Stony Brook’s math department in 1971, Laufer focused on complex variables and algebraic geometry, veering away from classical areas of complex analysis to develop insights into more contemporary problems.

In 1992, Laufer joined Simons as a full-time employee.  Zuckerman:

Laufer made an early decision that would prove extraordinarily valuable: Medallion would employ a single trading model rather than maintain various models for different investments and market conditions, a style most quantitative firms would embrace.  A collection of trading models was simpler and easier to pull off, Laufer acknowledged.  But, he argued, a single model could draw on Straus’s vast trove of pricing data, detecting correlations, opportunities, and other signals across various asset classes.  Narrow, individual models, by contrast, can suffer from too little data.

Just as important, Laufer understood that a single, stable model based on some core assumptions about how prices and markets behave would make it easier to add new investments later on.

Zuckerman continues:

Simons wondered if there might be a better way to parse their data trove.  Perhaps breaking the day up into finer segments might enable the team to dissect intraday pricing information and unearth new, undetected patterns.  Laufer… eventually decided five-minute bars were the ideal way to carve things up.  Crucially, Straus now had access to improved computer-processing power, making it easier for Laufer to compare small slices of historic data…

Laufer’s five-minute bars gave the team the ability to identify new trends, oddities, and other phenomena, or, in their parlance, nonrandom trading effects.  Straus and others conducted tests to ensure they hadn’t mined so deeply into their data that they had arrived at bogus trading strategies, but many of the new signals seemed to hold up.

Zuckerman adds:

Simons was challenging them to solve yet another vexing problem: Given  the range of possible trades they had developed and the limited amount of money that Medallion managed, how much should the bet on each trade?  And which moves should they pursue and prioritize?  Laufer began developing a computer program to identify optimal trades throughout the day, something Simons began calling his betting algorithm.  Laufer decided it would be “dynamic,” adapting on its own along the way and relying on real-time analysis to adjust the fund’s mix of holdings given the probabilities of future market moves—an early form of machine learning.

Simons, with about a dozen employees, realized that he still needed to hire more people in order to take on D. E. Shaw and other top funds.  One person Simons hired was mathematician and programmer Nick Patterson.  His coding ability gave him an advantage over other mathematicians.  He graduated from the University of Cambridge.

(Photo by Cristian Saulean)

Patterson was a strong chess player and was also a stud at poker.

(Photo by Krastiu Vasilev)

After completing graduate school, Patterson worked as a cryptologist for the British government, where he made use of Bayes’s theorem of probability.

Because computing power was expanding exponentially, Patterson thought that Simons had a chance to revolutionize investing by applying high-level math and statistics.

Patterson began working on how to reduce trading costs:

…Laufer and Patterson began developing sophisticated approaches to direct trades to various futures exchanges to reduce the market impact of each trade.  Now Medallion could better determine which investments to pursue, a huge advantage as it began trading new markets and investments.  They added German, British, and Italian bonds, then interest-rate contracts in London, and, later, futures on Nikkei Stock Average, Japanese government bonds, and more.

The fund began trading more frequently.  Having first sent orders to a team of traders five times a day, it eventually increased to sixteen times a day, reducing the impact on prices by focusing on the periods when there was the most volume.

Medallion increased 71 percent in 1994.  This was especially impressive because the Federal Reserve had hiked interest rates several times, which led to losses for many investors.  Simons still didn’t know why their models were working.  He told a colleague: “I don’t know why planets orbit the sun.  That doesn’t mean I can’t predict them.”  Zuckerman:

At the time, most academics were convinced markets were inherently efficient, suggesting that there were no predictable ways to beat the market’s return, and that the financial decision-making of individuals was largely rational.  Simons and his colleagues sensed the professors were wrong.  They believed investors are prone to cognitive biases, the kinds that lead to panics, bubbles, boom, and busts.

Simons didn’t realize it, but a new strain of economics was emerging that would validate his instincts.  In the 1970s, Israeli psychologists Amos Tversky and Daniel Kahneman had explored how individuals make decisions, demonstrating how prone most are to act irrationally.  Later, economist Richard Thaler used psychological insights to explain anomalies in investor behavior, spurring the growth of the field of behavioral economics, which explored the cognitive biases of individuals and investors.

Link to my blog post on cognitive biases:

(Illustration by Alain Lacroix)

Link to my blog post on the psychology of misjudgment:

Simons saw how successful Medallion’s updated approach was, and he remembered how difficult it was for Baum, Ax, and himself to profit from instincts.  As a result, Simons committed to not overriding the model.  Medallion’s model profited from the errors and overreactions of other investors.  Medallion’s core assumption was that investors will behave in the future like they did in the past.

Other investors at long last began noticing Medallion’s stellar results.  Some of these investors made an investment in Medallion.  By that point, Medallion was sharing its track record, but it wasn’t sharing much about how their trading system worked because it didn’t want rivals to catch on.

By the end of 1993, Medallion had $280 million under management.  Simons grew concerned the they wouldn’t be as profitable if they got too big.  So he decided, for the time being, not to let new investors into the fund.  Furthermore, Simons got even more secretive about Medallion’s trading model.  Simons even pressured his investors not to divulge any information about the fund’s operations.

Zuckerman writes:

Medallion was still on a winning streak.  It was scoring big profits trading futures contracts and managed $600 million, but Simons was convinced the hedge fund was in a serious bind.  Laufer’s models, which measured the fund’s impact on the market with surprising precision, concluded that Medallion’s returns would wane if it managed much more money.  Some commodity markets, such as grains, were just too small to handle additional buying and selling by the fund without pushing prices around.  There were also limitations to how much more Medallion could do in bigger bond and currency markets.


Simons worried his signals were getting weaker as rivals adopted similar strategies.

Yet Simons continued to search for ways to grow the fund.  There was only one way to expand:  start investing in stocks.  The trouble was that Medallion had never developed any model to invest in stocks profitably.

Simons’s son Paul battled a birth disorder—ectodermal dysplasia.  Paul worked almost every day to strengthen his body, constantly doing push-ups and pull-ups.  Paul was also an accomplished skier and endurance bicycle rider.  One day Paul was taking a fast ride through Old Field Road in Setauket, near the home where he grew up.  An elderly woman backed out of her driveway, out of nowhere, and crushed Paul, killing him instantly.  Jim Simons and Barbara were completely devastated.



Zuckerman comments:

When it came to stocks, Simons seemed well out of his depth.

Most successful stock investors at the time focused on fundamentals, poring over financial statements in order to understand things such as assets, liabilities, sales, earnings, and cash flows.

Nick Patterson began a side job he liked: recruiting talent for Renaissance.  Zuckerman:

One day, after reading in the morning paper that IBM was slashing costs, Patterson became intrigued.  He was aware of the accomplishments of the computer giant’s speech-recognition group and thought their work bore similarity to what Renaissance was doing.  In early 1993, Patterson sent separate letters to Peter Brown and Robert Mercer, deputies of the group, inviting them to visit Renaissance’s offices to discuss potential positions.

Zuckerman explains that Robert Mercer’s passion for computers had been sparked by his father Thomas:

It began the very moment Thomas showed Robert the magnetic drum and punch cards of an IBM 650, one of the earliest mass-produced computers.  After Thomas explained the computer’s inner workings to his son, the ten-year-old began creating his own programs, filling up an oversize notebook.  Bob carried that notebook around for years before he ever had access to an actual computer.

At Sandia High School and the University of New Mexico, Mercer was a member of the chess, auto, and Russian clubs.  He was low-key, but he came alive for mathematics.  In 1964, he and two classmates won top honors in a national mathematics contest.  Zuckerman:

While studying physics, chemistry, and mathematics at the University of New Mexico, Mercer got a job at a weapons laboratory at the Kirkland Air Force Base eight miles away, just so he could help program the base’s supercomputer.


…Mercer spent the summer on the lab’s mainframe computer rewriting a program that calculated electromagnetic fields generated by nuclear fusion bombs.  In time, Mercer found ways to make the program one hundred times faster, a real coup.  Mercer was energized and enthused, but his bosses didn’t seem to care about his accomplishment.  Instead of running the old computations at the new, faster speed, they instructed Mercer to run computations that were one hundred times the size.  It seemed Mercer’s revved-up speed made little difference to them, an attitude that helped mold the young man’s worldview.


He turned cynical, viewing government as arrogant and inefficient.

Mercer earned a PhD in computer science from the University of Illinois, then joined IBM and its speech-recognition group in 1972.


Peter Brown’s father Henry Brown introduced the world’s first money-market mutual fund.  Few investors showed any interest, however.  Henry worked every day except Christmas in 1972, when Peter was seventeen.  Zuckerman:

His lucky break came the next year in the form of a New York Times article about the fledgling fund.  Clients began calling, and soon Henry and his partner were managing $100 million in their Reserve Primary Fund.  The fund grew, reaching billions of dollars, but Henry resigned, in 1985, to move [with his wife] to the Brown family’s farm in a Virginia hamlet, where he raised cattle on five hundred acres.

Zuckerman continues:

Peter reserved his own ambitions for science and math.  After graduating from Harvard University with an undergraduate degree in mathematics, Brown joined a unit of Exxon that was developing ways to translate spoken language into computer text, an early form of speech-recognition technology.  Later, he’d earn a PhD in computer science from Carnegie Mellon University in Pittsburgh.

In 1984, Brown joined IBM’s speech-recognition group.  Zuckerman:

Brown, Mercer, and their fellow mathematicians and scientists, including the group’s hard-driving leader, Fred Jelinek, viewed language very differently from the traditionalists.  To them, language could be modeled like a game of chance.  At any point in a sentence, there exists a certain probability of what might come next, which can be estimated based on past, common usage…

Photo by Alexander Bogdanovich

Their goal was to feed their computers with enough data of recorded speech and written text to develop a probabilistic, statistical model capable of predicting likely word sequences based on sequences of sounds…

In mathematical terms, Brown, Mercer, and the rest of Jelinek’s team viewed sounds as the output of a sequence in which each step along the way is random, yet dependent on the previous step—a hidden Markov model.  A speech-recognition system’s job was to take a set of observed sounds, crunch the probabilities, and make the best possible guess about the “hidden” sequences of words that could have generated those sounds.  To do that, the IBM researchers employed the Baum-Welch algorithm—codeveloped by Jim Simons’s early trading partner Lenny Baum—to zero in on the various language probabilities.  Rather than manually programming in static knowledge about how language worked, they created a program that learned from data.

Mercer jumped rope to stay in shape.  He had a hyper-efficient style of communication, usually saying nothing and otherwise saying only a few words.  Brown, by contrast, was more approachable and animated.

Jelinek created a ruthless and fierce culture.  Zuckerman:

Researchers would posit ideas and colleagues would do everything they could to eviscerate them, throwing personal jabs along the way.  They’d fight it out until reaching a consensus on the merits of the suggestion… It was no-holds-barred intellectual combat.

…Brown stood out for having unusual commercial instincts, perhaps the result of his father’s influence.  Brown urged IBM to use the team’s advances to sell new products to customers, such as a credit-evaluation service, and even tried to get management to let them manage a few billion dollars of IBM’s pension-fund investments with their statistical approach, but failed to garner much support.


At one point, Brown learned of a team of computer scientists, led by a former Carnegie Mellon classmate, that was programming a computer to play chess.  He set out to convince IBM to hire the team.  One winter day, while Brown was in an IBM men’s room, he got to talking with Abe Peled, a senior IBM research executive, about the exorbitant cost of the upcoming Super Bowl’s television commercials.  Brown said he had a way to get the company exposure at a much lower cost—hire the Carnegie Mellon team and reap the resulting publicity when their machine beat a world champion in chess…

The IBM brass loved the idea and hired the team, which brought its Deep Thought program along.

Mercer was skeptical that hedge funds create good for society.  But he agreed to visit Renaissance and he was impressed that the company seemed to be about science.  Also, both Mercer and Brown were not paid much at IBM.  Brown, for his part, became more interested when he learned that Simons’s had worked with Lenny Baum, coinventor of the Baum-Welch algorithm.

Simons offered to double their salaries.  Mercer and Brown joined Renaissance in 1993.



Brown suggested that Renaissance interview David Magerman, whom Brown knew from IBM.  Magerman’s specialty was programming, which Renaissance needed.  Soon Renaissance hired Magerman.

Magerman had a difficult upbringing, growing up without much money.  His father was a math whiz who’d never been able to develop his talents.  He took it out on his son.  Magerman was left with a desire to earn praise from people in power, some of whom Magerman saw as father figures.  He also seemed to pick fights unnecessarily.  Zuckerman:

“I needed to right wrongs and fight for justice, even if I was turning molehills into mountains,” Magerman acknowledges.  “I clearly had a messiah complex.”

Magerman studied mathematics and computer science at the University of Pennsylvania.  He excelled.  Zuckerman:

At Stanford University, Magerman’s doctoral thesis tackled the exact topic Brown, Mercer, and other IBM researchers were struggling with: how computers could analyze and translate language using statistics and probability.

Although Simons had initially had Brown and Mercer working in different areas of Renaissance, the pair had been working together in their spare time on how to fix Renaissance’s stock-trading system.  Soon Mercer figured out the key problem.  Simons let Mercer join Brown in the stock-research area.  Zuckerman:

The Brown-Mercer reunion represented a new chapter in an unusual partnership between two scientists with distinct personalities who worked remarkably well together.  Brown was blunt, argumentative, persistent, loud, and full of energy.  Mercer conserved his words and rarely betrayed emotion, as if he was playing a never-ending game of poker.  The pair worked, though, yin with yang.


…[While at IBM, they] developed a certain work style—Brown would quickly write drafts of their research and then pass them to Mercer, a much better writer, who would begin slow and deliberate rewrites.

Brown and Mercer threw themselves into their new assignment [to revamp Renaissance’s stock-trading model.]  They worked late into the evening and even went home together; during the week they shared a living space in the attic of a local elderly woman’s home, returning to their families on weekends.  Over time, Brown and Mercer discovered methods to improve Simons’s stock-trading system.


[Brown and Mercer] decided to program the necessary limitations and qualifications into a single trading system that could automatically handle all potential complications.  Since Brown and Mercer were computer scientists, and they had spent years developing large-scale software projects at IBM and elsewhere, they had the coding chops to build a single automated system for trading stocks.


Brown and Mercer treated their challenge as a math problem, just as they had with language recognition at IBM.  Their inputs were the fund’s trading costs, its various leverages, risk parameters, and assorted other limitations and requirements.  Given all those factors, they built the system to solve and construct an ideal portfolio, making optimal decisions, all day long, to maximize returns.

Zuckerman writes:

The beauty of the approach was that, by combining all their trading signals and portfolio requirements into a single, monolithic model, Renaissance could easily test and add new signals, instantly knowing if the gains from a potential new strategy were likely to top its costs.  They also made their system adaptive, or capable of learning and adjusting on its own…

Photo by Weerapat Wattanapichayakul

If the model’s recommended trades weren’t executed, for whatever reason, it self-corrected, automatically searching for buy-or-sell orders to nudge the portfolio back where it needed to be… The system repeated on a loop several times an hour, conducting an optimization process that weighed thousands of potential trades before issuing electronic trade instructions.  Rivals didn’t have self-improving models; Renaissance now had a secret weapon, one that would prove crucial to the fund’s future success.

Eventually, Brown and Mercer developed an elaborate stock-trading system that featured a half million lines of code, compared to tens of thousands of lines in [the] old system.  The new system incorporated all necessary restrictions and requirements; in many ways, it was just that kind of automated trading system Simons had dreamed of years earlier.  Because [the] fund’s stock trades were now less sensitive to the market’s fluctuations, it began holding on to shares a bit longer, two days or so, on average.

…[The model] continued to identify enough winning trades to make serious money, usually by wagering on reversions after stocks got out of whack.  Over the years, Renaissance would add twists to this bedrock strategy, but, for more than a decade, those would just be second order complements to the firm’s core reversion-to-the-mean predictive signals.

However, there were problems early on.  Zuckerman:

It soon became clear that the new stock-trading system couldn’t handle much money, undermining Simons’s original purpose in pushing into equities.  Renaissance placed a puny $35 million in stocks; when more money was traded, the gains dissipated… Even worse, Brown and Mercer couldn’t figure out why their system was running into so many problems.

Brown and Mercer began putting together the old team from IBM, including Magerman.

Simons gave Brown and Mercer six months to get the fund’s trading system working.

Magerman started spending all his free time trying to fix the trading system.  He basically lived at the office.  Eventually he identified two errors and was able to fix them.  Magerman told Brown, who didn’t seem too excited.  Magerman then showed Mercer, who had written all the code.  Mercer checked Magerman’s work and then told him that he was right.

Brown and Mercer restarted the system and profits immediately started coming in.



Renaissance’s trading system gained 21 percent in 1997, lower than the 32, 38, and 71 percent gained in 1996, 1995, and 1994.  The system still had issues.

Simons drew on his experiences with the IDA and also from when he managed talented mathematicians at Stony Brook.  One lesson Simons had learned was the importance of having researchers work together.  Zuckerman:

…Medallion would have a single, monolithic trading system.  All staffers enjoyed full access to each line of the source code underpinning their moneymaking algorithms, all of it readable in cleartext on the firm’s internal network.  There would be no corners of the code accessible only to top executives; anyone could make experimental modifications to improve the trading system.  Simons hoped his researchers would swap ideas, rather than embrace private projects…

Simons created a culture of unusual openness.  Staffers wandered into colleagues’ offices offering suggestions and initiating collaborations.  When they ran into frustrations, the scientists tended to share their work and ask for help, rather than move on to new projects, ensuring the promising ideas weren’t “wasted,” as Simons put it.  Groups met regularly, discussing intimate details of their progress and fielding probing questions from Simons… Once a year, Simons paid to bring employees and their spouses to exotic vacation locales, strengthening the camaraderie.

Furthermore, peer pressure was used.  Employees were often working on presentations, and felt the need to try to impress one another.

Simons gave all employees the chance to share in Renaissance’s profits based on clear and transparent formulas.  Zuckerman notes:

Simons began sharing equity, handing a 10 percent stake in the firm to Laufer and, later, giving sizable slices to Brown, Mercer, and Mark Silber, who was now the firm’s chief financial officer, and others, steps that reduced Simons’s ownership to just over 50 percent.  Other top-performing employees could buy shares, which represented equity in the firm.  Staffers also could invest in Medallion, perhaps the biggest perk of them all.

Hiring wasn’t easy.  To attract scientists and mathematicians, Renaissance employees would emphasize certain positive parts of their job.  Solving puzzles was fun.  There was camaraderie.  And things moved at a fast pace.  Zuckerman:

“You have money in the bank or not, at the end of the day,” Mercer told science writer Sharon McGrayne.  “You don’t have to wonder if you succeeded… it’s just a very satisfying thing.”

Zuckerman continues:

By 1997, Medallion’s staffers had settled on a three-step process to discover statistically significant moneymaking strategies, or what they called their trading signals.  Identify anomalous patterns in historic pricing data; make sure the anomalies were statistically significant, consistent over time, and nonrandom; and see if the identified pricing behavior could be explained in a reasonable way.


…more than half of the trading signals Simons’s team was discovering were nonintuitive, or those they couldn’t fully understand.  Most quant firms ignore signals if they can’t develop a reasonable hypothesis to explain them, but Simons and his colleagues never liked spending too much time searching for the causes of market phenomena.  If their signals met various measures of statistical strength, they were comfortable wagering on them.  They only steered clear of the most preposterous ideas.

Zuckerman adds:

It’s not that they wanted trades that didn’t make any sense; it’s just that these were the statistically valid strategies they were finding.  Recurring patterns without apparent logic to explain them had an added bonus: They were less likely to be discovered and adopted by rivals, most of whom wouldn’t touch these kinds of trades.

The danger was that unexplainable signals could be simple coincidences.  Zuckerman:

Often, the Renaissance researchers’ solution was to place such head-scratching signals in their trading system, but to limit the money allocated to them, at least at first, as they worked to develop an understanding of why the anomalies appeared.  Over time, they frequently discovered reasonable explanations, giving Medallion a leg up on firms that had dismissed the phenomena.  They ultimately settled on a mix of sensible signals, surprising trades with strong statistical results, and a few bizarre signals so reliable they couldn’t be ignored.

Zuckerman again:

By then, Medallion increasingly was relying on strategies that its system taught itself, a form of machine learning.  The computers, fed with enough data, were trained to spit out their own answers.  A consistent winner, for example, might automatically receive more cash, without anyone approving the shift or even being aware of it.

Photo by SJMPhotos

Simons developed enough enthusiasm and confidence in the future of Renaissance that he moved the firm to a new office.  The compound was wood and glass.  Each office had a view of the woods.  There was a gym, lighted tennis courts, a library with a fireplace, and a large auditorium where Simons hosted biweekly seminars from visiting scholars (usually having little to do with finance).  The cafeteria and common areas were large, with plenty of whiteboards, so that staffers could meet, discuss, and debate.

Zuckerman describes Brown and Mercer:

Intense and energetic, Brown hustled from meeting to meeting, riding a unicycle through the halls and almost running over colleagues.  Brown worked much of the night on a computer near the Murphy bed in his office, grabbing a nap when he tired.


Analytical and unemotional, Mercer was a natural sedative for his jittery partner.  Mercer worked hard, but he liked to go home around six p.m.

One important factor of Renaissance’s success, as Zuckerman explains:

If a strategy wasn’t working, or when market volatility surged, Renaissance’s system tended to automatically reduce positions and risk.

This contrasted with many other hedge funds, including Long-Term Capital Management (LTCM), which tended to hold or to increase positions that had gone against them in the belief that they would ultimately be proved right.  In the summer of 1998, LTCM blew up when its positions went against the fund.  Traders at LTCM were more right than wrong, but because they had a great deal of leverage and because they maintained or increased their bets, they simply didn’t have enough capital to survive.  Renaissance wouldn’t make this mistake because, as noted, the fund typically reduced positions and risk if a strategy wasn’t working or if market volatility picked up.

Despite all the improvements, Renaissance’s stock-trading system still only generated about 10 percent of the fund’s profits.

In March 2000, Medallion encountered major difficulties.  Zuckerman:

It wasn’t just the mounting losses that had everyone concerned—it was the uncertainty over why things were so bad.  The Medallion portfolio held commodities, currencies, and bond futures, and its stock portfolio was largely composed of offsetting positions aimed at sidestepping broad market moves.  The losses shouldn’t be happening.  But because so many of the system’s trading signals had developed on their own through a form of machine learning, it was hard to pinpoint the exact cause of the problems or when they might ebb; the machines seemed out of control.

Brown was freaking out and barely sleeping.  Magerman felt nauseous.  Mercer tried to keep his composure.  But many were very worried.

Simons urged his team to stick with the model.  After more all-nighters, a few researchers thought they figured out the problem.  If certain stocks rallied in the preceding weeks, Medallion’s system would buy them on the expectation that the rally would continue.  But as the market sold off, this momentum strategy wasn’t working at all.  So the team put a halt to the fund’s momentum strategy.  Immediately, Medallion began making money again.  Zuckerman comments:

By the fall of 2000, word of Medallion’s success was starting to leak out.  That year, Medallion soared 99 percent, even after it charged clients 20 percent of their gains and 5 percent of the money invested with Simons.  Over the previous decade, Medallion and its 140 employees had enjoyed a better performance than funds managed by George Soros, Julian Robertson, Paul Tudor Jones, and other investing giants.  Just as impressive, Medallion had recorded a Sharpe ratio of 2.5 in its most recent five-year period, suggesting the fund’s gains came with low volatility and risk compared with those of many competitors.


Part Two: Money Changes Everything


Zuckerman writes:

Something unusual was going on at Jim Simons’s hedge fund in 2001.

Profits were piling up as Renaissance began digesting new kinds of information.  The team collected every trade order, including those that hadn’t been completed, along with annual and quarterly earnings reports, records of stock trades by corporate executives, government reports, and economic predictions and papers.


Soon, researchers were tracking newspaper and newswire stories, internet posts, and more obscure data—such as offshore insurance claims—racing to get their hands on pretty much any information that could be quantified and scrutinized for its predictive value.  The Medallion fund became something of a data sponge, soaking up a terabyte, or one trillion bytes, of information annually, buying expensive disc drives and processors to digest, store, and analyze it all, looking for reliable patterns.

“There’s no data like more data,” Mercer told a colleague, an expression that became the firm’s hokey mantra.

Renaissance’s goal was to predict the price of a stock or other investment “at every point in the future,” Mercer later explained.  “We want to know in three seconds, three days, three weeks, and three months.”

Zuckerman again:

It became clear to Mercer and others that trading stocks bore similarities to speech recognition, which was part of why Renaissance continued to raid IBM’s computational linguistics team.  In both endeavors, the goal was to create a model capable of digesting uncertain jumbles of information and generating reliable guesses about what might come next—while ignoring traditionalists who employed analysis that wasn’t nearly as data driven.


…After soaring 98.5 percent in 2000, the Medallion fund rose 33 percent in 2001.  By comparison, the S&P 500, the commonly used barometer of the stock market, managed a measly average gain of 0.2 percent over those two years, while rival hedge funds gained 7.3 percent.

Zuckerman adds:

Investment professionals generally judge a portfolio’s risk by its Sharpe ratio, which measures returns in relation to volatility; the higher one’s Sharpe, the better.  For most of the 1990s, Medallion had a strong Sharpe ratio of about 2.0, double the level of the S&P 500.  But adding foreign-market algorithms and improving Medallion’s trading techniques sent its Sharpe soaring to about 6.0 in early 2003, about twice the ratio of the largest quant firms and a figure suggesting there was nearly no risk of the fund losing money over a whole year.

(Holy grail, Photo by Charon)

Simons’s team appeared to have discovered something of a holy grail in investing: enormous returns from a diversified portfolio generating relatively little volatility and correlation to the overall market.  In the past, a few others had developed investment vehicles with similar characteristics.  They usually had puny portfolios, however.  No one had achieved what Simons and his team had—a portfolio as big as $5 billion delivering this kind of astonishing performance.

Zuckerman later says:

Brown and Mercer’s staffers often spent the night programming their computers, competing to see who could stay in the office the longest, then rushing back in the morning to see how effective their changes had been.  If Brown was going to push himself all day and sleep by his computer keyboard at night, his underlings felt the need to keep up…


By 2003, the profits of Brown and Mercer’s stock-trading group were twice those of Laufer’s futures team, a remarkable shift in just a few years.  Rewarding his ascending stars, Simons announced that Brown and Mercer would become executive vice presidents of the entire firm, co-managing all of Renaissance’s trading, research, and technical activities.


In 2002, Simons increased Medallion’s investor fees to 36 percent of each year’s profits, raising hackles among some clients.  A bit later, the firm boosted the fees to 44 percent.  Then, in early 2003, Simons began kicking all his investors out of the fund.  Simons had worried that performance would ebb if Medallion grew too big, and he preferred that he and his employees kept all the gains.  But some investors had stuck with Medallion through difficult periods and were crushed.

Nick Simons, Jim Simons’s third-eldest son shared his father’s passions for hiking and adventure.  He was taking a trip around the world before he was going to learn organic chemistry and apply for medical school.  A week before he was scheduled to come home, he went freediving near Amed, a coastal strip of fishing villages in eastern Bali.  While freediving, one person would be down, and the other up, in order to keep track of one another.  Nick and a friend thus took turns.  At one point, Nick’s friend had to go ashore to unfog his mask.  When he got back out, he found Nick’s body near the bottom.  They weren’t able to resuscitate him.  Nick had drowned.

When Jim and Marilyn found out, they were inconsolable.  Zuckerman:

In September, Jim, Marilyn, and other family members traveled to Nepal for the first time, joining some of Nick’s friends in searching for a way to continue Nick’s legacy.  Nick had been drawn to Kathmandu and had an interest in medicine, so they funded a maternity ward at a hospital in the city.  Later, Jim and Marilyn would start the Nick Simons Institute, which offers healthcare assistance to those living in Nepal’s rural areas, most of whom don’t have basic emergency services.



Medallion paid out its profits to investors—mostly its own employees by now—once per year.  That helped keep the fund small enough to maintain the same high level of profitability.  Simons, Henry Laufer, and others were sure that if the fund got much bigger, its profitability would suffer.

As a result, there were some profitable investments—mostly of a longer-term nature—that the fund couldn’t implement.  So Simons thought that they should launch a second fund.  This fund couldn’t be as profitable as Medallion; however, it could manage a lot more money than Medallion, giving outside investors a chance to invest with Renaissance.

One reason Simons wanted to launch the new fund was to give his scientists and mathematicians a new challenge.  Many of them now had more money than they ever imagined, so keeping them challenged and energized was something Simons worried about.

Photo by Feng Yu


His researchers settled on one that would trade with little human intervention, like Medallion, yet would hold investments a month or even longer.  It would incorporate some of Renaissance’s usual tactics, such as finding correlations and patterns in prices, but would add other, more fundamental strategies, including buying inexpensive shares based on price-earnings ratios, balance-sheet data, and other information.

After thorough testing, the scientists determined the new hedge fund could beat the stock market by a few percentage points each year, while generating lower volatility than the overall market.  It could produce the kinds of steady returns that hold special appeal for pension funds and other large institutions.  Even better, the prospective fund could score those returns even if it managed as much as $100 billion, they calculated, an amount that would make it history’s largest hedge fund.

The new fund was called Renaissance Institutional Equities Fund, or RIEF.  The firm emphasized to investors that the new fund wouldn’t resemble Medallion.  Yet many investors ignored the warning, thinking it was the same scientists and the same research.  Soon RIEF had $14 billion under management.  Zuckerman:

Other than making the occasional slipup, Simons was an effective salesman, a world-class mathematician with a rare ability to connect with those who couldn’t do stochastic differential equations.  Simons told entertaining stories, had a dry sense of humor, and held interests far afield from science and moneymaking.  He also demonstrated unusual loyalty and concern for others, qualities the investors may have sensed.

Zuckerman adds:

By the spring of 2007, it was getting hard to keep investors away.  Thirty-five billion dollars had been ploughed into RIEF, making it one of the world’s largest hedge funds… Simons made plans for other new funds, initiating work on the Renaissance Institutional Futures Fund, RIFF, to trade futures contracts on bonds, currencies, and other assets in a long-term style.  A new batch of scientists was hired, while staffers from other parts of the company lent a hand, fulfilling Simons’s goal of energizing and unifying staffers.

During the summer of 2007, a few quant funds were in trouble.  There was selling that seemed to be impacting all the quant funds, despite the fact that these funds thought their investment strategy was unique.  The downturn became known as the “quant quake.”

Medallion was approaching the point where it could face a margin call.  During the technology-stock meltdown in 2000, Brown hadn’t known what to do.  But this time, he knew: Medallion should maintain, and maybe even increase, its bets.  Mercer agreed with Brown.  So did Henry Laufer, who said: “Trust the models—let them run.”  Simons, however, disagreed.  Zuckerman:

Simons shook his head.  He didn’t know if his firm could survive much more pain.  He was scared.  If losses grew, and they couldn’t come up with enough collateral, the banks would sell Medallion’s positions and suffer their own huge losses.  If that happened, no one would deal with Simons’s fund again.  It would be a likely death blow, even if Renaissance suffered smaller financial losses than its bank lenders.

Medallion needed to sell, not buy, he told his colleagues.

“Our job is to survive,” Simons said.  “If we’re wrong, we can always add [positions] later.”

Brown seemed shocked by what he was hearing.  He had absolute faith in the algorithms he and his fellow scientists had developed…

Medallion began reducing its positions.  Zuckerman again:

Some rank-and-file senior scientists were upset—not so much by the losses, but because Simons had interfered with the trading system and reduced positions…

“You’re dead wrong,” a senior researcher emailed Simons.

“You believe in the system, or you don’t,” another scientist said, with some disgust.

But Simons continued to oversee a reduction in Medallion’s positions.  He wondered aloud how far the selloff could go, and he wanted to make sure Medallion survived.

Soon Simons ordered his firm to stop selling.  Their positions seemed to be stabilizing.  Some Renaissance scientists complained that their gains would have been larger if they had maintained their positions rather than lightening up.  But Simons claimed that he would make the same decision again.  Remarkably, by the end of 2007, Medallion had gained 86 percent.


Zuckerman writes about 2008:

…[The] Medallion fund thrived in the chaos, soaring 82 percent that year, helping Simons make over $2 billion in personal profits.  The enormous gains sparked a call from a House of Representatives committee asking Simons to testify as part of its investigation into the causes of the financial collapse.  Simons prepped diligently with his public-relations advisor Jonathan Gasthalter.  With fellow hedge-fund managers George Soros to his right and John Paulson on his left, Simons told Congress that he would back a push to force hedge funds to share information with regulators and that he supported higher taxes for hedge-fund managers.

Simons was something of an afterthought, however, both at the hearings and in the finance industry itself.  All eyes were on Paulson, Soros, and a few other investors who, unlike Simons, had successfully anticipated the financial meltdown.  They did it with old-fashioned investment research, a reminder of the enduring potential and appeal of those traditional methods.

Medallion was now managing $10 billion, and had averaged 45 percent returns—net of fees—since 1988.  Simons had created the best long-term track record in the world.  Now, at age seventy-two, Simons decided to turn the reins over to Brown and Mercer.




Jim Simons liked making money.  He enjoyed spending it, too.

Stepping down from Renaissance gave Simons—who, by then, was worth about $11 billion—more time on his 220-foot yacht, Archimedes.  Named for the Greek mathematician and inventor, the $100 million vessel featured a formal dining room that sat twenty, a wood-burning fireplace, a spacious Jacuzzi, and a grand piano.  Sometimes, Simons flew friends on his Gulfstream G450 to a foreign location, where they’d join Jim and Marilyn on the super-yacht.

Zuckerman continues:

Years earlier, Marilyn had carved out space in her dressing room to launch a family foundation.  Over time, she and Jim gave over $300 million to Stony Brook University, among other institutions.  As Simons edged away from Renaissance, he became more personally involved in their philanthropy.  More than anything, Simons relished tackling big problems.  Soon, he was working with Marilyn to target two areas in dire need of solutions: autism research and mathematics education.

In 2004, Simons helped launch Math for America, a nonprofit focused on math education and supporting outstanding teachers with annual stipends of $15,000.

Simons remained Renaissance’s chairman.  At first, the transition was difficult.  But soon Simons found his philanthropic goals to be challenging and engaging.  Zuckerman:

…[Renaissance] now employed about 250 staffers and over sixty PhDs, including experts in artificial intelligence, quantum physicists, computational linguists, statisticians, and number theorists, as well as other scientists and mathematicians.

Astronomers, who are accustomed to scrutinizing large, confusing data sets and discovering evidence of subtle phenomena, proved especially capable of identifying overlooked market patterns…

Medallion still did bond, commodity, and currency trades, and it made money from trending and reversion-predicting signals… More than ever, though, it was powered by complex equity trades featuring a mix of complex signals, rather than simple pair trades…

Zuckerman adds:

The gains on each trade were never huge, and the fund only got it right a bit more than half the time, but that was more than enough.

“We’re right 50.75 percent of the time… but we’re 100 percent right 50.75 percent of the time,” Mercer told a friend.  “You can make billions that way.”

Zuckerman continues:

Driving these reliable gains was a key insight: Stocks and other investments are influenced by more factors and forces than even the most sophisticated investors appreciated…


…By analyzing and estimating hundreds of financial metrics, social media feeds, barometers of online traffic, and pretty much anything that can be quantified and tested, they uncovered new factors, some borderline impossible for most to appreciate.

“The inefficiencies are so complex they are, in a sense, hidden in the markets in code,” a staffer says.  “RenTec decrypts them.  We find them across time, across risk factors, across sectors and industries.”

Even more important: Renaissance concluded that there are reliable mathematical relationships between all these forces.  Applying data science, the researchers achieved a better sense of when various factors were relevant, how they interrelated, and the frequency with which they influenced shares.  They also tested and teased out subtle, nuanced mathematical relationships between various shares—what staffers call multidimensional anomalies—that other investors were oblivious to or didn’t fully understand.

Photo by Funtap P

Zuckerman quotes a Renaissance employee:

“There is no individual bet we make that we can explain by saying we think one stock is going to go up or another down,” a senior staffer says.  “Every bet is a function of all the other bets, our risk profile, and what we expect to do in the near and distant future.  It’s a big, complex optimization based on the premise that we predict the future well enough to make money from our predictions, and that we understand risk, cost, impact, and market structure well enough to leverage the hell out of it.”

Renaissance also excelled at disguising its buying and selling so that others wouldn’t know what it was doing.  This helped to minimize the costs of trading.


Bob Mercer, while being highly intelligent, surprised his colleagues with political views that didn’t seem to be based on much evidence or critical thinking.  Of course, part of this could be that Mercer’s colleagues tended to have a liberal bias.  In any case, Mercer was a Rational Rifle Association member who believed in guns and gold.  Mercer didn’t like taxes and he was a skeptic of climate change.

Furthermore, Mercer emerged as a key donor to right-wing causes.  Unusually, while most big contributors wanted something in return from politicians, Mercer didn’t ask for anything in return.  Zuckerman:

Mercer’s penchant for privacy limited his activity, however, as did his focus on Renaissance.  It was his second-oldest daughter, Rebekah, who started showing up at conservative fund-raising events and other get-togethers, becoming the family’s public face, and the one driving its political strategy.

Zuckerman continues:

…in 2011, the Mercers met conservative firebrand Andrew Breitbart at a conference.  Almost immediately, they were intrigued with his far-right news organization, Breitbart News Network, expressing interest in funding its operations.  Breitbart introduced the Mercers to his friend, Steve Bannon, a former Goldman Sachs banker, who drew up a term sheet under which the Mercer family purchased nearly 50 percent of Breitbart News for $10 million.

In March 2012, Breitbart collapsed on a Los Angeles sidewalk and died of heart failure at the age of forty-three.  Bannon and the Mercers convened an emergency meeting in New York to determine the network’s future, and decided that Bannon would become the site’s executive chairman.  Over time, the site became popular with the “alt-right,” a loose conglomeration of groups, some of which embraced tenets of white supremacy and viewed immigration and multiculturalism as threats.


Bannon helped broker a deal for Mercer to invest in an analytics firm called Cambridge Analytica, the US arm of the British behavioral research company SCL Group.  Cambridge Analytica specialized in the kinds of advanced data Mercer was accustomed to parsing at Renaissance, and the type of information that Rebekah said the GOP lacked.

Zuckerman writes:

In February 2014, Mercer and other conservative political donors gathered at New York’s Pierre hotel to strategize about the 2016 presidential election.  He told attendees he had seen data indicating that mainstream Republicans, such as Jeb Bush and Marco Rubio, would have difficulty winning.  Only a true outsider with a sense of the voters’ frustrations could emerge victorious, Mercer argued.

At a gathering of GOP backers who were going to meet Trump, Rebekah walked straight to Trump.  Zuckerman:

“It’s bad,” Trump acknowledged.

“No, it’s not bad—it’s over,” she told Trump.  “Unless you make a change.”

Rebekah told Trump to bring in Steve Bannon and Kellyanne Conway.  Zuckerman:

Before long, Bannon was running the campaign, and Conway was its manager, becoming a ubiquitous and effective television presence.  Bannon helped instill order on the campaign, making sure Trump focused on two things—disparaging Clinton’s character and promoting a form of nationalism that Bannon branded “America First” …


Meanwhile, Jim Simons was supporting Democrats.  Zuckerman says:

Ever since he and his childhood friend, Jim Harpel, had driven across the country and witnessed some of the hardships experienced by minorities and others, Simons had leaned left politically…. By the middle of 2016, Simons had emerged as the most important supporter of the Democratic Party’s Priorities USA Action super PAC and a key backer of Democratic House and Senate candidates.  By the end of that year, Simons would donate more than $27 million to Democratic causes.  Marilyn Simons was even more liberal than her husband, and Jim’s son, Nathaniel, had established a nonprofit foundation focused on climate change mitigation and clean-energy policy, issues the Trump campaign generally mocked or ignored.

Associates and others at Renaissance began asking Jim Simons if he could do anything about Mercer’s political activities.  But Mercer had always been pleasant and respectful to Simons.  Zuckerman:

“He’s a nice guy,” he insisted to a friend.  “He’s allowed to use his money as he wishes.  What can I do?”



Zuckerman writes:

Rebekah Mercer was emerging as a public figure in her own right… GQ named Mercer the seventeenth most powerful person in Washington, DC, calling her “the First Lady of the alt-right.”  The family’s political clout, along with its ongoing support for the president-elect, seemed assured.

David Magerman was unhappy with Bob Mercer’s support of Trump.  Magerman thought that much of what Trump was doing was harming the country.  Eventually, he did an interview with Gregory Zuckerman of The Wall Street Journal.  Magerman didn’t hold back.  Soon after that, he got a call from Renaissance telling him he was suspended without pay.

The Mercers were facing other consequences of their political activities.  Zuckerman:

At one point, the New York State Democratic Committee ran a television advertisement flashing Bob and Rebekah Mercer’s faces on the screen, saying they were the “same people who bankrolled Trump’s social media bot army and Steve Bannon’s extremist Breitbart News.”

In March 2017, about sixty demonstrators gathered outside Mercer’s home, decrying his funding of far-right causes and calling for higher taxes on the wealthy.  A week later, a second group held a protest, some holding signs reading: “Mercer Pay Your Taxes.”


The Mercers received death threats, friends said, forcing the family to hire security.  For a family that relished its privacy, their growing infamy was both shocking and disturbing.

Magerman felt some uncertainty about whether his public criticism of his boss was the right move.  Zuckerman:

Magerman had mixed feelings of his own.  He had made so much money at the firm that he didn’t have to worry about the financial pain of getting fired.  He loathed what Mercer was doing to the country and wanted to stop his political activity.  But Magerman also remembered how kind Mercer and his wife had been to him when he first joined the firm, inviting him to dinners at Friendly’s and movie nights with their family.  Magerman respected Bob for his intelligence and creativity, and a big part of him still yearned to please the powerful men in his life.  At that point, Magerman had spent two decades at Renaissance and he felt an appreciation for the firm.  He decided that if he could go on speaking about Mercer’s politics, he’d return to his old job.

Magerman decided to attend a poker tournament at New York’s St. Regis hotel benefitting Math for America.

Photo by Ali Altug Kirisoglu

He wanted to reintroduce himself to key Renaissance people.  Zuckerman:

The event was a highly anticipated annual showdown for quants, professional poker players, and others.  Magerman knew Simons, Mercer, Brown, and other Renaissance executives would be there.

Rebekah Mercer was there, too.  Magerman tried to approach her, and she started yelling at him.  Zuckerman:

“How could you do this to my father?  He was so good to you,” she said.


“You’re pond scum,” Mercer told him, repeatedly.  “You’ve been pond scum for twenty-five years.  I’ve always known it.”

Get out of here, she told Magerman.

Soon security forced Magerman to leave.  Shortly thereafter, Renaissance fired Magerman.

Meanwhile, criticism of Renaissance grew, focused on Mercer.  Zuckerman:

…nearly fifty protesters picketed the hedge fund itself, saying Mercer was their target, adding to the discomfort of executives, who weren’t accustomed to such negative publicity.

By October 2017, Simons was worried the controversy was jeopardizing Renaissance’s future.  The firm’s morale was deteriorating.

Finally, Simons made the difficult decision to ask Mercer to step down from his co-CEO role.

Overall, while the Mercers pulled back on many activities, they seemed to be happy with Trump.  Zuckerman:

The Mercers told friends they were happy the Trump administration had cut taxes and chosen conservative judges, among other moves, suggesting they didn’t regret becoming so involved in national politics.



In late December 2018, the stock market was collapsing.  The S&P 500 Index fell almost 10 percent during that month, the worst December since 1931.  Simons was worried.  He called Ashvin Chhabra, who runs Euclidean Capital, a firm that manages Simons’s money.  Simons asked Chhabra whether they should sell short.  Chhabra suggested they wait until the market calmed down before considering whether to do anything.  Simons agreed.

Zuckerman comments:

Hanging up, neither Simons nor Chhabra focused on the rich irony of their exchange.  Simons had spent more than three decades pioneering and perfecting a new way to invest.  He had inspired a revolution in the financial world, legitimizing a quantitative approach to trading.  By then, it seemed everyone in the finance business was trying to invest the Renaissance way: digesting data, building mathematical models to anticipate the direction of various investments, and employing automated trading systems.  The establishment had thrown in the towel.  Today, even banking giant JP Morgan Chase puts hundreds of its new investment bankers and investment professionals through mandatory coding lessons.  Simons’s success had validated the field of quantitative investing.


The goal of quants like Simons was to avoid relying on emotions and gut instinct.  Yet, that’s exactly what Simons was doing after a few difficult weeks in the market.  It was a bit like Oakland A’s executive Billy Beane scrapping his statistics to draft a player with the clear look of a star.

Zuckerman continues:

Simons’s phone call is a stark reminder of how difficult it can be to turn decision-making over to computers, algorithms, and models—even, at times, for the inventors of these very approaches.  His conversation with Chhabra helps explain the faith investors have long placed in stock-and-bond pickers dependent on judgment, experience, and old-fashioned research.

By 2019, however, confidence in the traditional approach had waned.  Years of poor performance had investors fleeing actively managed stock-mutual funds, or those professing an ability to beat the market’s returns.  At that point, these funds, most of which embrace traditional approaches to investing, controlled just half of the money entrusted by clients in stock-mutual funds, down from 75 percent a decade earlier.  The other half of the money was in index funds and other so-called passive vehicles, which simply aim to match the market’s returns, acknowledging how challenging it is to top the market.

Zuckerman later notes:

There are reasons to think the advantages that firms like Renaissance enjoy will only expand amid an explosion of new kinds of data that their computer-trading models can digest and parse.  IBM has estimated that 90 percent of the world’s data sets have been created in the last two years alone, and that forty zettabytes—or forty-four trillion gigabytes—of data will be created by 2020, a three-hundred-fold increase from 2005.

Today, almost every kind of information is digitized and made available as part of huge data sets, the kinds that investors once only dreamed of tapping.  The rage among investors is for alternative data, which includes just about everything imaginable, including instant information from sensors and satellite images around the world.  Creative investors test for money-making correlations and patterns by scrutinizing the tones of executives on conference calls, traffic in the parking lots of retail stores, records of auto-insurance applications, and recommendations by social media influencers.

Rather than wait for figures on agricultural production, quants examine sales of farm equipment or satellite images of crop yields.  Bills of lading for cargo containers can give a sense of global shifts.  Systematic traders can even get cell phone-generated data on which aisles, and even which shelfs, consumers are pausing to browse within stores.  If you seek a sense of the popularity of a new product, Amazon reviews can be scraped.  Algorithms are being developed to analyze the backgrounds of commissioners and others at the Food and Drug Administration to predict the likelihood of a new drug’s approval.

Zuckerman adds:

Years after Simons’s team at Renaissance adopted machine-learning techniques, other quants have begun to embrace these methods.  Renaissance anticipated a transformation in decision-making that’s sweeping almost every business and walk of life.  More companies and individuals are accepting and embracing models that continuously learn from their successes and failures.  As investor Matthew Granade has noted, Amazon, Tencent, Netflix, and others that rely on dynamic, ever-changing models are emerging dominant.  The more data that’s fed to the machines, the smarter they’re supposed to become.

All that said, quantitative models don’t have as much of an advantage—if any—when making longer-term investment decisions.  If you’re going to hold a stock for a year, you only have 118 data points back to the year 1900.  As a result, some fundamental investors who hold stocks for 1 to 5 years (or longer) will likely continue to prosper, at least until artificial intelligence for long-term stock picking gets a lot better than it is now.  Most quant funds are focused on investments with short or very short holding periods.


As quantitative trading models become ever more widely adopted, the nature of the financial markets itself could change, introducing new risks.  Zuckerman:

Author and former risk manager Richard Bookstaber has argued that risks today are significant because the embrace of quant models is “system-wide across the investment world,” suggesting that future troubles for these investors would have more impact than in the past.  As more embrace quantitative trading, the very nature of financial markets could change.  New types of errors could be introduced, some of which have yet to be experienced, making them harder to anticipate.  Until now, markets have been driven by human behavior, reflecting the dominant roles played by traders and investors.  If machine learning and other computer models become the most influential factors in the markets, they may become less predictable and maybe even less stable, since human nature is roughly constant while the nature of this kind of computerized trading can change rapidly.

On the other hand, computerized trading could very well make markets more stable, argues Zuckerman:

The dangers of computerized trading are generally overstated, however.  There are so many varieties of quant investing that it is impossible to generalize about the subject.  Some quants employ momentum strategies, so they intensify the selling by other investors in a downturn.  But other approaches—including smart beta, factor investing, and style investing—are the largest and fastest-growing investment categories in the quant world.  Some of these practitioners have programmed their computers to buy when stocks get cheap, helping to stabilize the market.

It’s important to remember that market participants have always tended to pull back and do less trading during crises, suggesting that any reluctance by quants to trade isn’t so very different from past approaches.  If anything, markets have become more placid as quant investors have assumed dominant positions.  Humans are prone to fear, greed, and outright panic, all of which tend to sow volatility in financial markets.  Machines could make markets more stable, if they elbow out individuals governed by biases and emotions.  And computer-driven decision-making in other fields, such as the airline industry, has generally led to fewer mistakes.


Zuckerman notes Renaissance’s three-decade track record:

By the summer of 2019, Renaissance’s Medallion fund had racked up average annual gains, before investors fees, of about 66 percent since 1988, and a return after fees of approximately 39 percent.  Despite RIEF’s early stumbles, the firm’s three hedge funds open for outside investors have also outperformed rivals and market indexes.  In June 2019, Renaissance managed a combined $65 billion, making it one of the largest hedge-fund firms in the world, and sometimes represented as much as 5 percent of daily stock-trading volume, not including high-frequency traders.

Renaissance relies on the scientific method rather than human intuition and judgment:

…staffers embrace the scientific method to combat cognitive and emotional biases, suggesting there’s value to this philosophical approach when tackling challenging problems of all kinds.  They propose hypotheses and then test, measure, and adjust their theories, trying to let data, not intuition and instinct, guide them.

“The approach is scientific,” Simons says.  “We use very rigorous statistical approaches to determine what we think is underlying.”

Another lesson of the Renaissance experience is that there are more factors and variables influencing financial markets and individual investments than most realize or can deduce.  Investors tend to focus on the most basic forces, but there are dozens of factors, perhaps whole dimensions of them, that are missed.  Renaissance is aware of more of the forces that matter, along with the overlooked mathematical relationships that affect stock prices and other investments, than most anyone else.

However, Zuckerman observes:

For all the unique data, computer firepower, special talent, and trading and risk management expertise Renaissance has gathered, the firm only profits on barely more than 50 percent of its trades, a sign of how challenging it is to try to beat the market—and how foolish it is for most investors to try.

Simons and his colleagues generally avoid predicting pure stock moves.  It’s not clear any expert or system can reliably predict individual stocks, at least over the long term, or even the direction of financial markets.  What Renaissance does is try to anticipate stock moves relative to other stocks, to an index, to a factor model, and to an industry.

I disagree that no one can predict individual stocks.  Especially when their assets under management weren’t too large, Warren Buffett and his partner Charlie Munger clearly were more often right than wrong in predicting how individual businesses—and their associated stocks—were going to perform over time.  However, now that Berkshire Hathaway has hundreds of billions of dollars, including over one hundred billion in cash, Buffett and Munger are forced to look only at larger businesses, the stocks of which tend to be more efficiently priced.

In brief, if an investor focuses on microcap stocks (market caps of $300 million or less)—which are ignored by the vast majority of professional investors—it’s quite possible for someone who is patient, diligent, and reasonably intelligent to successfully pick stocks so as to generate performance in excess of the S&P 500 Index and the Russell Microcap Index.  Or an investor could apply a quantitative value approach to microcap stocks and also do better than the S&P 500 and the Russell Microcap Index.  That’s what the Boole Microcap Fund does.

Zuckerman concludes the chapter:

The Simons Foundation, with an annual budget of $450 million, had emerged as the nation’s second-largest private funder of research in basic science.  Math for America, the organization Simons helped found, provided annual stipends of $15,000 to over one thousand top math and science teachers in New York City.  It also hosted hundreds of annual seminars and workshops, creating a community of skilled and enthusiastic teachers.  There were signs the initiative was helping public schools retain the kinds of teachers who previously had bolted for private industry.



Zuckerman notes that by Spring 2019, Simons was focused on his two greatest challenges—curing autism, and discovering the origins of the universe and life itself:

True breakthroughs in autism research hadn’t been achieved and time was ticking by.  Six years earlier, the Simons Foundation had hired Louis Reichardt, a professor of physiology and neuroscience who was the first American to climb both Mount Everest and K2.  Simons handed Reichardt an even more daunting challenge: improve the lives of those with autism.

The foundation helped establish a repository of genetic samples from 2,800 families with at least one child on the autism spectrum, accelerating the development of animal models, a step towards potential human treatments.  By the spring of 2019, Simons’s researchers had succeeded in gaining a deeper understanding of how the autistic brain works and were closing in on drugs with the potential to help those battling the condition.

Zuckerman continues:

Simons was just as hopeful about making headway on a set of existential challenges that have confounded humankind from its earliest moments.  In 2014, Simons recruited Princeton University astrophysicist David Spergel, who is known for groundbreaking work measuring the age and composition of the universe.  Simons tasked Spergel with answering the eternal question of how the universe began.  Oh, a please try to do it in a few years, while I’m still around, Simons said.

Simons helped fund a $75 million effort to build an enormous observatory with an array of ultrapowerful telescopes in Chile’s Atacama Desert, a plateau 17,000 feet above sea level featuring especially clear, dry skies.  It’s an ideal spot to measure cosmic microwave radiation and get a good look into creation’s earliest moments.  The project, led by a group of eight scientists including Spergel and Brain Keating—an astrophysicist who directs the Simons Observatory and happens to be the son of Simons’s early partner, James Ax—is expected to be completed by 2022…

Many scientists assume the universe instantaneously expanded after creation, something they call cosmic inflation.  That event likely produced gravitational waves and twisted light, or what Keating calls “the fingerprint of the Big Bang.”…

…Subscribing to a view that time never had a starting point, Simons simultaneously supports work by Paul Steinhardt, the leading proponent of the noninflationary, bouncing model, an opposing theory to the Big Bang.



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:

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:




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.

How Not to Be Wrong

(Image:  Zen Buddha Silence by Marilyn Barbone.)

December 8, 2019

Jordan Ellenberg has written a wonderful book, How Not to Be Wrong: The Power of Mathematical Thinking (Penguin, 2014).

Steven Pinker comments:

Like Lewis Carroll, George Gamow, and Martin Gardner before him, Jordan Ellenberg shows how mathematics can delight and stimulate the mind.  But he also shows that mathematical thinking should be in the toolkit of every thoughtful person—of everyone who wants to avoid fallacies, superstitions, and other ways of being wrong.

Here’s the outline:

    • When Am I Going to Use This?


    • One.  Less Like Sweden
    • Two.  Straight Locally, Curved Globally
    • Three.  Everyone is Obese
    • Four.  How Much Is That in Dead Americans?
    • Five.  More Pie Than Plate


    • Six.  The Baltimore Stockbroker and the Bible Code
    • Seven.  Dead Fish Don’t Read Minds
    • Eight.  Reductio Ad Unlikely
    • NineThe International Journal of Haruspicy
    • Ten.  Are You There God?  It’s Me, Bayesian Inference


    • Eleven.  What to Expect When You’re Expecting to Win the Lottery
    • Twelve.  Miss More Planes!
    • Thirteen.  Where the Train Tracks Meet


    • Fourteen.  The Triumph of Mediocrity
    • Fifteen.  Galton’s Ellipse
    • Sixteen.  Does Lung Cancer Make You Smoke Cigarettes?


    • Seventeen.  There Is No Such Thing as Public Opinion
    • Eighteen.  “Out of Nothing I Have Created a Strange New Universe”
    • How to Be Right
Illustration by Sergii Pal



Ellenburg tells the story of Abraham Wald.

This story, like many World War II stories, starts with the Nazis hounding a Jew out of Europe and ends with the Nazis regretting it…. He was the grandson of a rabbi and the son of a kosher baker, but the younger Wald was a mathematician almost from the start.  His talent for the subject was quickly recognized, and he was admitted to study mathematics at the University of Vienna, where he was drawn to subjects abstract and recondite even by the standards of pure mathematics: set theory and metric spaces.

In the mid-1930s, Austria was in economic distress and it wasn’t possible for a foreigner to be hired as a professor.  Wald ended up taking a job at the Cowles Commission, an economic institute then in Colorado Springs.  A few months later, Wald was offered a professorship of statistics at Columbia University.  When World War II came, the Statistical Research Group was formed.

The Statistical Research Group (SRG), where Wald spent much of World War II, was a classified program that yoked the assembled might of American statisticians to the war effort—something like the Manhattan Project, except the weapons being developed were equations, not explosives.

The mathematical talent at SRG was extraordinary.

Frederick Mosteller, who would later found Harvard’s statistics department, was there.  So was Leonard Jimmy Savage, the pioneer of decision theory and great advocate of the field that came to be called Bayesian statistics.  Norbert Wiener, the MIT mathematician and the creator of cybernetics, dropped by from time to time.  This was a group where Milton Friedman, the future Nobelist in economics, was often the fourth-smartest person in the room.

The smartest person in the room was usually Abraham Wald.

Planes need armor in order to lessen their chance of being shot down.  You don’t want too much armor due to the added weight, but you also don’t want too little armor, causing more planes to be shot down.  So there was a question about where to put extra armor on the planes.

The planes that came back had bullet holes per square foot distributed as follows:  1.11 for the engine, 1.73 for the fuselage, 1.55 for the fuel system, and 1.8 for the rest of the plane.

Given this knowledge, where should you put the extra armor?  At first glance, you might think the added armor should not be on the engine or fuel system, but on the fuselage and the rest of the plane.  However, Wald said the extra armor should go not where the bullet holes are, but where they aren’t:  on the engines.  Why?  Because you have to consider the planes that never made it back.  Those planes had bullet holes on the engines, which is why they never made it back.  Ellenberg comments:

If you go to the recover room at the hospital, you’ll see a lot more people with bullet holes in their legs than people with bullet holes in their chests.

That’s because the people shot in the chest generally don’t recover.  Ellenberg again:

One thing the American defense establishment has traditionally understood very well is that countries don’t win wars just by being braver… The winners are usually the guys who get 5% fewer of their planes shot down, or use 5% less fuel, or get 5% more nutrition into their infantry at 95% of the cost.  That’s not the stuff war movies are made of, but it’s the stuff wars are made of.  And there’s math every step of the way.

The reason Wald figured out where to put the armor was due to his mathematical training.

A mathematician is always asking, “What assumptions are you making?  Are they justified?”

The officers were assuming that the planes that returned were a random sample of all planes.  But this wasn’t the case.  There was no reason to assume the planes had an equal chance of survival no matter where they had been hit.

Ellenberg asserts:

Mathematics is the extension of common sense by other means.

Ellenberg again:

Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.

For example:

…we have built-in mental systems for assessing the likelihood of an uncertain outcome.  But those systems are pretty weak and unreliable, especially when it comes to events of extreme rarity.  That’s when we shore up our intuition with a few sturdy, well-placed theorems and techniques, and make out of it a mathematical theory of probability.

(Gaussian normal distribution, Illustration by Peter Hermes Furian)

The specialized language in which mathematicians converse with one another is a magnificent tool for conveying complex ideas precisely and swiftly.  But its foreignness can create among outsiders the impression of a sphere of thought totally alien to ordinary thinking.  That’s exactly wrong.

Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.  Despite the power of mathematics, and despite its sometimes forbidding notation and abstraction, the actual mental work involved is little different from the way we think about more down-to-earth problems.

Ellenberg continues:

Mathematics is not settled.  Even concerning the basic objects of study, like numbers and geometric figures, our ignorance is much greater than our knowledge.  And the things we do know were arrived at only after massive effort, contention, and confusion.  All this sweat and tumult is carefully screened off in your textbook.

Ellenberg then explains that mathematical facts can be simple or complicated, and they can be shallow or profound.  The focus of the book is on the simple yet profound.  These tools can help you not be wrong.

Ellenberg mentions his own experience:

As a graduate student, I dedicated myself to number theory, what Gauss called “the queen of mathematics,” the purest of the pure subjects, the sealed garden at the center of the convent, where we contemplated the same questions about numbers and equations that troubled the Greeks and have gotten hardly less vexing in the twenty-five hundred years since.

Ellenberg then adds:

But something funny happened.  The more abstract and distant from lived experience my research got, the more I started to notice how much math was going on in the world outside the walls.  Not Galois representations or cohomology, but ideas that were simpler, older, and just as deep… I started writing articles for magazines and newspapers about the way the world looked through a mathematical lens, and I found, to my surprise, that even people who said they hated math were willing to read them.




Daniel J. Mitchell of the libertarian Cato Institute wrote a blog post in 2012, during the battle over the Affordable Care Act.  The blog post was titled, “Why Is Obama Trying to Make America More Like Sweden when Swedes Are Trying to Be Less Like Sweden?”  Mitchell writes: “If Swedes have learned from their mistakes and are now trying to reduce the size and scope of government, why are American politicians determined to repeat those mistakes?”

The answer has to do with a nonlinear relationship between prosperity and Swedishness, writes Ellenberg.  It’s false to say that reducing the size of government automatically increases prosperity, just as it’s false to say that increasing the size of government automatically increases prosperity.  Arguably, there’s an optimal size for government that America is below and that Sweden is above.

Illustration by Oleksandra Drypsiak



Ellenberg continues the theme of nonlinearity:

You might not have thought you needed a professional mathematician to tell you that not all curves are straight lines.  But linear reasoning is everywhere.  You’re doing it every time you say that if something is good to have, having more of it is even better.

Ellenberg points out that if you look at one section of a curve very closely, it looks like a line.  Ellenberg explains calculus by using the example of a missile that follows a curved path due to gravity:

Now here’s the conceptual leap.  Newton said, look, let’s go all the way.  Reduce your field of view until it’s infinitesimal—so small that it’s smaller than any size you can name, but not zero.  You’re studying the missile’s arc, not over a very short time interval, but at a single moment.  What was almost a line becomes exactly a line.  And the slope of this line is what Newton called the fluxion, and what we’d now call the derivative.

(Projectile motion illustration by Ayush12gupta, via Wikimedia Commons)

Ellenberg introduces Zeno’s paradox.  In order to walk from one place to another place, first you must walk half the distance.  Then you must walk half of the remaining distance.  After that, in order to get to your destination, you must walk half of the remaining distance.  Thus it seems you can never reach your destination.  To go anywhere, first you must go half way there, then you must cover half the remaining distance, ad infinitum.  Ellenberg observes that all motion is ruled out.  To wave your hand, first you must raise your hand.  To do that, first you must raise it halfway, etc.

What’s the solution?  Ellenberg says to consider the infinite series:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ….

If you add up the first twenty terms, you get 0.999999.  If you keep adding terms, you get 0.9999….., which doesn’t end.  But is 0.9999… equal to 1?

Ellenberg says to consider that

0.3333…. = 1/3

Multiply both sides by 3:

0.9999…. = 1

You could also do it this way, multiplying 0.9999…. by 10:

10 x (0.9999…) = 9.9999….

Subtract the decimal from both sides:

10 x (0.9999…) – 1 x (0.9999…) = 9.9999… – 0.9999…

9 x (0.9999…) = 9

This still doesn’t fully answer the question because we assume that 0.3333… = 1/3, which one could argue is not quite true.

Ellenberg presents another brain teaser:

1 + 2 + 4 + 8 + 16 + 32 + ….

What does this series equal?  It must be infinite, right?

What if you multiplied it by 2:

2 x (1 + 2 + 4 + 8 + …) = 2 + 4 + 8 + 16 + …

It looks like the same series but without the 1 at the beginning.  That implies:

2 x (1 + 2 + 4 + 8 + …) – 1 x (1 + 2 + 4 + 8 + …) = –1

Which means:

1 + 2 + 4 + 8 + … = –1

Really?  Here’s another interesting example:

1 – 1 + 1 – 1 + 1 – 1 + …

It looks like:

(1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + … = 0

But it also can be written:

1 – (1 – 1) – (1 – 1) – (1 – 1) = 1 – 0 – 0 – 0 = 1

If you define T as:

T = 1 – 1 + 1 – 1 + 1 – 1 + …

Then take the negative of both sides:

–T = – 1 + 1 – 1 + 1 … = T – 1

Then you get –2T = –1, or T = 1/2.

Now back to the infinite series:


Augustin-Louis Cauchy introduced the idea of limit into calculus in the 1820s.  Ellenberg:

The sum 0.9 + 0.09 + 0.009 + … gets closer and closer to 1 the more terms you add.  And it never gets any farther away.  No matter how tight a cordon we draw around the number 1, the sum will eventually, after some finite number of steps, penetrate it, and never leave.  Under those circumstances, Cauchy said, we should simply define the value of the infinite sum to be 1.  And then he worked very hard to prove that committing oneself to this definition didn’t cause horrible contradictions to pop up elsewhere.  By the time this labor was done, he’s constructed a framework that made Newton’s calculus completely rigorous.  When we say a curve looks locally like a straight line at a certain angle, we now mean more or less this: as you zoom in tighter and tighter, the curve resembles the given line more and more closely.

So we can take 1 and 0.999… to be equal.  Ellenberg concludes:

One of the great joys of mathematics is the incontrovertible feeling that you’ve understood something the right way, all the way down to the bottom; it’s a feeling I haven’t experienced in any other sphere of mental life.



Ellenberg read an article about Americans getting more obese.  The article concludes that if this trend continues, ALL Americans will be obese by 2048.

As another example of linear regression, Ellenberg asks the reader to consider the relationship between SAT scores and tuition.  Generally, the higher the average SAT score, the higher the tuition.  What is linear regression?

…if you replace the actual tuition with the estimate the line suggests, and then you compute the difference between the estimated and the actual tuition for each school, and then you square each of these numbers, and you add all those squares up, you get some kind of total measure of the extent to which the line misses the points, and you choose the line that makes this measure as small as possible.  This business of summing squares smells like Pythagoras, and indeed the underlying geometry of linear regression is no more than Pythagoras’s theorem transposed and upgraded to a much-higher-dimensional setting…

Here’s an example of linear regression:

(Least squares fit by Prof. Boykin, via Wikimedia Commons)

Ellenberg continues:

Linear regression is a marvelous tool, versatile, scalable, and as easy to execute as clicking a button on your spreadsheet.  You can use it for data involving two variables… but it works just a well for three variables, or a thousand.  Whenever you want to understand which variables drive which other variables, and in which direction, it’s the first thing you reach for.  And it works on any data set at all.

However, linear regression is a tool that can be misused if it is applied to phenomena that aren’t linear.

Back to obesity: Overweight means having a body-mass index of 25 or higher.

In the early 1970s, just under half of Americans had a BMI that high.  By the early 1990s that figure had risen to almost 60%, and by 2008 almost three-quarters of the U.S. population was overweight.

If you extrapolate the trend, you conclude that ALL Americans will be overweight by 2048.  That doesn’t make sense.



Ellenberg writes:

How bad is the conflict in the Middle East?  Counterterrorism specialist Daniel Byman of Georgetown University lays down some cold, hard numbers in Foreign Affairs: “The Israeli military reports that from the start of the second intifada [in 2000] through the end of October 2005, Palestinians killed 1,074 Israelis and wounded 7,520—astounding figures for such a small country, the proportional equivalent of more than 50,000 dead and 300,000 wounded for the United States.”

Ellenberg adds:

Eventually (or perhaps immediately?) this reasoning starts to break down.  When there are two men left in the bar at closing time, and one of them coldclocks the other, it is not the equivalent in context to 150 million Americans getting simultaneously punched in the face.

Or: when 11% of the population of Rwanda was wiped out in 1994, all agree that it was among the worst crimes of the century.  But we don’t describe the bloodshed there by saying, “In the context of 1940s Europe, it was nine times as bad as the Holocaust.”

As a proportion of the population, South Dakota has a high degree of brain cancer while North Dakota has a low degree of it.  Similarly, Maine has a high degree of brain cancer while Vermont has a low degree of it.  This seems strange.

To understand what’s doing on, Ellenberg considers coin flipping.  If you flip one coin ten times, you may get 8 or 9 heads, which is a proportion of 80% or 90%.  But if you flip a coin one hundred times, you’ll never get 80% heads unless you repeat the flipping a few billion times.

Photo by Ronstik

What causes a large number of coin tosses to move toward 50%?  The Law of Large Numbers.  Ellenberg:

…if you flip enough coins, there’s only the barest chance of getting as many as 51%.  Observing a highly unbalanced result in ten flips is unremarkable; getting the same proportional imbalance in a hundred flips would be so startling as to make you wonder whether someone has mucked with your coins.

In 1756, Abraham de Moivre published The Doctrine of Chances.  De Moivre wanted to know how close to 50% heads you would get if you flipped a large number of coins.

De Moivre’s insight is that the size of the typical discrepancy is governed by the square root of the number of coins you toss.  Toss a hundred times as many coins as before and the typical discrepancy grows by a factor of 10—at least, in absolute terms.  As a proportion of the total number of tosses, the discrepancy shrinks as the number of coins grows, because the square root of the number of coins grows much more slowly than does the number of coins itself.

…if you want to know how impressed to be by a good run of heads, you can ask how many square roots away from 50% it is.  The square root of 100 is 10.  So when I got 60 heads in 100 tries, that was exactly one square root away from 50-50.  The square root of 1,000 is about 31; so when I got 538 heads in 1,000 tries, I did something even more surprising, even though I got only 53.8% heads in the latter case and 60% heads in the former.

But de Moivre wasn’t done.  He found that the discrepancies from 50-50, in the long run, always tend to form themselves into a perfect bell curve, or, as we call it in the biz, the normal distribution.

Ellenberg continues:

The bell curve… is tall in the middle and very flat near the edges, which is to say that the farther a discrepancy is from zero, the less likely it is to be encountered.  And this can be precisely quantified.

One common misperception is the so-called law of averages (which Ellenberg points out is not well-named because laws should be true but the law of averages is false).  If you flip ten heads in a row, what are the odds that the next flip will be heads?  Based on the Law of Large Numbers, you might think the next flip is extremely likely to be tails.  But it’s not.  It’s still only 50% likely to be tails and 50% likely to be heads.  Ellenberg explains:

The way the overall proportion settles down to 50% isn’t that fate favors tails to compensate for the heads that have already landed; it’s that those first ten flips become less and less important the more flips we make.  If I flip the coin a thousand more times, and get about half heads, then the proportion of heads in the first 1,010 flips is also going to be close to 50%.

So can wars and other violent events be compared based on proportion of population?  Ellenberg concludes the chapter:

Most mathematicians would say that, in the end, the disasters and the atrocities of history form what we call a partially ordered set.  That’s a fancy way of saying that some pairs of disasters can be meaningfully compared and others cannot.  This isn’t because we don’t have accurate enough death counts, or firm enough opinions as to the relative merits of being annihilated by a bomb versus dying of war-induced famine.  It’s because the question of whether one war was worse than another is fundamentally unlike the question of whether one number is bigger than another.  The latter question always has an answer.  The former does not.



Ellenberg writes:

Between 1990 and 2008, the U.S. economy gained a net 27.3 million jobs.  Of those, 26.7 million, or 98%, came from the “nontradable sector”: the part of the economy including things like government, health care, retail, and food service, which can’t be outsourced and which don’t produce goods to be shipped overseas.

…So [is] growth as concentrated in the nontradable part of the economy as it could possibly be?  That’s what it sounds like—but that’s not quite right.  Jobs in the tradable sector grew by a mere 620,000 between 1990 and 2008, that’s true.  But it could have been worse—they could have declined!  That’s what happened between 2000 and 2008; the tradable sector lost about 3 million jobs, while the nontradable sector added 7 million.  So the nontradable sector accounted for 7 million jobs out of the total gain of 4 million, or 175%!

The slogan to live by here is:

Don’t talk about percentages of numbers when the numbers might be negative.

Ellenberg gives the example of a coffee shop.  Say he lost $500 on coffee, but made $750 on a pastry case and $750 on a CD Rack.  Overall, he made $1,000.  75% of profits came from the pasty case.  Or you could say that 75% of profits came from the CD rack.  To say either one is misleading.

This problem doesn’t happen with numbers that have to be positive, like revenues.

Consider growth in income from 2009 to 2010, says Ellenberg.  93% of additional income went to the top 1% of taxpayers.  17% of additional income went to those in the top 10%, but not the top 1%, of taxpayers.  How does that make sense?  Again, because of a negative number: the bottom 90% of taxpayers saw their incomes move lower.




Ellenberg observes that mathematics is used for a wide range of things, from when the next bus can be expected to what the universe looked like three trillionths of a second after the Big Bang.  But what about questions concerning God and religion?  Ellenberg:

Never underestimate the territorial ambitions of mathematics!  You want to know about God?  There are mathematicians on the case.

The rabbinical scholar focuses on the Torah, notes Ellenberg.  A group of researchers at Hebrew University—senior professor of mathematics Eliyahu Rips, graduate student in computer science Yoav Rosenberg, and physicist Doron Witztum—started examining the Torah.  Specifically, they looked at “equidistant letter sequence,” or ELS.  The question they asked was:

Do the names of the rabbis appear in equidistant letter sequences unusually close to their birth and death dates?

Put differently: Did the Torah know the future?


First they searched the book of Genesis for ELSs spelling out the rabbis’ names and dates, and computed how close in the text the sequences yielding the names were to the ones yielding the corresponding dates.  Then they shuffled the thirty-two dates, so that each one was now matched with a random rabbi, and they ran the test again.  Then they did the same thing a million times.  If there were no relation in the Torah’s text between the names of the rabbis and the corresponding dates, you’d expect the true matching between rabbis and dates to do about as well as one of the random shuffles.  That’s not what they found.  The correct association ended up very near the top of the rankings, notching the 453rd highest score among the 1 million contenders.

American journalist Michael Drosnin heard about the Witztum paper.  Drosnin started looking for ELSs, but without scientific constraint.  Drosnin published The Bible Code, which claimed to predict Yitzhak Rabin’s assassination, the Gulf War, and the 1994 collision of Comet Shoemaker-Levy 9 with Jupiter.  Rips, Rosenberg, and Witztum denounced Drosnin’s ad hoc method.  But The Bible Code became a bestseller.

At the same time as the bible codes were being accepted by the public, the Witztum paper came under severe criticism from mathematicians including Shlomo Sternberg of Harvard University.  To understand the criticism, Ellenberg tells the story of the Baltimore stockbroker.

What if you got an unsolicited letter from a Baltimore stockbroker saying that a certain stock would rise, and then it does?  Further assume that each week, for ten weeks, you get a new letter predicting some stock to rise or fall, and each week, the prediction is correct.  The eleventh week, the Baltimore stockbroker asks you to invest money with him.

Now, what are the odds that the Baltimore stockbroker got ten straight predictions right due to chance alone?  Those odds can be computed:

(1/2) x (1/2) x (1/2) x (1/2) x (1/2) x (1/2) x (1/2) x (1/2) x (1/2) x (1/2) = 1/1024

However, what you didn’t know was that the Baltimore stockbroker mailed 10,240 letters the first week, half predicting a rise in a specific stock and half predicting a fall in that same stock.  So that first week, there were 5,120 correct predictions.  At that point, the stockbroker mails out 5,120 new letters, half of which predict a rise in some stock and half of which predict a fall.  After that second week, there are 2,560 people who have seen two correct predictions in a row.  And thus the Baltimore stockbroker continues until, after ten weeks, there are 10 people who received by mail 10 straight correction predictions.

Some companies will launch multiple mutual funds at the same time, experimenting with different strategies before advertising any fund to the public.  The funds that do well during this initial period—called incubation—are then marketed to the public, while the funds that don’t do well are quietly closed (and often these funds are not noticed by the public).  Subsequently, the surviving funds do not perform better than the median, which suggests the part of their early outperformance was simply due to chance.

(Photo by Martinlisner)

The general lesson, writes Ellenberg, is:

Improbable things happen a lot.

In other words, a great many things are due largely to chance, therefore there will always be improbable things happening.

Criticism of the Witztum paper came from Brendan McKay, an Australian computer scientist, and Dror Bar-Natan, an Israeli mathematician then at Hebrew University.  They pointed out that the rabbis didn’t have birth certificates or passports, so they were known by many different names.  Why did the Witztum paper, for each rabbi studied, use specific appellations but not others?  Ellenberg:

McKay and Bar-Natan found that wiggle room in the choices of names led to drastic changes in the quality of the results.  They made a different set of choices about the appellations of the rabbis; their choices, according to biblical scholars, make just as much sense as the ones picked by Witztum… And they found that with the new list of names, something quite amazing transpired.  The Torah no longer seemed to detect the birth and death dates of the rabbinic notables.  But the Hebrew edition of War and Peace nailed it, identifying the rabbis with their correct dates about as well as the book of Genesis did in the Witztum paper.

What’s going on?  Ellenberg:

It is very unlikely that any given set of rabbinic appellations is well matched to birth and death dates in the book of Genesis.  But with so many ways of choosing the names, it’s not at all improbable that among all the choices there would be one that made the Torah look uncannily prescient.

Obviously it’s also not the case that Tolstoy composed his novel with the names of rabbis concealed in it, to be later revealed when modern Hebrew was invented and the novel was translated into it.



At the 2009 Organization for Human Brain Mapping conference in San Francisco, UC Santa Barbara neuroscientist Craig Bennett presented his finding that a dead salmon was able to read human emotions.

A dead fish, scanned in an fMRI device, was shown a series of photographs of human beings, and was found to have a surprisingly strong ability to correctly assess the emotions the people in the pictures displayed.

What?!  Here’s what’s going on:

…the nervous system is a big place, with tens of thousands of voxels to choose from.  The odds that one of those voxels provides data matching up well with the photos is pretty good… The point of Bennett’s paper is to warn that the standard methods of assessing results, the way we draw our thresholds between a real phenomenon and random static, come under dangerous pressure in this era of massive data sets, effortlessly obtained.  We need to think very carefully about whether our standards for evidence are strict enough, if the empathetic salmon makes the cut.

Ellenberg adds:

The really surprising result of Bennett’s paper isn’t that one or two voxels in a dead fish passed a statistical test; it’s that a substantial proportion of the neuroimaging articles he surveyed didn’t use statistical safeguards (known as “multiple comparisons correction”) that take into account the ubiquity of the improbable.  Without those corrections, scientists are at serious risk of running the Baltimore stockbroker con, not only on their colleagues but on themselves.

The null hypothesis is the hypothesis that the intervention you’re studying has no effect, notes Ellenberg.

Illustration by Hafakot

To test your intervention, you have to run a null hypothesis significance test.  Ellenberg:

It goes like this.  First, you have to run an experiment.  You might start with a hundred subjects, then randomly select half to receive your proposed wonder drug while the other half gets a placebo…

From here, the protocol might seem simple: if you observe fewer deaths among the drug patients than the placebo patients, declare victory and file a marketing application with the FDA.  But that’s wrong.  It’s not enough that the data be consistent with your theory; they have to be inconsistent with the negation of your theory, the dreaded null hypothesis.

Assuming the null hypothesis, the chance of death (assume 10%) is exactly the same for the patients who got the drug and the patients who got the placebo.  That doesn’t mean that exactly five patients will die in each category.  Under the null hypothesis, Ellenberg says there is:

    • 13.3% chance equally many drug and placebo patients die
    • 43.3% chance fewer placebo patients than drug patients die
    • 43.3% chance fewer drug patients than placebo patients die

So drug patients doing better than placebo patients is not necessarily significant.  But what if drug patients do A LOT better, asks Ellenberg.  Assume, for example, that none of the drug patients die.  What are the odds that the could happen under the null hypothesis?

Under the null hypothesis, here’s a 90% chance that a patient will survive.  What are the odds that all fifty survive?

0.9 x 0.9 x 0.9 x … fifty times! … x 0.9 x 0.9 = 0.00515…

Under the null hypothesis, there is only one chance in two hundred of getting results this good, observes Ellenberg.

So here’s the procedure for ruling out the null hypothesis…:

    • Run an experiment.
    • Suppose the null hypothesis is true, and let p be the probability (under that hypothesis) of getting results as extreme as those observed.
    • The number p is called the p-value. If it is very small, rejoice; you get to say your results are statistically significant.  If it is large, concede that the null hypothesis has not been ruled out.



Ellenberg points out that assuming the null hypothesis, which we believe is false, might seem questionable.  But the reductio ad absurdum goes all the way back to Aristotle.  If a hypothesis implies a falsehood, then the hypothesis must be false.  The reduction ad absurdum looks like this:

    • Suppose the hypothesis H is true.
    • It follows from H that a certain fact F cannot be the case.
    • But F is the case.
    • Therefore, H is false.

Ellenberg then describes what he calls a reductio ad unlikely:

    • Suppose the null hypothesis H is true.
    • It follows from H that a certain outcome O is very improbable (say, less than Fisher’s 0.05 threshold).
    • But O was actually observed.
    • Therefore, H is very improbable.
Illustration by Ctitze



Ellenberg writes about haruspicy:

…If the null hypothesis is always true—that is, if haruspicy is undiluted hocus-pocus—then only 1 in 20 experiments will be publishable.

And yet there are hundreds of haruspices, and thousands of ripped-open sheep, and even one in twenty divinations provides plenty of material to fill each issue of the journal with novel results, demonstrating the efficacy of the methods and the wisdom of the gods.  A protocol that worked in one case and gets published usually fails when another haruspex tries it; but experiments without statistically significant results don’t get published, so no one ever finds out about the failure to replicate.  And even if word starts getting around, there are always small differences the experts can point to that explain why the follow-up study didn’t succeed; after all, we know the protocol works, because we tested it and it had a statistically significant effect!

Ellenberg then makes his main point:

Modern medicine and social science are not haruspicy.  But a steadily louder drum circle of dissident scientists has been pounding out an uncomfortable message in recent years: there’s probably a lot more entrail reading in the sciences than we’d like to admit.

The loudest drummer is John Ioannidis, a Greek high school math star turned biomedical researcher whose 2005 paper “Why Most Published Research Findings Are False” touched off a fierce bout of self-criticism (and a second wave of self-defense) in the clinical sciences… Ioannidis takes seriously the idea that entire specialties of medical research are “null fields,” like haruspicy, in which there are simply no actual effects to be found.  “It can be proven,” he writes, “that most claimed research findings are false.”

Photo by Ekaterina79


…In a 2012 study, scientists at the California biotech company Amgen set out to replicate some of the most famous experimental results in the biology of cancer, fifty-three studies in all.  In their independent trials, they were able to reproduce only six.

How can this have happened?  It’s not because genomicists and cancer researchers are dopes.  In part, the replicability crisis is simply a reflection of the fact that science is hard and that most ideas we have are wrong—even most of those ideas that survive a first round of prodding.

Ellenberg again:

Suppose you tested twenty genetic markers to see whether they were associated with some disorder of interest, and you found just one result that achieved p < .05 significance.  Being a mathematical sophisticate, you’d recognize that one success in twenty is exactly what you’d expect if none of the markers had any effect…

All the more so if you tested the same gene, or the green jelly bean, twenty times and got a statistically significant effect just once.

But what if the green jelly bean were tested twenty times by twenty different research groups in twenty different labs?  Nineteen of the labs find no significant statistical effect.  They don’t write up their results… The scientists in the twentieth lab, the lucky ones, find a statistically significant effect, because they got lucky—but they don’t know they got lucky.

It can be difficult for scientists when the results seem to be statistically insignificant.  Ellenberg:

If you run your analysis and get a p-value of .06, you’re supposed to conclude that your results are statistically insignificant.  But it takes a lot of mental strength to stuff years of work in the file drawer… Give yourself license to tweak and shade the statistical tests you carry out on your results, and you can often get that .06 down to a .04.  Uri Simonsohn, a professor at Penn who’s a leader in the study of replicability, calls these practices “p-hacking.”  Hacking the p isn’t usually as crude as I’ve made it out to be, and it’s seldom malicious.  The p-hackers truly believe in their hypotheses, just as the Bible coders do, and when you’re a believer, it’s easy to come up with reasons that the analysis that gives a publishable p-value is the one you should have done in the first place.

But everybody knows it’s not really right.

Replication is central to science.  Ellenberg comments:

But even studies that could be replicated often aren’t.  Every journal wants to publish a breakthrough finding, but who wants to publish the paper that does the same experiment a year later and gets the same result?  Even worse, what happens to papers that carry out the same experiment and don’t find a significant result?  For the system to work, those experiments need to be made public.  Too often they end up in the file drawer instead.

But the culture is changing.  Reformers with loud voices like Ioannidis and Simonsohn, who speak both to the scientific community and to the broader public, have generated a new sense of urgency about the danger of descent into large-scale haruspicy.  In 2013, the Association for Psychological Science announced that they would start publishing a new genre of article, called Registered Replication Reports.  These reports, aimed at reproducing the effects reported in widely cited studies, are treated differently from usual papers in a crucial way: the proposed experiment is accepted for publication before the study is carried out.



While we can predict the course of an asteroid better as we get ever more data, there may be hard limits on how far into the future meteorologists can predict the weather.  The weather is chaotic, and a sea gull flapping its wings could alter the weather forever.  Ellenberg:

Is human behavior more like an asteroid or more like the weather?  It surely depends on what aspect of human behavior you’re talking about.  In at least one respect, human behavior ought to be even harder to predict than the weather.  We have a very good mathematical model for weather, which allows us at least to get better at short-range predictions when given access to more data, even if the inherent chaos of the system inevitably wins out.  For human action we have no such model and may never have one.  That makes the prediction problem massively harder.

Ellenberg asks whether Facebook can predict that someone is a terrorist.

on Facebook list Not on list
terrorist 10 9,990
not terrorist 99,990 199,890,010

What if your neighbor ends up on the Facebook list?  Ellenberg:

The null hypothesis is that your neighbor is not a terrorist.  Under that hypothesis—that is, presuming his innocence—the chance of him showing up on the Facebook red list is a mere 0.05%, well below the 1-in-20 threshold of statistical significance.  In other words, under the rules that govern the majority of contemporary science, you’d be justified in rejecting the null hypothesis and declaring your neighbor a terrorist.

Except there’s a 99.99% chance he’s not a terrorist.

Ellenberg continues:

On the one hand, there’s hardly any chance that an innocent person will be flagged by the algorithm.  At the same time, the people the algorithm points to are almost all innocent.  It seems like a paradox, but it’s not.  It’s just how things are…

Here’s the crux.  There are really two questions you can ask.  They sound kind of the same, but they’re not.

Question 1: What’s the chance that a person gets put on Facebook’s list, given that they’re not a terrorist?

Question 2: What’s the chance that a person’s not a terrorist, given that they’re on Facebook’s list?

One way you can tell these two questions are different is that they have different answers.  Really different answers.  We’ve already seen that the answer to the first question is about 1 in 2,000, while the answer to the second is 99.99%.  And it’s the answer to the second question that you really want.


The p-value is the answer to the question

“The chance that the observed experimental result would occur, given that the null hypothesis is correct.”

But what we want to know is the other conditional probability:

“The chance that the null hypothesis is correct, given that we observed a certain experimental result.”

Ellenberg adds:

The danger arises precisely when we confuse the second quantity for the first.  And this confusion is everywhere, not just in scientific studies.  When the district attorney leans into the jury box and announces, “There is only a one in five million, I repeat, a ONE IN FIVE MILLLLLLLION CHANCE that an INNOCENT MAN would match the DNA sample found at the scene,” he is answering question 1, How likely would an innocent person be to look guilty?  But the jury’s job is to answer question 2, How likely is this guilty-looking defendant to be innocent?

In Bayesian inference, you start out with your best guess—the a priori probability that something is true—and then, once you get further evidence, you end up with a posterior probability.  Consider the example of the neighbor being on Facebook’s terrorist list.

The neighbor’s presence on the list really does offer some evidence that he’s a potential terrorist.  But your prior for that hypothesis ought to be very small, because most people aren’t terrorists.  So, despite the evidence, your posterior probability remains small as well, and you don’t—or at least shouldn’t—worry.

Ellenberg writes:

For those who are willing to adopt the view of probability as degree of belief, Bayes’s theorem can be seen not as a mere mathematical equation but as a form of numerically flavored advice.  It gives us a rule, which we may choose to follow or not, for how we should update our beliefs about things in the light of new observations.  In this new, more general form, it is naturally the subject of much fiercer disputation.  There are hard-core Bayesians who think that all our beliefs should be formed by strict Bayesian computations, or at least as strict as our limited cognition can make them; others think of Bayes’s rule as more of a loose qualitative guideline.

Here is Bayes’s rule:

(Photo by mattbuck, via Wikimedia Commons)

Ellenberg quotes Sherlock Holmes:

“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth.”

Ellenberg says Holmes should have said the following:

“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth, unless the truth is a hypothesis it didn’t occur to you to consider.”

Ellenberg gives the example of GOD vs. NO GOD.  There are other possibilities, like GODS (plural), which could help explain creation.  Ellenberg:

…Another theory with some adherents is SIMS, where we’re not actually people at all, but simulations running on an ultracomputer built by other people.  That sounds bizarre, but plenty of people take the idea seriously (most famously, the Oxford philosopher Nick Bostrom), and on Bayesian grounds, it’s hard to see why you shouldn’t.  People like to build simulations of real-world events; surely, if the human race doesn’t extinguish itself, our power to simulate will only increase, and it doesn’t seem crazy to imagine that those simulations might one day include conscious entities that believe themselves to be people.

All that said, Ellenberg notes that it’s probably best to arrive at faith—or to discard it—in a non-quantitative way.  And people should stick to, “I do believe in God,” or “I don’t believe in God,” or “I’m not sure.”




Ellenberg notes:

During the Revolutionary War, both the Continental Congress and the governments of the states established lotteries to fund the fight against the British.  Harvard, back in the days before it enjoyed a nine-figure endowment, ran lotteries in 1794 and 1810 to fund two new college buildings.

Like “statistical significance,” the term “expected value” is misleading.  If you bet $10 on a dog that has a 10% chance of winning, then the expected value is:

(10% x $100) + (90% x $0) = $10

The expected value for the $10 bet is $10.  But that’s not what you expect.  You expect either $0 or $100.  Ellenberg states that “average value” might be a better term than “expected value.”  If you make one thousand $10 bets—with each bet having a 10% chance of winning $100 and a 90% chance of winning $0—then you would expect to make about $10,000, which equals the total amount that you bet.  Over time, if you keep repeating this bet, you expect to come out even.

What is your expected value from playing Powerball?  Assuming the jackpot is $100 million, your expected value is:

100 million / 175 million + 1 million / 5 million + 10,000 / 650,000 + 100 / 19,000 + 100 / 12,000 + 7 / 700 + 7 / 360 + 4 / 110 + 4 / 55

All the different terms represent different amounts that you can win playing Powerball with the jackpot at $100 million.  (These smaller prizes keep people feeling that the game is worth playing.)  If you work out that math, your expected value from playing is just under 94 cents per $2 ticket.  For each $2 ticket you buy, on average you expect to get back just under 94 cents.  It would seem that the game is not worth playing, but what if the jackpot, call it J, is higher?  Your expected value is:

J / 175 million + 1 million / 5 million + 10,000 / 650,000 + 100 / 19,000 + 100 / 12,000 + 7 / 700 + 7 / 360 + 4 / 110 + 4 / 55

This simplifies to:

J / 175 million + 36.7 cents

The breakeven threshold is a bit over J = $285 million.  So if the jackpot is greater than $285 million, then you expect to win more than $2 for each $2 bet.  It makes sense not only to play, but to buy as many $2 tickets as you can reasonably afford.

But this assumes that all the other players fail to win the jackpot.  Assume there are 75 million players total.  Assume you win and that the jackpot is $337 million.  What are the odds that everyone else loses?  It is (174,999,999 / 175,000,000) multiplied by itself 75 million times, which is 0.651… (65.1%).  This means there’s about a 35% chance that someone else will win, which means you would have to share the jackpot.  Your expected payoff is:

65% x $337 million + 35% x $168 million = $278 million

$278 million is below the $285 million threshold, so the possibility of sharing the jackpot means that the game is no longer worth playing (assuming a $337 million jackpot and 75 million players).

Photo by Gajus

Are lotteries always bad bets?  No.  Consider the story of Cash WinFall in Massachusetts.  Here’s the prize distribution on a normal day:

match all 6 numbers 1 in 9.3 million variable jackpot
match 5 of 6 1 in 39,000 $4,000
match 4 of 6 1 in 800 $150
match 3 of 6 1 in 47 $5
match 2 of 6 1 in 6.8 free lottery ticket

Assume the jackpot is $1 million.  Then the expected return on a $2 ticket is:

($1 million / 9.3 million) + ($4,000 / 39,000) + ($150 / 800) + ($5 / 47) + ($2 / 6.8) = 79.8 cents

Each $2 bet would return about 80 cents, which is not a good bet.  However, roll-down days are different.  (Roll-downs happen when nobody wins the jackpot, so the prize money is rolled down.)  On February 7, 2005, nobody won the $3 million dollar jackpot.  So the money was rolled down:

The state’s formula rolled $600,000 to the match-5 and match-3 prize pools and $1.4 million into the match-4s.  The probability of getting 4 out of 6 WinFall numbers right is about 1 in 800, so there must have been about 600 match-4 winners that day out of the 470,000 players.  That’s a lot of winners, but $1.4 million dollars is a lot of money… In fact, you’d expect the payout for matching 4 out of 6 numbers that day to be around $2,385.  That’s a much more attractive proposition than the measly $150 you’d win on a normal day.  A 1-in-800 chance of a $2,385 payoff has an expected value of

$2385 / 800 = $2.98

The match-4 prize alone makes the game profitable.  If you add in the other payoffs, the ticket is worth

$50,000 / 39,000 + $2385 / 800 + $60 / 47 = $5.53

So for each $2 you invest, you expect to get back $5.53 on average.  To be clear: If you only bought one ticket, you probably wouldn’t win even though the average payoff is positive.  However, if you bought one thousand tickets, or ten thousand, then you would almost certainly earn a profitable return of about $5.53 per $2 ticket purchased.

February 7, 2005, is when James Harvey, an MIT senior doing an independent study on the merits of various state lottery games, realized that Massachusetts had accidentally created a highly profitable investment opportunity.  Harvey got a group of MIT friends together, and they purchased a thousand tickets.  Overall, they tripled their investment.  Ellenberg:

It won’t surprise you to hear that Harvey and his co-investors didn’t stop playing Cash WinFall.  Or that he never did get around to finishing that independent study—at least not for course credit.  In fact, his research project quickly developed into a thriving business.  By summer, Harvey’s confederates were buying tens of thousands of tickets at a time… They called their team Random Strategies, though their approach was anything but scattershot; the name referred to Random Hall, the MIT dorm where Harvey had originally cooked up his plan to make money on WinFall.

And the MIT students weren’t alone.  At least two more betting clubs formed up to take advantage of the WinFall windfall.  Ying Zhang, a medical researcher in Boston with a PhD from Northeastern, formed the Doctor Zhang Lottery Club… Before long, the group was buying $300,000 worth of tickets for each roll-down.  In 2006, Doctor Zhang quit doctoring to devote himself full-time to Cash WinFall.

Still another betting group was led by Gerald Selbee, a retiree in his seventies with a bachelor’s degree in math.



Assume that if you arrive at the airport 2 hours early, you have a 2% chance of missing the plane; if you arrive 1.5 hours early, you have a 5% chance of missing the plane; and if you arrive 1 hour early, you have a 15% chance of missing the plane.  Then, if you think of missing a plane as costing six hours of time, you can calculate the expected cost in utils of arriving at the airport 2 hours, 1.5 hours, and 1 hour early.

Option 1 –2 + 2% x (–6) = –2.12 utils
Option 2 –1.5 + 5% x (–6) = –1.8 utils
Option 3 –1 + 15% x (–6) = –1.9 utils

Under these assumptions, you should arrive at the airport 1.5 hours early.  Of course, maybe you really hate missing a plane.  Perhaps the cost of missing a plane is –20 utils.  In that case, if you redo the expected value, then Option 1 works out to be your best choice.

Ellenberg discusses Pascal’s wager.  Say that living a life of piety costs 100 utils.  But if the God of Christianity is real, and if you believe it and live accordingly, then the payoff is infinite joy.  Say there’s a 5% chance that the God of Christianity is real.  Then Pascal’s wager can be written:

(5%) x infinity + (95%) x (–100) = infinity

Illustration by Mariia Domnikova

No matter how small the odds of the Christian God’s existence, a tiny number times infinity is still infinity.  Ellenberg comments:

Pascal’s argument has serious flaws.  The gravest is that it suffers from… failing to consider all possible hypotheses.  In Pascal’s setup, there are only two options: that the God of Christianity is real and will reward that particular sector of the faithful, or that God doesn’t exist.  But what if there’s a God who damns Christians eternally?  Such a God is surely possible too, and this possibility alone suffices to kill the argument…

Utils can be useful for problems that don’t have well-defined dollar values.  But they can also be useful for problems that are stated in dollar values.  In 1738, Daniel Bernoulli put forward the St. Petersburg Paradox: “Peter tosses a coin and continues to do so until it should land ‘heads’ when it comes to the ground.  He agrees to give Paul one ducat if he gets ‘heads’ on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled.”  The question is: How much should Paul pay in order to play this game?  Paul’s expected return is:

(1/2) x 1 + (1/4) x 2 + (1/8) x 4 + (1/16) x 8 + (1/32) x 16 + …

This can be written as:

(1/2) + (1/2) + (1/2) + (1/2) + (1/2) + …

It would seem that the expected dollar value of playing the game is infinite, so Paul should be willing to spend any number of ducats in order to play.  Ellenberg:

The mistake, Bernoulli said, is to say that a ducat is a ducat is a ducat… having two thousand ducats isn’t twice as good as having one thousand; it is less than twice as good, because a thousand ducats is worth less to a person who already has a thousand ducats than it is to the person who has none…

Bernoulli thought that utility grew like the logarithm, so that the kth prize of 2k ducats was worth just k utils…

In Bernoulli’s formulation, the expected utility of the St. Petersburg game is the sum

(1/2) x 1 + (1/4) x 2 + (1/8) x 3 + (1/16) x 4 + ….

This series equals 2.  To see this, consider:

(1/2) + (1/4) + (1/8) + (1/16) + (1/32) + … = 1

    (1/4) + (1/8) + (1/16) + (1/32) + … = 1/2

        (1/8) + (1/16) + (1/32) + … = 1/4

            (1/16) + (1/32) + … = 1/8

                  (1/32) + … = 1/16

The top row is Zeno’s paradox.  The series converges to 1.  The second row is just the top row, but without the 1/2 at the beginning, so it must equal 1/2.  The third row is the same as the second, but without the 1/4 at the beginning, so it must equal 1/4.  And so forth.  If you look at the series of what each row equals, you get:

1 + (1/2) + (1/4) + (1/8) + (1/16) + (1/32) + … = 2

If you sum up the columns of the five rows above, you get:

1/2 + 2/4 + 3/8 + 4/16 + 5/32 + … = 2

This series converges to 2.   So the expected utility of the St. Petersburg game is 2.  Paul should be willing to pay 2 ducats in order to play this game.  2 is substantially less than infinity.



Ellenberg writes:

Mathematical elegance and practical utility are close companions, as the history of science has shown again and again.  Sometimes scientists discover the theory and leave it to mathematicians to figure out why it’s elegant, and other times mathematicians develop an elegant theory and leave it to scientists to figure out what it’s good for.

Ellenberg introduces the projective plane, which is governed by two axioms:

    • Every pair of points is contained in exactly one common line.
    • Every pair of lines contains exactly one common point.

If you trace out two parallel lines traveling away from you, those two lines meet on the horizon.  Call that point P.  But that vanishing point is defined to be the same point P if you turned around and looked the opposite direction.  In this way, if you think about a graph, the y-axis is a circle going vertically, while the x-axis is a circle going horizontally.

Are there other geometries besides the projective plane that satisfy the two axioms?  Yes.  For example, the Fano plane:


Ellenberg comments:

For Fano and his intellectual heirs, it doesn’t matter whether a line “looks like” a line, a circle, a mallard duck, or anything else—all that matters is that lines obey the laws of lines, set down by Euclid and his successors.  If it walks like geometry, and it quacks like geometry, we call it geometry.  To one way of thinking, this move constitutes a rupture between mathematics and reality, and is to be resisted.  But that view is too conservative.  The bold idea that we can think geometrically about systems that don’t look like Euclidean space, and even call these systems “geometries” with head held high, turned out to be critical to understanding the geometry of relativistic space-time we live in; and nowadays we use generalized geometric ideas to map Internet landscapes, which are even further removed from anything Euclid would recognize.  That’s part of the glory of math; we develop a body of ideas, and once they’re correct, they’re correct, even when applied far, far outside the context in which they were first conceived.

Looking at Fano’s plane again, there are seven lines (one of which is the circle), each of which has three points on it.

    • 124
    • 135
    • 167
    • 257
    • 347
    • 236
    • 456

Look familiar?  Ellenberg:

This is none other than the seven-ticket combo we saw in the last section, the one that hits each pair of numbers exactly once, guaranteeing a minimum payoff…

…it’s simply geometry.  Each pair of numbers appears on exactly one ticket, because each pair of points appears on exactly one line.  It’s just Euclid, even though we’re speaking now of points and lines Euclid would not have recognized as such.

What about the Massachusetts lottery?  There is no geometry that fits the precise requirements.  Ellenberg says we should consider the theory of digital signal processing.  Say we’re trying to send the following digital message:


Assume this is a communication to a satellite that says, “Turn on right thruster.”  But what if the message get garbled and sends the following instead:


This could mean, “Turn on the left thruster.”  That would be a serious problem.  A solution is to send each digit twice.  So the original message looks like:

11 11 11 00 11 00 11…

However, this still doesn’t solve the issue of potential garbling because if 11 gets turned into 01, we don’t know if 01 is supposed to be 11 or 00.  This can be solved by repeating each digit three times.  So the message looks like:

111 111 111 000 111 000 111…

Now if 111 gets garbled and turns into 101, the satellite knows it’s supposed to be 111.  That’s not a guarantee, of course, but there’s at least a high probability that the original message was 111.  Ellenberg asks whether there can really be a mathematical theory of communication.  Ellenberg comments:

Understand this: I warmly endorse, in fact highly recommend, a bristly skepticism in the face of all claims that such-and-such an entity can be explained, or tamed, or fully understood, by mathematical means.

And yet the history of mathematics is a history of aggressive territorial expansion, as mathematical techniques get broader and richer, and mathematicians find ways to address questions previously thought of as outside their domain.  “A mathematical theory of probability” sounds unexceptional now, but once it would have seemed a massive overreach; math was about the certain and the true, not the random and the maybe-so!  All that changed when Pascal, Bernoulli, and others found mathematical laws that governed the workings of chance.  A mathematical theory of infinity?  Before the work of Georg Cantor in the nineteenth century, the study of the infinite was as much theology as science; now we understand Cantor’s theory of multiple infinities, each one infinitely larger than the last, well enough to teach it to first-year math majors.

These mathematical formalisms don’t capture every detail of the phenomena they describe, and aren’t intended to.  There are questions about randomness, for instance, about which probability theory is silent…

People are working on mathematical theories that can explain and describe consciousness, society, aesthetics, and other areas.  Though success has thus far been limited, mathematics may end up getting some important points right, notes Ellenberg.

Regarding information theory, a colleague of Claude Shannon, Richard Hamming, was trying to run his programs on the weekend, but any error would halt the computation, with no one to get the machine running again until Monday morning.  So Hamming thought of a way for the machine to correct its own errors.  First he broke the message into blocks of three symbols.  Then he invented a way for each three-digit block to be transformed into a seven-digit string.  This is the Hamming code:

    • 000 -> 0000000
    • 001 -> 0010111
    • 010 -> 0101011
    • 011 -> 0111100
    • 101 -> 1011010
    • 110 -> 1100110
    • 100 -> 1001101
    • 111 -> 1110001

If the receiver gets anything that isn’t a code word, something has gone wrong.  Also, if a message is only different from a code word by one digit, you can safely infer which code word was intended.

Look back at the lines in the Fano plane.

…the seven nonzero code words in the Hamming code match up exactly to the seven lines in the Fano plane [124 is 0010111, for instance, while 135 is 0101011].  The Hamming code and the Fano plane… are exactly the same mathematical object in two different outfits!

Note that the Hamming code sends just seven bits for every three bits of your original message, a more efficient ratio of 2.33 (vs. 3 for the repeat-three-times code presented earlier).  There’s also the idea of Hamming distance, which is the number of bits you need to alter to go from one code word to another.  Two different code words are at a Hamming distance of at least 4 from each other.

Ellenberg continues:

Hamming’s notion of “distance” follows Fano’s philosophy… But why stop there?  The set of points at distance less than or equal to 1 from a given central point… is called a circle, or, if we are in higher dimensions, a sphere.  So we’re compelled to call the set of strings at Hamming distance at most 1 from a code word a “Hamming sphere,” with the code word at the center.  For a code to be an error-correcting code, no string—no point, if we’re to take this geometric analogy seriously—can be within distance 1 of two different code words; in other words, we ask that no two of the Hamming spheres centered at the code words share any points.

So the problem of constructing error-correcting codes has the same structure as a classical geometric problem, that of sphere packing: how do we fit a bunch of equal-sized spheres as tightly as possible into a small space, in such a way that no two spheres overlap?  More succinctly, how many oranges can you stuff into a box?

If playing the lottery is fun, then it’s OK to spend a few dollars regularly playing.  Also, you won’t end up in a lower class by playing, but if you get lucky and win, you could end up in a higher class.  The bottom line, as Ellenberg says, is that mathematics gives you permission to go ahead and play Powerball if it’s fun for you.

Ellenberg concludes by comparing an entrepreneur to someone who plays the lottery:

…That’s the nature of entrepreneurship: you balance a very, very small probability of making a fortune against a modest probability of eking out a living against a substantially larger probability of losing your pile, and for a large proportion of potential entrepreneurs, when you crunch the numbers, the expected financial value, like that of a lottery ticket, is less than zero… And yet society benefits from a world in which people, against their wiser judgment, launch businesses.

Perhaps the biggest part of the utility of running a business, notes Ellenberg, is the act of realizing a dream, or even trying to realize it.




Horace Secrist was a professor of statistics and director of the Bureau for Business Research at Northwestern.  Since 1920 and into the great crash, Secrist had been tracking a number of statistics on various businesses.  What he found was reversion to the mean, i.e., regression to mediocrity:

Secrist found the same phenomenon in every kind of business.  Hardware stores regressed to mediocrity; so did grocery stores.  And it didn’t matter what metric you used.  Secrist tried measuring his companies by the ratio of wages to sales, the ratio of rent to sales, and whatever other economic stat he could put his hands on.  It didn’t matter.  With time, the top performers started to look and behave just like the members of the common mass.

Similarly, Secrist found that the bottom performers improved and became more like the average.  Secrist’s views descend from those of the nineteenth-century British scientist, Francis Galton.

Galton had a peripatetic education; he tried studying mathematics at Cambridge but was defeated by the brutal Tripos exam, and devoted himself intermittently to the study of medicine, the line of work his parents had planned for him.  But after his father died in 1844, leaving him a substantial fortune, he found himself suddenly less motivated to pursue a traditional career.  For a while Galton was an explorer, leading expeditions into the African interior.  But the epochal publication of The Origin of Species in 1859 catalyzed a drastic shift in his interests: …from then on, the greater share of Galton’s work was devoted to the heredity of human characteristics, both physical and mental.

(Sir Francis Galton in the 1850s or early 1860s, scanned from Karl Pearson’s biography, via Wikimedia Commons)

Galton discovered regression to the mean.  For example, tall parents were likely to have tall children, but those children were usually not as tall as the parents.  Similarly, short parents were likely to have short children, but those children were usually not as share as the parents.  In both cases, there is regression to the mean.  Ellenberg:

So, too, Galton reasoned, must it be for mental achievement.  And this conforms with common experience; the children of a great composer, or scientist, or political leader, often excel in the same field, but seldom so much so as their illustrious parent.  Galton was observing the same phenomenon that Secrist would uncover in the operations of business.  Excellence doesn’t persist; time passes, and mediocrity asserts itself.

People are tall due to a combination of genetics and chance.  Genetics persist, but chance does not persist.  That’s why the children of tall parents tend to be tall, but not as tall as their parents: the factor of chance does not persist.  Ellenberg explains:

It’s just the same for businesses.  Secrist wasn’t wrong about the firms that had the fattest profits in 1922; it’s likely that they ranked among the most well-managed companies in their sectors.  But they were lucky, too.  As time went by, their management might well have remained superior in wisdom and judgment.  But the companies that were lucky in 1922 were no more likely than any other companies to be lucky ten years later.



A scatterplot of the heights of fathers versus the heights of sons arranges itself in an elliptical pattern, where the heights of sons are closer to the mean than the heights of fathers.  Ellenberg:

…we have ellipses of various levels of skinniness.  That skinniness, which the classical geometers call the eccentricity of the ellipse, is a measure of the extent to which the height of the father determines that of the son.  High eccentricity means that heredity is powerful and regression to the mean is weak; low eccentricity means the opposite, that regression to the mean holds sway.  Galton called his measure correlation, the term we still use today.  If Galton’s ellipse is almost round, the correlation is near 0; when the ellipse is skinny, lined up along the northeast-southwest axis, the correlation comes close to 1.

It’s important to note that correlation is not transitive.  A may be correlated with B, and B with C, but that doesn’t mean A is correlated with C.  Ellenberg:

If correlation were transitive, medical research would be a lot easier than it actually is.  Decades of observation and data collection have given us lots of known correlations to work with.  If we had transitivity, doctors could just chain these together into reliable interventions.  We know that women’s estrogen levels are correlated with lower risk of heart disease, and we know that hormone replacement therapy can raise those levels, so you might expect hormone replacement therapy to be protective against heart disease.  And, indeed, that used to be conventional clinical wisdom.  But the truth, as you’ve probably heard, is a lot more complicated.  In the early 2000s, the Women’s Health Initiative, a long-term study involving a gigantic randomized clinical trial, reported that hormone replacement therapy with estrogen and progestin appeared actually to increase the risk of heart disease in the population they studied…

In the real world, it’s next to impossible to predict what effect a drug will have on a disease, even if you know a lot about how it affects biomarkers like HDL or estrogen level.  The human body is an immensely complex system, and there are only a few of its features we can measure, let alone manipulate.  But on the correlations we can observe, there are lots of drugs that might plausibly have a desired health effect.  And so you try them out in experiments, and most of them fail dismally.  To work in drug development requires a resilient psyche, not to mention a vast pool of capital.



Correlation is not causation:

(Photo by Alain Lacroix)

Ellenberg writes:

Teasing apart correlations that come from causal relationships from those that don’t is a maddeningly hard problem, even in cases you might think of as obvious, like the relation between smoking and lung cancer.  At the turn of the twentieth century, lung cancer was an extremely rare disease.  But by 1947, the disease accounted for nearly a fifth of cancer deaths among British men, killing fifteen times as many people as it had a few decades earlier.  At first, many researchers thought that lung cancer was simply being diagnosed more effectively than before, but it soon became clear that the increase in cases was too big and too fast to be accounted for by any such effect.  Lung cancer really was on the rise.  But no one was sure what to blame.  Maybe it was smoke from factories, maybe increased levels of car exhaust, or maybe some substance not even thought of as a pollutant.  Or maybe it was cigarette smoking, whose popularity had exploded during the same period.

In the 1950s, there were some large studies that showed a correlation between smoking and lung cancer.  That doesn’t establish causality.  Perhaps lung cancer causes smoking, or perhaps there is a common cause that leads to both smoking and lung cancer.  It’s not very reasonable to assert that lung cancer causes smoking, because the tumor would have to reach back years into the past to start causing smoking.  But it is possible that there is some cause responsible for both smoking and lung cancer.  Keep in mind, notes Ellenberg, that in the 1950s, no chemical component of tobacco had yet been shown to produce tumors in the lab.  Today, things are different.  We know that smoking does cause lung cancer:

We know a lot more now about cancer and how tobacco brings it about.  That smoking gives you cancer is no longer in serious dispute.

Back in the 1950s, the evidence was not so clear.  But this seemed to change in the 1960s.  Ellenberg:

By 1964, the association between smoking and cancer had appeared consistently across study after study.  Heavier smokers suffered more cancer than lighter smokers, and cancer was most likely at the point of contact between tobacco and human tissue; cigarette smokers got more lung cancer, pipe smokers more lip cancer.  Ex-smokers were less prone to cancer than smokers who kept up the habit.  All these factors combined to lead the surgeon general’s committee to the conclusion that smoking was not just correlated with lung cancer, but caused lung cancer, and that efforts to reduce tobacco consumption would be likely to lengthen American lives.

From a public policy point of view, whether to come out against smoking depends on the expected value of doing so:

…So we don’t and can’t know the exact expected value of launching a campaign against…tobacco.  But often we can say with confidence that the expected value is positive.  Again, that doesn’t mean the campaign is sure to have good effects, only that the sum total of all similar campaigns, over time, is likely to do more good than harm.  The very nature of uncertainty is that we don’t know which of our choices will help, like attacking tobacco, and which will hurt, like recommending hormone replacement therapy.  But one thing’s for certain: refraining from making recommendations at all, on the grounds that they might be wrong, is a losing strategy.  It’s a lot like George Stigler’s advice about missing planes.  If you never give advice until you’re sure it’s right, you’re not giving enough advice.




Americans report that they would rather cut government programs—and make government smaller—than pay more taxes.  Ellenberg:

But which government programs?  That’s where things get sticky.  It turns out the things the U.S. government spends money on are things people kind of like.  A Pew Research poll from February 2011 asked Americans about thirteen categories of government spending: in eleven of those categories, deficit or no deficit, more people wanted to increase spending than dial it down.  Only foreign aid and unemployment insurance—which, combined, accounted for under 5% of 2010 spending—got the ax.  That, too, agrees with years of data; the average American is always eager to slash foreign aid, occasionally tolerant of cuts of welfare or defense, and pretty gung ho for increased spending on every single other program our taxes fund.

Oh, yeah, and we want small government.

(Photo by David Watmough)

Ellenberg again:

The average American thinks there are plenty of non-worthwhile federal programs that are wasting our money and is ready and willing to put them on the chopping block to make ends meet.  The problem is, there’s no consensus on which programs are the worthless ones.  In large part, that’s because most Americans think the programs that benefit them personally are the ones that must, at all costs, be preserved…

Ellenberg continues:

The “majority rules” system is simple and elegant and feels fair, but it’s at its best when deciding between just two options.  Any more than two, and contradictions start to seep into the majority’s preferences.  As I write this, Americans are sharply divided over President Obama’s signature domestic policy accomplishment, the Affordable Care Act.  In an October 2010 poll of likely voters, 52% of respondents said they opposed the law, while only 41% supported it.  Bad news for Obama?  Not once you break down the numbers.  Outright repeal of health care reform was favored by 37%, with another 10% saying the law should be weakened; but 15% preferred to leave it as is, and 36% said the ACA should be expanded to change the current health care system more than it currently does.  That suggests that many of the law’s opponents are to Obama’s left, not his right.  There are (at least) three choices here: leave the health care law alone, kill it, or make it stronger.  And each of the three choices is opposed by most Americans.

Ellenberg adds:

The incoherence of the majority creates plentiful opportunities to mislead.  Here’s how Fox News might report the poll results above:

Majority of Americans oppose Obamacare!

And this is how it might look to MSNBC:

Majority of Americans want to preserve or strengthen Obamacare!

These two headlines tell very different stories about public opinion.  Annoyingly enough, both are true.

But both are incomplete.  The poll watcher who aspires not to be wrong has to test each of the poll’s options, to see whether it might break down into different-colored pieces.  Fifty-six percent of the population disapproves of President Obama’s policy in the Middle East?  That impressive figure might include people from both the no-blood-for-oil left and the nuke-‘em-all right, with a few Pat Buchananists and devoted libertarians in the mix.  By itself, it tells us just about nothing about what the people really want.

Ellenberg later writes about experiments studying the decision making of slime mold.  Note that slime mold likes to eat oats and likes to avoid light.  (Note also that you can train a slime mold to navigate through a maze by using oats.)  In one experiment, the slime mold is faced with a choice: 3 grams of oats in the dark (3-dark) versus 5 grams of oats in the light (5-light).  In this scenario, the slime mold picks 3-dark half the time and 5-light half the time.

Now, if you replace the 5 grams of oats with 10 grams (10-light), the slime mold chooses 10-light every time.

However, something strange happens if you have 3-dark and 5-light options, but then add a third option: 1-dark (1 gram of oats in the dark).  You might think that the slime mold would never pick 1-dark.  That’s true.  But you might also think that the slime mold continues to pick 3-dark half the time and 5-light half the time.  But that’s not what the slime mold does.  Faced with 1-dark, 3-dark, and 5-light, the slime mold picks 3-dark more than three times as often as 5-light.  Ellenberg:

The mathematical buzzword in play here is “independence of irrelevant alternatives.”  That’s a rule that says, whether you’re a slime mold, a human being, or a democratic nation, if you have a choice between two options, A and B, the presence of a third option, C, shouldn’t affect which of A and B you like better.

Ellenberg gives an example.  In Florida in the 2000 election, the majority preferred Gore over Bush.  However, the presence of an irrelevant alternative—Ralph Nader—tipped the election to Bush.  Bush got 48.85% of the vote, Gore got 48.84%, while Nader got 1.6%.  Back to the slime mold:

…the slime mold likes the small, unlit pile of oats about as much as it likes the big, brightly lit one.  But if you introduce a really small unlit pile of oats, the small dark pile looks better by comparison; so much so that the slime mold decides to choose it over the big bright pile almost all the time.

This phenomenon is called the “asymmetric domination effect,” and slime molds are not the only creatures subject to it.  Biologists have found jays, honeybees, and hummingbirds acting in the same seemingly irrational way.

Not to mention humans!

Ellenberg tries to explain human irrationality:

Maybe individual people seem irrational because they aren’t really individuals!  Each one of us is a little nation-state, doing our best to settle disputes and broker compromises between the squabbling voices that drive us.  The results don’t always make sense.  But they somehow allow us, like the slime molds, to shamble along without making too many terrible mistakes.  Democracy is a mess—but it kind of works.



Ellenberg poses an interesting question:

Are we trying to figure out what’s true, or are we trying to figure out what conclusions are licensed by our rules and procedures?  Hopefully the two notions frequently agree; but all the difficulty, and all the conceptually interesting stuff, happens at the points where they diverge.

You might think it’s obvious that figuring out what’s true is always our proper business.  But that’s not always the case in criminal law, where the difference presents itself quite starkly in the form of defendants who committed the crime but who cannot be convicted (say, because evidence was obtained improperly) or who are innocent of the crime but are convicted anyway.  What’s justice here—to punish the guilty and free the innocent, or to follow criminal procedure wherever it leads us?  In experimental science, we’ve already seen the dispute with R.A. Fisher on one side and Jerzy Neyman and Egon Pearson on the other.  Are we, as Fisher thought, trying to figure out which hypotheses we should actually believe are true?  Or are we to follow the Neyman-Pearson philosophy, under which we resist thinking about the truth of hypotheses at all and merely ask: Which hypotheses are we to certify as correct, whether they’re really true or not, according to our chosen rules of inference?

You might think that mathematics itself doesn’t have such problems.  But it does.  Consider the parallel postulate, Euclid’s fifth axiom:  “If P is a point and L is a line not passing through P, there is exactly one line through P parallel to L.”  Even Euclid was thought to dislike his fifth axiom.  He proved the first twenty-eight propositions in the Elements using only the first four axioms.

In 1820, the Hungarian noble Farkas Bolyai, who had given years of his life to the problem, wrote a letter to his son trying to dissuade him from attempting to solve the same problem.  However, Janos Bolyai ignored his father’s advice.  By 1823, he had an outline of the solution.  He wrote to his father, “out of nothing I have created a strange new universe.”

Jonas Bolyai inverted the problem and asked: If the parallel axiom were false, would a contradiction follow?  Bolyai realized the answer was no:

…there was another geometry, not Euclid’s but something else, in which the first four axioms were correct but the parallel postulate was not.  Thus, there can be no proof of the parallel postulate from the other axioms; such a proof would rule out the possibility of Bolyai’s geometry.  But there it was.

Sometimes, a mathematical development is “in the air”—for reasons only poorly understood, the community is ready for a certain advance to come, and it comes from several sources at once.  Just as Bolyai was constructing his non-Euclidean geometry in Austria-Hungary, Nikolai Lobachevskii was doing the same in Russia.  And the great Carl Friedrich Gauss, an old friend of the senior Bolyai, had formulated many of the same ideas in work that had not yet seen print…

A few decades later, Bernhard Riemann pointed out that there’s a simpler non-Euclidean geometry: the geometry of the sphere.  Consider Euclid’s first four axioms:

    • There is a Line joining any two Points.
    • Any Line segment can be extended to a Line segment of any desired length.
    • For every Line segment L, there is a Circle which has L as a radius.
    • All Right Angles are congruent to each other.

Ellenberg on Riemann’s spherical geometry:

A Point is a pair of points on the sphere which are antipodal, or diametrically opposite each other.  A Line is a “great circle”—that is, a circle on the sphere’s surface—and a Line segment is a segment of such a circle.  A Circle is a circle, now allowed to be of any size.

Using these definitions, Euclid’s first four axioms are true.  Ellenberg comments:

Here’s the thing; once you understand that the first four axioms apply to many different geometries, then any theorem Euclid proves from only those axioms must be true, not only in Euclid’s geometry, but in all the geometries where those axioms hold.  It’s a kind of mathematical force multiplier; from one proof, you get many theorems.

And these theorems are not just about abstract geometries made up to prove a point.  Post-Einstein, we understand that non-Euclidean geometry is not just a game; like it or not, it’s the way space-time actually looks.

This is a story told in mathematics again and again: we develop a method that works for one problem, and if it is a good method, one that really contains a new idea, we typically find that the same proof works in many different contexts, which may be as different from the original as a sphere is from a plane, or more so.

Ellenberg continues:

The tradition is called “formalism.”  It’s what G. H. Hardy was talking about when he remarked, admiringly, that mathematicians of the nineteenth century finally began to ask what things like

1 – 1 + 1 – 1 + …

should be defined to be, rather than what they were… In the purest version of this view, mathematics becomes a kind of game played with symbols and words.  A statement is a theorem precisely if it follows by logical steps from the axioms.  But what the axioms and theorems refer to, what they mean, is up for grabs.  What is a Point, or a Line…?  It can be anything that behaves the way the axioms demand, and the meaning we should choose is whichever one suits our present needs.  A purely formal geometry is a geometry you can in principle do without ever having seen or imagined a point or a line; it is a geometry in which it’s irrelevant what points and lines, understood in the usual way, are actually like.

Mathematical formalism shares similarities with legal formalism.

In Scalia’s view, when judges try to understand what the law intends—its spirit—they’re inevitably bamboozled by their own prejudices and desires.  Better to stick to the words of the Constitution and the statutes, treating them as axioms from which judgments can be derived by something like logical deduction.

Ellenberg continues:

Formalism has an austere elegance.  It appeals to people like G. H. Hardy, Antonin Scalia, and me, who relish that feeling of a nice rigid theory shut tight against contradiction.  But it’s not easy to hold to principles like this consistently, and it’s not clear it’s even wise.  Even Justice Scalia has occasionally conceded that when the literal words of the law seem to require an absurd judgment, the literal words have to be set aside in favor of a reasonable guess as to what Congress must have meant.  In just the same way, no scientist really wants to be bound strictly by the rules of significance, no matter what they say their principles are.  When you run two experiments, one testing a clinical treatment that seems theoretically promising and the other testing whether dead salmon respond emotionally to romantic photos, and both experiments succeed with p-values of .03, you don’t really want to treat the two hypotheses the same.

Ellenberg then notes that the German mathematician David Hilbert was formalism’s greatest champion in mathematics.

(David Hilbert before 1912, via Wikimedia Commons)

Hilbert wanted to create a purely formal mathematics.  To say that a statement was true was to say that it could be derived logically from the axioms.  However, says Ellenberg:

Mathematics has a nasty habit of showing that, sometimes, what’s obviously true is absolutely wrong.

For an example, consider set theory.  An ouroboric set has itself as a member.

Let NO be the set of all non-ouroboric sets…

Is NO ouroboric or not?  That is, is NO an element of NO?  By definition, if NO is ouroboric, then NO cannot be in NO, which consists only of non-ouroboric sets.  But to say NO is not an element of NO is precisely to say NO is non-ouroboric; it does not contain itself.

But wait a minute—if NO is non-ouroboric, then it is an element of NO, which is the set of all non-ouroboric sets.  Now NO is an element of NO after all, which is to say that NO is ouroboric.

If NO is ouroboric, it isn’t, and if it isn’t, it is.

But could finite arithmetic be proved consistent?  Ellenberg:

Hilbert sought a finitary proof of consistency, one that did not make reference to any infinite sets, one that a rational mind couldn’t help but wholly believe.

But Hilbert was to be disappointed.  In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finite proof of the consistency of arithmetic.  He had killed Hilbert’s program with a single stroke.


Hilbert’s style of mathematics survived the death of his formalist program.



Nearly everyone has heard of Theodore Roosevelt’s speech “Citizenship in a Republic,” which he delivered in Paris in 1910.  Here’s the part people love to quote:

It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better.  The credit belongs to the man who is actually in the arena, whose faced is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.

Ellenberg writes:

And yet, when Roosevelt says, “The closet philosopher, the refined and cultured individual who from his library tells how men ought to be governed under ideal conditions, is of no use in actual governmental work,” I think of Condorcet, who spent his time in the library doing just that, and who contributed more to the French state than most of his time’s more practical men.  And when Roosevelt sneers at the cold and timid souls who sit on the sidelines and second-guess the warriors, I come back to Abraham Wald, who as far as I know went his whole life without lifting a weapon in anger, but who nonetheless played a serious part in the American war effort, precisely by counseling the doers of deeds how to do them better.  He was unsweaty, undusty, and unbloody, but he was right.  He was a critic who counted.

Mathematics not only deals with certainties, but also allows us to deal with uncertainty.  Ellenberg:

Math gives us a way of being unsure in a principled way: not just throwing up our hands and saying “huh,” but rather making a firm assertion: “I’m not sure, this is why I’m not sure, and this is roughly how not-sure I am.”  Or even more: “I’m unsure, and you should be too.”

Ellenberg comments:

The paladin of principled uncertainty in our time is Nate Silver, the online-poker-player-turned-baseball-statistics-maven-turned-political analyst…

What made Silver so good?  In large part, it’s that he was willing to talk about uncertainty, willing to treat uncertainty not as a sign of weakness but as a real thing in the world, a thing that can be studied with scientific rigor and employed to good effect.  If it’s September 2012 and you ask a bunch of political pundits, “Who’s going to be elected president in November?” a bunch of them are going to say, “Obama is,” and a somewhat smaller bunch are going to say, “Romney is,” and the point is that all of those people are wrong, because the right answer is the kind of answer that Silver, almost alone in the broad-reach media, was willing to give: “Either one might win, but Obama is substantially more likely to win.”

One important piece of advice for mathematicians who are trying to prove a theorem is to divide your time between trying to prove the theorem and trying to disprove it.  First, the theorem could be wrong, in which case the sooner you realize that, the better.  Second, if the theorem is true and you try to disprove it, eventually you will get a better idea of how to prove that the theorem is true.

This self-critical attitude applies to other areas besides mathematics, says Ellenberg:

Proving by day and disproving by might is not just for mathematics.  I find it’s a good habit to put pressure on all your beliefs, social, political, scientific, and philosophical.  Believe whatever you believe by day; but at night, argue against the propositions you hold most dear.  Don’t cheat!  To the greatest extent possible you have to think as though you believe what you don’t believe.  And if you can’t talk yourself out of your existing beliefs, you’ll know a lot more about why you believe what you believe.  You’ll have come a little closer to a proof.

Ellenberg concludes:

What’s true is that the sensation of mathematical understanding—of suddenly knowing what’s going on, with total certainty, all the way to the bottom—is a special thing, attainable in few if any other places in life.  You feel you’ve reached in to the universe’s guts and put your hand on the wire.  It’s hard to describe to people who haven’t experienced it.

We are not free to say whatever we like about the wild entities we make up.  They require definition, and having been defined, they are no more psychedelic than trees and fish; they are what they are.  To do mathematics is to be, at once, touched by fire and bound by reason.  This is no contradiction.  Logic forms a narrow channel through which intuition flows with vastly augmented force.

The lessons of mathematics are simple ones and there are no numbers in them: that there is structure in the world; that we can hope to understand some of it and not just gape at what our senses present to us; that our intuition is stronger with a formal exoskeleton than without one.  And that mathematical certainty is one thing, the softer convictions we find attached to us in everyday life another, and we should keep track of the difference if we can.

Every time you observe that more of a good thing is not always better; or you remember that improbable things happen a lot, given enough chances, and resist the lure of the Baltimore stockbroker; or you make a decision based not just on the most likely future, but on the cloud of all possible futures, with attention to which ones are likely and which ones are not; or you let go of the idea that the beliefs of groups should be subject to the same rules as beliefs of individuals; or, simply, you find that cognitive sweet spot where you can let your intuition run wild on the network of tracks formal reasoning makes for it; without writing down an equation or drawing a graph, you are doing mathematics, the extension of common sense by other means.  When are you going to use it?  You’ve been using mathematics since you were born and you’ll probably never stop.  Use it well.



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:

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.


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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.


(Image:  Zen Buddha Silence by Marilyn Barbone.)

November 17, 2019

Philip Tetlock is the world expert on the topic of making forecasts.  Tetlock and Dan Gardner have written an excellent book, Superforecasting: The Art and Science of Prediction, which summarizes the mental habits of people who have consistently been able to make useful predictions about the future.  Everyone would benefit if we could predict the future better.  This book teaches us how we can do this if we’re willing to put in the hard work required.

Here’s an outline:

    • An Optimistic Skeptic
    • Illusions of Knowledge
    • Keeping Score
    • Superforecasters
    • Supersmart?
    • Superquants?
    • Supernewsjunkies?
    • Perpetual Beta
    • Superteams
    • The Leader’s Dilemma
    • Are They Really So Super?
    • What’s Next?
    • Ten Commandments for Aspiring Superforecasters
Illustration by Maxim Popov.



Tetlock writes that we are all forecasters.  Whenever we think about making an important decision—changing jobs, getting married, buying a home, making an investment, launching a product, or retiring—we make forecasts about the future.  Tetlock:

Often we do our own forecasting.  But when big events happen—markets crash, wars loom, leaders tremble—we turn to the experts, those in the know.  We look to people like Tom Friedman.

If you are a White House staffer, you might find him in the Oval Office with the president of the United States, talking about the Middle East.  If you are a Fortune 500 CEO, you might spot him in Davos, chatting in the lounge with hedge fund billionaires and Saudi princes.  And if you don’t frequent the White House or swanky Swiss hotels, you can read his New York Times columns and bestselling books that tell you what’s happening now, why, and what will happen next.  Millions do.

Tetlock compares Friedman to Bill Flack, who forecasts global events.  Bill recently retired from the Department of Agriculture.  He is fifty-five and spends some of his time forecasting.

Bill has answered roughly three hundred questions like “Will Russia officially annex additional Ukrainian territory in the next three months?” and “In the next year, will any country withdraw from the eurozone?”  They are questions that matter.  And they’re difficult.  Corporations, banks, embassies, and intelligence agencies struggle to answer such questions all the time.  “Will North Korea detonate a nuclear device before the end of this year?”  “How many additional countries will report cases of the Ebola virus in the next eight months?”  “Will India or Brazil become a permanent member of the UN Security Council in the next two years?”  Some of the questions are downright obscure, at least for most of us.  “Will NATO invite new countries to join the Membership Action Plan (MAP) in the next nine months?”  “Will the Kurdistan Regional Government hold a referendum on national independence this year?”  “If a non-Chinese telecommunications firm wins a contract to provide Internet services in the Shanghai Free Trade Zone in the next two years, will Chinese citizens have access to Facebook and/or Twitter?”  When Bill first sees one of these questions, he may have no clue how to answer it.  “What on earth is the Shanghai Free Trade Zone?” he may think.  But he does his homework.  He gathers facts, balances clashing arguments, and settles on an answer.

Tetlock continues:

No one bases decisions on Bill Flack’s forecasts, or asks Bill to share his thoughts on CNN.  He has never been invited to Davos to sit on a panel with Tom Friedman.  And that’s unfortunate.  Because Bill Flack is a remarkable forecaster.  We know that because each one of Bill’s predictions has been dated, recorded, and assessed for accuracy by independent scientific observers.  His track record is excellent.

Bill is not alone.  There are thousands of others answering the same questions.  All are volunteers.  Most aren’t as good as Bill, but about 2% are.  They include engineers and lawyers, artists and scientists, Wall Streeters and Main Streeters, professors and students… I call them superforecasters because that is what they are.  Reliable evidence proves it.  Explaining why they’re so good, and how others can learn to do what they do, is my goal in this book.

Tetlock points out that it would be interesting to compare superforecasters to an expert like Tom Friedman.  However, Friedman’s track record has never been rigorously tested.  Of course, there are endless opinions about Friedman’s track record.  Tetlock:

Every day, the news media deliver forecasts without reporting, or even asking, how good the forecasters who made the forecasts really are.  Every day, corporations and governments pay for forecasts that may be prescient or worthless or something in between.  And every day, all of us—leaders of nations, corporate executives, investors, and voters—make critical decisions on the basis of forecasts whose quality is unknown.  Baseball managers wouldn’t dream of getting out the checkbook to hire a player without consulting performance statistics… And yet when it comes to the forecasters who help us make decisions that matter far more than any baseball game, we’re content to be ignorant.

That said, forecasting is a skill.  This book, says Tetlock, will show you how.

Prior to his work with superforecasters, Tetlock conducted a 20-year research project in which close to 300 experts made predictions about the economy, stocks, elections, wars, and other issues.  The experts made roughly 28,000 predictions.  On the whole, the experts were no better than chance.  The average expert was no better than a dart-throwing chimpanzee.  Tetlock says he doesn’t mind the joke about the dart-throwing chimpanzee because it makes a valid point:

Open any newspaper, watch any TV news show, and you find experts who forecast what’s coming.  Some are cautious.  More are bold and confident.  A handful claim to be Olympian visionaries able to see decades into the future.  With few exceptions, they are not in front of the cameras because they possess any proven skill at forecasting.  Accuracy is seldom even mentioned.  Old forecasts are like old news—soon forgotten—and pundits are almost never asked to reconcile what they said with what actually happened.  The one undeniable talent that talking heads have is their skill at telling a compelling story with conviction, and that is enough.  Many have become wealthy peddling forecasting of untested value to corporate executives, government officials, and ordinary people who would never think of swallowing medicine of unknown efficacy and safety but who routinely pay for forecasts that are as dubious as elixirs sold from the back of a wagon.

Tetlock is optimistic about the ability of people to learn to be superforecasters.  But he’s also a “skeptic” when it comes to how precisely and how far into the future people can predict.  Tetlock explains:

In a world where a butterfly in Brazil can make the difference between just another sunny day in Texas and a tornado tearing through a town, it’s misguided to think anyone can see very far into the future.

Tetlock on why he’s optimistic:

We know that in so much of what people want to predict—politics, economics, finance, business, technology, daily life—predictability exists, to some degree, in some circumstances.  But there is so much else we do not know.  For scientists, not knowing is exciting.  It’s an opportunity to discover; the more that is unknown, the greater the opportunity.  Thanks to the frankly quite amazing lack of rigor in so many forecasting domains, this opportunity is huge.  And to seize it, all we have to do is set a clear goal—accuracy!—and get serious about measuring.

Tetlock and his research (and life) partner Barbara Mellers launched the Good Judgment Project (GJP) in 2011.  Tetlock:

Cumulatively, more than twenty thousand intellectually curious laypeople tried to figure out if protests in Russia would spread, the price of gold would plummet, the Nikkei would close above 9,500, war would erupt on the Korean peninsula, and many other questions about complex, challenging global issues.  By varying the experimental conditions, we could gauge which factors improved foresight, by how much, over which time frames, and how good forecasts could become if best practices were layered on each other.

The GJP was part of a much larger research effort sponsored by the Intelligence Advanced Research Projects Activity (IARPA).

IARPA is an agency within the intelligence community that reports to the director of National Intelligence and its job is to support daring research that promises to make American intelligence better at what it does.  And a big part of what American intelligence does is forecast global political and economic trends.

(IARPA logo via Wikimedia Commons)

Tetlock continues:

…IARPA created a forecasting tournament in which five scientific teams led by top researchers in the field would compete to generate accurate forecasts on the sorts of tough questions intelligence analysts deal with every day.  The Good Judgment Project was one of those five teams… By requiring teams to forecast the same questions at the same time, the tournament created a level playing field—and a rich treasure trove of data about what works, how well, and when.  Over four years, IARPA posed nearly five hundred questions about world affairs… In all, we gathered over one million individual judgments about the future.

In year 1, GJP beat the official control group by 60%.  In year 2, we beat the control group by 78%.  GJP also beat its university-affiliated competitors, including the University of Michigan and MIT, by hefty margins, from 30% to 70%, and even outperformed professional intelligence analysts with access to classified data.  After two years, GJP was doing so much better than its academic competitors that IARPA dropped the other teams.

What Tetlock learned was two key things:  some people clearly can predict certain events; and the habits of thought of these forecasters can be learned and cultivated “by any intelligent, thoughtful, determined person.”

There’s a question about whether computers can be trained to outpredict superforecasters.  Probably not for some time.  It’s more likely, says Tetlock, the superforecasters working with computers will outperform computers alone and superforecasters alone.  Think of freestyle chess where chess experts using computers are often stronger than computers alone and than human experts alone.



Archie Cochrane was supposed to die from cancer.  A specialist had done surgery to remove a lump.  During the surgery, it appeared that the cancer had spread.  So the specialist removed the pectoralis minor during the surgery.  He then told Archie Cochrane that he didn’t have long to live.  However, when a pathologist later examined the tissue that had been removed during the surgery, he learned that Cochrane didn’t have cancer at all.  This was a good thing because Archie Cochrane eventually became a revered figure in medicine.  Tetlock comments:

We have all been too quick to make up our minds and too slow to change them.  And if we don’t examine how we make these mistakes, we will keep making them.  This stagnation can go on for years.  Or a lifetime.  It can even last centuries, as the long and wretched history of medicine illustrates.

(Photograph from 1898, via Wikimedia Commons)

Tetlock continues:

When George Washington fell ill in 1799, his esteemed physicians bled  him relentlessly, doused him with mercury to cause diarrhea, induced vomiting, and raised blood-filled blisters by applying hot cups to the old man’s skin.  A physician in Aristotle’s Athens, or Nero’s Rome, or medieval Paris, or Elizabethan London would have nodded at much of that hideous regime.

Washington died… It’s possible that the treatments helped but not enough to overcome the disease that took Washington’s life, or that they didn’t help at all, or that the treatments even hastened Washington’s death.  It’s impossible to know which of these conclusions is true merely by observing that one outcome.  Even with many such observations, the truth can be difficult or impossible to tease out.  There are just too many factors involved, too many possible explanations, too many unknowns.  And if physicians are already inclined to think the treatments work—which they are, or they wouldn’t prescribe them—all that ambiguity is likely to be read in favor of the happy conclusion that the treatments really are effective.

Rigorous experimentation was needed.  But that was never done.  Tetlock brings up the example of Galen, the second-century physician to Roman emperors.  Galen’s writings were “the indisputable source of medical authority for more than a thousand years.”  But Galen never did any experiments.  Galen was a tad overconfident, writing: “All who drink of this treatment recover in a short time, except those whom it does not help, who all die.  It is obvious, therefore, that it fails only in incurable cases.”  Tetlock comments:

Galen was an extreme example but he is the sort of figure who pops up repeatedly in the history of medicine.  They are men (always men) of strong conviction and profound trust in their own judgment.  They embrace treatments, develop bold theories for why they work, denounce rivals as quacks and charlatans, and spread their insights with evangelical passion.

Tetlock again:

Not until the twentieth century did the idea of randomized trial experiments, careful measurement, and statistical power take hold… Randomly assigning people to one group or the other would mean whatever differences there are among them should balance out if enough people participated in the experiment.  Then we can confidently conclude that the treatment caused any differences in observed outcomes.  It isn’t perfect.  There is no perfection in our messy world.  But it beats wise men stroking their chins.

The first serious trials were attempted only after World War II.

But still the physicians and scientists who promoted the modernization of medicine routinely found that the medical establishment wasn’t interested, or was even hostile to their efforts.  “Too much that was being done in the name of health care lacked scientific validation,” Archie Cochrane complained about medicine in the 1950s and 1960s… Physicians and the institutions they controlled didn’t want to let go of the idea that their judgment alone revealed the truth, so they kept doing what they did because they had always done it that way—and they were backed up by respected authority.  They didn’t need scientific validation.  They just knew.  Cochrane despised this attitude.  He called it “the God complex.”

Tetlock describes the two systems we have in our brains: System 1 and System 2.

System 2 is the familiar realm of conscious thought.  It consists of everything we choose to focus on.  By contrast, System 1 is largely a stranger to us.  It is the realm of automatic perceptual and cognitive operations—like those you are running right now to transform the print on this page into a meaningful sentence or to hold the book while reaching for a glass and taking a sip.  We have no awareness of these rapid-fire processes but we could not function without them.  We would shut down.

The numbering of the two systems is not arbitrary.  System 1 comes first.  It is fast and constantly running in the background.  If a question is asked and you instantly know the answer, it sprang from System 1.  System 2 is charged with interrogating that answer.  Does it stand up to scrutiny?  Is it backed by evidence?  This process takes time and effort, which is why the standard routine in decision making is this: System 1 delivers an answer, and only then can System 2 get involved, starting with an examination of what System 1 decided.

Tetlock notes that in the Paleolithic world in which our brains evolved, System 1’s ability to make quick decisions helped us to survive.  A shadow in the grass was immediately assumed to be dangerous.  There was no time for System 2 to second guess.  If System 1 was functioning properly, we would already be running by the time we were consciously aware of the shadow in the grass.

Although System 2 can be trained to think rationally, mathematically, and in terms of statistics, both System 1 and System 2 naturally look first for evidence that confirms a given hypothesis.  The tendency to look for and see only confirming evidence, while disregarding potentially disconfirming evidence, is called confirmation bias.

Confirmation bias is one reason we tend to be overconfident.  Tetlock quotes Daniel Kahneman:

“It is wise to take admissions of uncertainty seriously, but declarations of high confidence mainly tell you that an individual has constructed a coherent story in his mind, not necessarily that the story is true.”

One thing that System 1 does if it faces a question is to substitute an easier question and answer that.  Tetlock calls this bait and switch.  Tetlock gives an example.  The first question is, “Should I worry about the shadow in the long grass?”  System 1 automatically looks for an easier question, like, “Can I easily recall a lion attacking someone from the long grass?”  If the answer to the easier question is “yes,” then the answer to the original question is also “yes.”  Tetlock calls our reliance on the automatic operations of System 1 tip-of-your-nose perspective.

That said, there are areas in life where a human can develop expertise to the point where System 1 intuition can be trusted.  Examples include fire fighting and chess.  Tetlock:

It’s pattern recognition.  With training or experience, people can encode patterns deep in their memories in vast number and intricate detail—such as the estimated fifty thousand to one hundred thousand chess positions that top players have in their repertoire.  If something doesn’t fit a pattern—like a kitchen fire giving off more heat than a kitchen fire should—a competent expert senses it immediately.

Note that developing such expertise requires working in a world with valid cues.  Valid cues exist in the world of firefighting and chess, but much less so in the world of stocks, for example.  If there’s a lot of randomness and it takes a long time to get valid feedback from decisions—which is the case in the world of stocks—then it can take much longer to develop true expertise.



In order to tell whether someone is able to make forecasts with some accuracy, the forecasts must be specific enough to be measured.  Also, by measuring forecasts, it becomes possible for people to improve.

Photo by Redwall

Tetlock writes:

In 1984, with grants from the Carnegie and MacArthur foundations, the National Research Council—the research arm of the United States National Academy of Sciences—convened a distinguished panel charged with nothing less than “preventing nuclear war.”…

The panel did its due diligence.  It invited a range of experts—intelligence analysts, military officers, government officials, arms control experts, and Sovietologists—to discuss the issues.  They… were an impressive bunch.  Deeply informed, intelligent, articulate.  And pretty confident that they knew what was happening and where we were heading.

Both liberals and conservatives agreed that the next Soviet leader would be a Communist Party man.  Both liberals and conservative were confident of their views.

But then something unexpected happened.  The Politburo appointed Mikhail Gorbachev as the next general secretary of the Communist Party of the Soviet Union.  Tetlock:

Gorbachev changed direction swiftly and abruptly.  His policies of glasnost (openness) and perestroika (restructuring) liberalized the Soviet Union.  Gorbachev also sought to normalize relations with the United States and reverse the arms race…

Few experts saw this coming.  And yet it wasn’t long before most of those who didn’t see it coming grew convinced that they knew exactly why it had happened, and what was coming next.

Tetlock comments:

My inner cynic started to suspect that no matter what had happened the experts would have been just as adept at downplaying their predictive failures and sketching an arc of history that made it appear that they saw it coming all along.  After all, the world had just witnessed a huge surprise involving one of the most consequential matters imaginable.  If this didn’t induce a shiver of doubt, what would?  I was not questioning the intelligence or integrity of these experts, many of whom had won big scientific prizes or held high government offices… But intelligence and integrity are not enough.  The national security elites looked a lot like the renowned physicians from the prescientific era… But tip-of-your-nose delusions can fool anyone, even the best and the brightest—perhaps especially the best and the brightest.

In order for forecasts to be measurable, they have to have a specific time frame and what is expected to happen must be clearly defined.  Then there’s the issue of probability.  It’s easy to judge a forecast that something will definitely happen by some specific time.  Jonathan Schell predicted, in his influential book The Fate of the Earth, that a nuclear war would happen by 1985.  Since that didn’t happen, obviously Schell’s prediction was wrong.  But Tetlock asks, what if Schell had said that a nuclear war was “very likely”?  Tetlock:

The only way to settle this definitively would be to rerun history hundreds of times, and if civilization ends in piles of irradiated rubble in most of those reruns, we would know Schell was right.  But we can’t do that, so we can’t know.

Furthermore, phrases such as “very likely” are a problem.  Consider the experience of Sherman Kent.

In intelligence circles, Sherman Kent is a legend.  With a PhD in history, Kent left a faculty position at Yale to join the Research and Analysis Branch of the newly created Coordinator of Information (COI) in 1941.  The COI became the Office of Strategic Services (OSS).  The OSS became the Central Intelligence Agency (CIA).  By the time Kent retired from the CIA in 1967, he had profoundly shaped how the American intelligence community does what it calls intelligence analysis—the methodological examination of the information collected by spies and surveillance to figure out what it means, and what will happen next.

…forecasting is all about estimating the likelihood of something happening, which Kent and his colleagues did for many years at the Office of National Estimates—an obscure but extraordinarily influential bureau whose job was to draw on all information available to the CIA, synthesize it, and forecast anything and everything that might help top officeholders in the US government decide what to do next.

The stakes were high, and Kent weighted each word carefully.  Nonetheless, there was some confusion because they used phrases like “very likely” and “a serious possibility” instead of assigning precise probabilities.  When Kent asked each team member what they had in mind as a specific probability, it turned out that each person had a different probability in mind even though they had all agreed on “a serious possibility.”  The opinions ranged from 20 percent to 80 percent.  Kent grew concerned.

Kent was right to worry.  In 1961, when the CIA was planning to topple the Castro government by landing a small army of Cuban expatriates at the Bay of Pigs, President John F. Kennedy turned to the military for an unbiased assessment.  The Joint Chiefs of Staff concluded that the plan had a “fair chance” of success.  The man who wrote the words “fair chance” later said he had in mind odds of 3 to 1 against success.  But Kennedy was never told precisely what “fair chance” meant and, not unreasonably, he took it to be a much more positive assessment.  Of course we can’t be sure that if the Chiefs had said “We feel it’s 3 to 1 the invasion will fail” that Kennedy would have called it off, but it surely would have made him think harder about authorizing what turned out to be an unmitigated disaster.

In order to solve this problem, Kent suggested specific probabilities be associated with specific phrases.  For instance, “almost certain” would mean a 93% chance, plus or minus 6%, while “probable” would mean a 75% chance, plus or minus 12%.

Unfortunately, Kent’s scheme was never adopted.  Some felt specific probabilities were unnatural.  Others thought it made you sound like a bookie.  (Kent’s famous response: “I’d rather be a bookie than a goddamn poet.”)  Still others objected that specific probabilities seemed too much like objective facts instead of subjective judgments, which they were.  The answer to that objection was simply to make it understood that the estimates were just guesses—opinions—and nothing more.

Tetlock observes that there’s a fundamental obstacle to adopting specific probabilities.  The obstacle has to do with what Tetlock calls the wrong-side-of-maybe fallacy.

If a meteorologist says there’s a 70% chance of rain, is she wrong if it doesn’t rain?  Not necessarily.  To see how right or wrong the estimate is, we would have to rerun the day a hundred times.  If it rained in 70% of the reruns, then the meteorologist would have been exactly right with her 70% estimate.  Unfortunately, however, that’s not how people judge it.  If the meteorologist estimates a 70% chance of rain, then if it rains, she was right, but if it doesn’t rain, she was wrong, according to how most people think about it.  Tetlock writes:

The prevalence of this elementary error has a terrible consequence.  Consider that if an intelligence agency says there is a 65% chance that an event will happen, it risks being pilloried if it does not—and because the forecast itself says there is a 35% chance it will not happen, that’s a big risk.  So what’s the safe thing to do?  Stick with elastic language… If the event happens, “a fair chance” can retroactively be stretched to mean something considerably bigger than 50%—so the forecaster nailed it.  If it doesn’t happen, it can be shrunk to something much smaller than 50%—and again the forecaster nailed it.  With perverse incentives like these, it’s no wonder people prefer rubbery words over firm numbers.

Tetlock observes that it wasn’t until after the debacle regarding Saddam Hussein’s purported weapons of mass destruction, and the reforms that followed, that expressing probabilities with numbers became more accepted.  But that’s within the intelligence community.  In the broader community, especially in the media, very vague language is still common.  Tetlock:

If we are serious about measuring and improving, this won’t do.  Forecasts must have clearly defined terms and timelines.  They must use numbers.  And one more thing is essential: we must have lots of forecasts.

Consider the meteorologist again.  If she makes a new forecast each day, then over time her track record can be determined.  If she’s perfect, then 70% of the time she says there’s a 70% chance of rain, it rains, and so forth.  That would be perfect calibration.  If it only rains 40% of the time she says there’s a 70% chance of rain, then she’s overconfident.  If it rains 80% of the time she says there’s a 30% chance of rain, then she’s underconfident.

Of course, when you consider something like presidential elections, which happen every four years, it could take a long time to build about enough predictions for testing purposes.  And some events are even rarer than every four years.

Besides calibration, there is also “resolution.”  If someone assigns very high probabilities—80% to 100%—to things that happen, and very low probabilities—0% to 20%—to things that don’t happen, then they have good resolution.

The math behind this system was developed by Glenn W. Brier in 1950, hence results are called Brier scores.  In effect, Brier scores measure the distance between what you forecast and what actually happened… Perfection is 0.  A hedged fifty-fifty call, or random guessing in the aggregate, will produce a Brier score of 0.5.  A forecast that is wrong to the greatest possible extent—saying there is a 100% chance that something will happen and it doesn’t, every time—scores a disastrous 2.0, as far from The Truth as it is possible to get.

Tetlock writes about the 20 years he spent gathering roughly 28,000 predictions by 284 experts.  This was the Expert Political Judgment (EPJ) project.  Tetlock:

If you didn’t know the punch line of EPJ before you read this book, you do now: the average expert was roughly as accurate as a dart-throwing chimpanzee.

Tetlock then notes that there were two statistically distinguishable groups.  One group failed to do better than random guessing.  The second group beat the chimp, but not by much.

So why did one group do better than the other?  It wasn’t whether they had PhDs or access to classified information.  Nor was it what they thought—whether they were liberals or conservatives, optimists or pessimists.  The critical factor was how they thought.

Tetlock explains:

One group tended to organize their thinking around Big Ideas, although they didn’t agree on which Big Ideas were true or false… As ideologically diverse as they were, they were united by the fact that their thinking was so ideological.  They sought to squeeze complex problems into the preferred cause-effect templates and treated what did not fit as irrelevant distractions.  Allergic to wishy-washy answers, they kept pushing their analyses to the limit (and then some), using terms like “furthermore” and “moreover” while piling up reasons why they were right and others wrong.  As a result, they were unusually confident and likelier to declare things “impossible” or “certain.”  Committed to their conclusions, they were reluctant to change their minds even when their predictions clearly failed.  They would tell us, “Just wait.”

The other group consisted of more pragmatic experts who drew on many analytical tools, with the choice of tool hinging on the particular problem they faced.  These experts gathered as much information from as many sources as they could.  When thinking, they often shifted mental gears, sprinkling their speech with transition markers such as “however,” “but,” “although,” and “on the other hand.”  They talked about possibilities and probabilities, not certainties.  And while no one likes to say “I was wrong,” these experts more readily admitted it and changed their minds.

The first group above are called hedgehogs, while the second group are called foxes.  Hedgehogs know one big thing, while foxes know many things.  Foxes beat hedgehogs on both calibration and resolution.  Moreover, hedgehogs actually did slightly worse than random guessing.

Tetlock compares the hedgehog’s Big Idea to a pair of green-tinted glasses that he never takes off.

So the hedgehog’s one Big Idea doesn’t improve his foresight.  It distorts it.  And more information doesn’t help because it’s all seen through the same tinted glasses.  It may increase the hedgehog’s confidence, but not his accuracy.  That’s a bad combination.  The predictable result?  When hedgehogs in the EPJ research made forecasts on the subjects they knew the most about—their own specialties—their accuracy declined.

Perhaps not surprisingly, the more famous an expert in EPJ was, the less accurate.

That’s not because editors, producers, and the public go looking for bad forecasters.  They go looking for hedgehogs, who just happen to be bad forecasters.  Animated by a Big Idea, hedgehogs tell tight, simple, clear stories that grab and hold audiences… Better still, hedgehogs are confident… The simplicity and confidence of the hedgehog impairs foresight, but it calms nerves—which is good for the careers of hedgehogs.

What about foxes in the media?

Foxes don’t fare so well in the media.  They’re less confident, less likely to say something is “certain” or “impossible,” and are likelier to settle on shades of “maybe.”  And their stories are complex, full of “howevers” and “on the other hands,” because they look at problems one way, then another, and another.  This aggregation of many perspectives is bad TV.  But it’s good forecasting.  Indeed, it’s essential.



Tetlock writes:

After invading in 2003, the United States turned Iraq upside down looking for WMDs but found nothing.  It was one of the worst—arguably the worst—intelligence failure in modern history.

The question, however, is not whether the Intelligence Community’s conclusion was correct, but whether it was reasonable on the basis of known information.  Was it reasonable?  Yes.  It sure looked like Saddam was hiding something.  Else why play hide-and-seek with UN arms inspectors that risks triggering an invasion and your own downfall?

It’s difficult to evaluate whether the conclusion was reasonable because, looking back, we know that it was wrong.

This particular bait and switch—replacing “Was it a good decision?” with “Did it have a good outcome?”—is both popular and pernicious.

(Illustration by Alain Lacroix)

Think of it in terms of poker.  A beginner may overestimate his odds, bet big, get lucky and win.  But that doesn’t mean the bet was a good decision.  Similarly, a good poker pro may correctly estimate her odds, bet big, get unlucky and lose.  But that doesn’t mean the bet was a bad decision.

In this case, the evidence seems to show that the conclusion of the Intelligence Community (IC) regarding Iraq’s WMDs was reasonable on the basis of known information, even though it turned out later to be wrong.

However, even though the IC’s conclusion was reasonable, it could have been better because it would have expressed less certainty had all the information been carefully considered.  In other words, the IC would have reached the same conclusion, but it wouldn’t have been associated with a probability so close to 100%.  Tetlock:

The congressional resolution authorizing the use of force might not have passed and the United States might not have invaded.  Stakes rarely get much higher than thousands of lives and trillions of dollars.

The IC didn’t even admit the possibility that they could be wrong about Iraq’s WMDs.  Normally, if there was any doubt at all, you’d have some analysts required to present an opposing view.  But nothing like that happened because the IC was so certain of its conclusion.

In 2006 the Intelligence Advanced Research Projects Activity (IARPA) was created.  Its mission is to fund cutting-edge research with the potential to make the intelligence community smarter and more effective…

In 2008 the Office of the Director of National Intelligence—which sits atop the entire network of sixteen intelligence agencies—asked the National Research Council to form a committee.  The task was to synthesize research on good judgment and help the IC put that research to good use.  By Washington’s standards, it was a bold (or rash) thing to do.  It’s not every day that a bureaucracy pays one of the world’s most respected scientific institutions to produce an objective report that might conclude that the bureaucracy is clueless.

The report delivered was delivered two years later.

“The IC should not rely on analytical methods that violate well-documented behavioral principles or that have no evidence of efficacy beyond their intuitive appeal,” the report noted.  The IC should “rigorously test current and proposed methods under conditions that are as realistic as possible.  Such an evidence-based approach to analysis will promote the continuous learning need to keep the IC smarter and more agile than the nation’s adversaries.”

The IC does a good job teaching its analysts the correct process for doing research and reaching judgments.  The IC also does a good job holding analysts accountable for following the correct process.  However, the IC doesn’t hold analysts accountable for the accuracy of their judgments.  There’s no systematic tracking of the accuracy of judgments.  That’s the biggest problem.  IARPA decided to do something about this.

IARPA would sponsor a massive tournament to see who could invent the best methods of making the sorts of forecasts that intelligence analysts make every day.  Will the president of Tunisia flee to a cushy exile in the next month?  Will an outbreak of H5N1 in China kill more than ten in the next six months?  Will the euro fall below $1.20 in the next twelve months?


The research teams would compete against one another and an independent control group.  Teams had to beat the combined forecast—the “wisdom of the crowd”—of the control group, and by margins we all saw as intimidating.  In the first year, IARPA wanted teams to beat that standard by 20%—and it wanted that margin of victory to grow to 50% by the fourth year.

But that was only part of IARPA’s plan.  Within each team, researchers could run Archie Cochrane-style experiments to assess what really works against internal control groups.  Researchers might think, for example, that giving forecasters a basic training exercise would improve their accuracy… Give the training to one randomly chosen group of forecasters but not another.  Keep all else constant.  Compare results.

Tetlock put together a team of 3,200 forecasters.  His team was called the Good Judgment Project (GJP).  Tetlock called IARPA’s tournament “gutsy” because of what it could reveal.

Here’s one possible revelation: Imagine you get a couple of hundred ordinary people to forecast geopolitical events.  You see how often they revise their forecasts and how accurate those forecasts prove to be and use that information to identify the forty or so who are the best.  Then you have everyone makes lots more forecasts.  This time, you calculate the average forecast of the whole group—“the wisdom of the crowd”—but with extra weight given to those forty top forecasters.  Then you give the forecast a final tweak: You “extremize” it, meaning you push it closer to 100% or zero.  If the forecast is 70% you might bump it up to, say, 85%.  If it’s 30%, you might reduce it to 15%.

Now imagine that the forecasts you produce this way beat those of every other group and method available, often by large margins.  Your forecasts even beat those of professional intelligence analysts inside the government who have access to classified information—by margins that remain classified.

Think how shocking it would be to the intelligence professionals who have spent their lives forecasting geopolitical events—to be beaten by a few hundred ordinary people and some simple algorithms.

It actually happened.  What I’ve described is the method we used to win IARPA’s tournament.  There is nothing dazzlingly innovative about it.  Even the extremizing tweak is based on a pretty simple insight: When you combine the judgments of a large group of people to calculate the “wisdom of the crowd” you collect all the relevant information that is dispersed among all those people.  But none of those people has access to all that information… What would happen if every one of those people were given all the information?  They would become more confident—raising their forecasts closer to 100% or zero.  If you then calculated the “wisdom of the crowd” it too would be more extreme.  Of course it’s impossible to give every person all the relevant information—so we extremize to simulate what would happen if we could.

Tetlock continues:

Thanks to IARPA, we now know a few hundred ordinary people and some simple math can not only compete with professionals supported by a multibillion-dollar apparatus but also beat them.

And that’s just one of the unsettling revelations IARPA’s decision made possible.  What if the tournament discovered ordinary people who could—without the assistance of any algorithmic magic—beat the IC?  Imagine how threatening that would be.

Tetlock introduces one of the best forecasters on GJP: Doug Lorch, a retired computer programmer who “doesn’t look like a threat to anyone.”  Out of intellectual curiosity, Lorch joined GJP.

Note that in the IARPA tournament, a forecaster could update her forecast in real time.  She may have thought there was a 60% chance some event would happen by the six-month deadline, and then read something convincing her to update her forecast to 75%.  For scoring purposes, each update counts as a separate forecast.  Tetlock:

Over four years, nearly five hundred questions about international affairs were asked of thousands of GJP’s forecasters, generating well over one million judgments about the future.  But even at the individual level, the numbers quickly added up.  In year 1 alone, Doug Lorch made roughly one thousand separate forecasts.

Doug’s accuracy was as impressive as his volume.  At the end of the first year, Doug’s overall Brier score was 0.22, putting him in the fifth spot among the 2,800 competitors in the Good Judgment Project…

In year 2, Doug joined a superforecaster team and did even better, with a final Brier score of 0.14, making him the best forecaster of the 2,800 GJP volunteers.  He also beat by 40% a prediction market in which traders bought and sold futures contracts on the outcomes of the same questions.  He was the only person to beat the extremizing algorithm.  And Doug not only beat the control group’s “wisdom of the crowd,” he surpassed it by more than 60%, meaning that he single-handedly exceeded the fourth-year performance target that IARPA set for multimillion-dollar research programs that were free to use every trick in the forecasting textbook for improving accuracy.

Tetlock points out that Doug Lorch was not uniquely gifted:

There were 58 others among the 2,800 volunteers who scored at the top of the charts in year 1.  They were our first class of superforecasters.  At the end of year 1, their collective Brier score was 0.25, compared with 0.37 for all the other forecasters—and that gap grew in later years so that by the end of the four-year tournament, superforecasters had outperformed regulars by 60%.  Another gauge of how good superforecasters were is how much further they could see into the future.  Across all four years of the tournament, superforecasters looking out three hundred days were more accurate than regular forecasters looking out one hundred days.  In other words, regular forecasters needed to triple their foresight to see as far as superforecasters.

What if the superforecasters’ performance was due to luck?  After all, if you start with 2,800 people flipping coins, some of those people by sheer luck will flip a high proportion of heads.  If luck was a significant factor, then we should expect regression to the mean: the superforecasters in year 1 should perform less well, on the whole, in year 2.  But that didn’t happen.  That doesn’t mean luck isn’t a factor, because it is a factor in many of the questions that were asked.  Tetlock writes:

So we have a mystery.  If chance is playing a significant role, why aren’t we observing significant regression of superforecasters as a whole toward the overall mean?  An offsetting process must be pushing up superforecasters’ performance numbers.  And it’s not hard to guess what that was: after year 1, when the first cohort of superforecasters was identified, we congratulated them, anointed them “super,” and put them on teams with fellow superforecasters.  Instead of regressing toward the mean, their scores got even better.  This suggests that being recognized as “super” and placed on teams of intellectually stimulating colleagues improved their performance enough to erase regression to the mean we would otherwise have seen.  In years 3 and 4, we harvested fresh crops of superforecasters and put them to work in elite teams.  That gave us more apples-to-apples comparisons.  The next cohorts continued to do as well or better than they did in the previous year, again contrary to the regression hypothesis.

That’s not to say there was no regression to the mean.  Each year, roughly 30% of the individual superforecasters fall from the top 2% the next year.  This confirms that luck is a factor.  As Tetlock points out, even superstar athletes occasionally look less than stellar.



Tetlock introduces Sanford “Sandy” Sillman.  In 2008, he was diagnosed with multiple sclerosis, which was debilitating.  Walking was difficult.  Even typing was a challenge.  Sandy had to retire from his job as an atmospheric scientist.

How smart is Sandy?  He earned a double major in math and physics from Brown University, plus a master of science degree from MIT’s technology and policy program, along with a second master’s degree, in applied mathematics, from Harvard, and finally a PhD in applied physics from Harvard.  Furthermore, Sandy’s intelligence isn’t confined to math and physics.  He is fluent in French, Russian, Italian, and Spanish.

In year 1 of GJP, Sandy finished with an overall Brier score of 0.19, which put him in a tie for overall champion.

There’s an obvious question about whether superforecasters are simply more knowledgeable and intelligent than others.  Tetlock and Barbara Mellers tested forecasters.  It turns out that although superforecasters have well above average intelligence, they did not score off-the-charts high and most fall well short of so-called genius territory, which if often defined as the top 1%, or an IQ of 135 and up.

Tetlock concludes that knowledge and intelligence help, but they add little beyond a certain threshold.

But having the requisite knowledge and intelligence is not enough.  Many clever and informed forecasters in the tournament fell far short of superforecaster accuracy.  And history is replete with brilliant people who made forecasts that proved considerably less than prescient.

For someone to become a superforecaster, she must develop the right habits of thinking.   Tetlock gives an example.

DAVOS/SWITZERLAND, 01/28/2001 – President of the Palestinian Authority Yasser Arafat.  (Wikimedia Commons)

On October 12, 2004, Yasser Arafat became severely ill with vomiting and abdominal pain.  On November 11, 2004, Arafat was pronounced dead.  There was speculation that he had been poisoned.  In July 2012, scientists at Switzerland’s Lausanne University Institute of Radiation Physics announced that they had tested some of Arafat’s belonging and found high levels of Polonium-210, a radioactive element that can be deadly if ingested.

Two separate agencies, one in France and one in Switzerland, decided to test Arafat’s body.  So IARPA asked forecasters the following question: “Will either the French or Swiss inquiries find elevated levels of polonium in the remains of Yasser Arafat’s body?”

How would someone answer this question?  Most people would follow their hunch about the matter.  Some people might feel that “Israel would never do that!”, while others might feel that “Of course Israel did it!”  Following your hunch is not the right way to approach the question.  How did the superforecaster Bill Flack answer the question?

…Bill asked himself how Arafat’s remains could have been contaminated with enough polonium to trigger a positive result.  Obviously, “Israel poisoned Arafat” was one way.  But because Bill carefully broke the question down, he realized there were others.  Arafat had many Palestinian enemies.  They could have poisoned him.  It was also possible that there had been “intentional postmortem contamination by some Palestinian faction looking to give the appearance that Israel had done a Litvinenko on Arafat,” Bill told me later.  These alternatives mattered because each additional way Arafat’s body could have been contaminated with polonium increased the probability that it was.

What’s the next step?  Before getting to this, Tetlock describes in detail an American family, the Renzettis, who have one child and asks how likely it is that they have a pet.  The first step in answering that question is to learn what percentage of American households have a pet.

Statisticians call that the best rate—how common something is within a broader class.  Daniel Kahneman has a much more evocative visual term for it.  He calls it the “outside view”—in contrast to the “inside view,” which is the specifics of the particular case.  A few minutes with Google tells me about 62% of American households own pets.  That’s the outside view here.  Starting with the outside view means I will start by estimating that there is a 62% chance the Renzettis have a pet.  Then I will turn to the inside view—all those details about the Renzettis—and use them to adjust that initial 62% up or down.

Tetlock comments:

It’s natural to be drawn to the inside view.  It’s usually concrete and filled with engaging detail we can use to craft a story about what’s going on.  The outside view is typically abstract, bare, and doesn’t lend itself so readily to storytelling.  So even smart, accomplished people routinely fail to consider the outside view.

Tetlock writes:

Here we have a famous person who is dead.  Major investigative bodies think there is enough reason for suspicion that they are exhuming the body.  Under those circumstances, how often would the investigation turn up evidence of poisoning?  I don’t know and there is no way to find out.  But I do know there is at least a prima facie case that persuades courts and medical investigators that this is worth looking into.  It has to be considerably above zero.  So let’s say it’s at least 20%.  But the probability can’t be 100% because if it were that clear and certain the evidence would have been uncovered before burial.  So let’s so the probability cannot be higher than 80%.  That’s a big range.  The midpoint is 50%.  So that outside view can serve as our starting point.

Someone might wonder why you couldn’t start with the inside view and then consider the outside view.  This wouldn’t work, however, due to anchoring.  When we make estimates, we tend to start with some number and adjust.  If we start with the outside view, we have reasonable starting point, but if we start with the inside view, we may end up anchoring on a number that is not a reasonable starting point.

Having established the outside view on the Arafat question, we next turn to the inside view.  What are the hypotheses?  Israel could have poisoned Arafat.  Arafat’s Palestinian enemies could have poisoned him.  Arafat’s remains could have been contaminated to make it look like he was poisoned.  Tetlock:

Start with the first hypothesis: Israel poisoned Yassar Arafat with polonium.  What would it take for that to be true?

    • Israel had, or could obtain, polonium.
    • Israel wanted Arafat dead badly enough to take a big risk.
    • Israel had the ability to poison Arafar with polonium.

Each of these elements could then be researched—looking for evidence pro and con—to get a sense of how likely they are to be true, and therefore how likely the hypothesis is to be true.  Then it’s on to the next hypothesis.  And the next.

This sounds like detective work because it is—or to be precise, it is detective work as real investigators do it, not the detectives on TV shows.  It’s methodical, slow, and demanding.

Tetlock concludes:

A brilliant puzzle solver may have the raw material for forecasting, but if he doesn’t also have an appetite for questioning basic, emotionally charged beliefs he will often be at a disadvantage relative to a less intelligent person who has a greater capacity for self-critical thinking.  It’s not the raw crunching power you have that matters most.  It’s what you do with it.


For superforecasters, beliefs are hypotheses to be tested, not treasures to be guarded.



Perhaps superforecasters are exceptionally good using math to make their forecasts?  Although occasionally a superforecaster will consult a mathematical model, the vast majority of the time a superforecaster uses very little math.

That said, superforecasters tend to be granular in their probability estimates.  Is this justified?  Tetlock:

So how can we know that the granularity we see among superforecasters is meaningful?… The answer lies in the tournament data.  Barbara Mellers has shown that granularity predicts accuracy: the average forecaster who sticks with the tens—20%, 30%, 40%—is less accurate than the finer-grained forecaster who uses fives—20%, 25%, 30%—and still less accurate than the even finer-grained forecaster who uses ones—20%, 21%, 22%.




Superforecasting isn’t a paint-by-numbers method but superforecasters often tackle questions in a roughly similar way—one that any of us can follow: Unpack the question into components.  Distinguish as sharply as you can between the known and unknown and leave no assumptions unscrutinized.  Adopt the outside view and put the problem into a comparative perspective that downplays its uniqueness and treats it as a special case of a wider class of phenomena.  Then adopt the inside view that plays up the uniqueness of the problem.  Also explore the similarities and differences between your views and those of others—and pay special attention to prediction markets and other methods of extracting wisdom from crowds.  Synthesize all these different views into a single vision as acute as that of a dragonfly.  Finally, express your judgment as precisely as you can, using a finely grained scale of probability.

This is just the beginning.  The next step, which is typically repeated many times, is to update your predictions as you get new information.

Photo by Marek Uliasz.

Superforecasters update their forecasts much more frequently than regular forecasters.  One might think that superforecasters are super because they are news junkies.  However, their initial forecasts were at least 50% more accurate than those of regular forecasters.  Furthermore, properly updating forecasts on the basis of new information requires the same skills used in making the initial forecasts.   Tetlock:

…there are two dangers a forecaster faces after making the initial call.  One is not giving enough weight to new information.  That’s underreaction.  The other danger is overreacting to new information, seeing it as more meaningful than it is, and adjusting a forecast too radically.

Both under- and overreaction can diminish accuracy.  Both can also, in extreme cases, destroy a perfectly good forecast.

One typical reason for underreaction to new information is that people often become committed to their beliefs.  Especially when they have publicly committed to their beliefs and when the have an ego investment.  Superforecasters, by contrast, have no trouble changing their minds on the basis of new information.

While superforecasters update their forecasts much more often than regular forecasters, they do so in small increments.  They tend not to overreact.



Superforecasters have a growth mindset.  They believe they can get better with work, and they do.

(Illustration by Tereza Paskova)

Tetlock notes that John Maynard Keynes failed twice as an investor, and was almost wiped out during the first failure, before he settled on a value investing approach.  Keynes came to understand value investing through his own reflection.  He didn’t learn about it from Ben Graham, who is regarded as the father of value investing.  In any case, Keynes turned out to be enormously successful as an investor despite investing during the Great Depression of the 1930s.

Keynes had a growth mindset: try, fail, analyze, adjust, try again.

Improving requires getting good feedback.  Meteorologists get feedback on whether their forecasts were correct on a daily basis.  Bridge players also get fairly immediate feedback on how well they’re playing.  The trouble for forecasters is twofold:  first, the forecasts must be specific enough to be testable; and second, the time lag between the forecast and the result is often long, unlike for meteorologists or bridge players.

Pulling It All Together—Portrait of a Superforecaster

In philosophic outlook, they tend to be:

    • Cautious: Nothing is certain.
    • Humble: Reality is infinitely complex.
    • Nondeterministic: What happens is not meant to be and does not have to happen.

In their abilities and thinking styles, they tend to be:

    • Actively Open-minded: Beliefs are hypotheses to be tested, not treasures to be protected.
    • Intelligent and Knowledgeable, with a “Need for Cognition”: Intellectually curious, enjoy puzzles and mental challenges.
    • Reflective: Introspective and self-critical.
    • Numerate: Comfortable with numbers.

In their methods of forecasting, they tend to be:

    • Pragmatic: Not wedded to any idea or agenda.
    • Analytical: Capable of stepping back from the tip-of-your-nose perspective and considering other views.
    • Dragonfly-eyed: Value diverse views and synthesize them into their own.
    • Probabilistic: Judge using many grades of maybe.
    • Thoughtful Updaters: When facts change, they change their minds.
    • Good Intuitive Psychologists: Aware of the value of checking thinking for cognitive and emotional biases.

In their work ethic, they tend to have:

    • A Growth Mindset: Believe it’s possible to get better.
    • Grit: Determined to keep at it however long it takes.

Tetlock concludes by noting that the strongest predictor of a forecaster becoming a superforecaster is perpetual beta, the degree to which one is committed to belief updating and self-improvement.  Perpetual beta is three times as powerful a predictor as its closet rival, intelligence.



(Photo by Chrisharvey)

Tetlock writes:

[Groups] let people share information and perspectives.  That’s good.  It helps make dragonfly eye work, and aggregation is critical to accuracy.  Of course aggregation can only do its magic when people form judgments independently, like the fairgoers guessing the weight of the ox.  The independence of judgments ensures that errors are more or less random, so they cancel each other out.  When people gather and discuss in a group, independence of thought and expression can be lost.  Maybe one person is a loudmouth who dominates the discussion, or a bully, or a superficially impressive talker, or someone with credentials that cow others into line.  In so many ways, a group can get people to abandon independent judgment and buy into errors.  When that happens, the mistakes will pile up, not cancel out.

The GJP randomly assigned several hundred forecasters to work alone and several hundred to work in teams.  At the end of the first year, the teams had been 23% more accurate than the individuals.  GJP kept experimenting.

The results speak for themselves.  On average, when a forecaster did well enough in year 1 to become a superforecaster, and was put on a superforecaster team in year 2, that person became 50% more accurate.  An analysis in year 3 got the same result.  Given that these were collections of strangers tenuously connected in cyberspace, we found that result startling.

Tetlock adds:

How did superteams do so well?  By avoiding the extremes of groupthink and Internet flame wars.  And by fostering minicultures that encouraged people to challenge each other respectfully, admit ignorance, and request help.



Good leaders have to be confident and decisive.  But good leaders also need good forecasts in order to make good decisions.  We’ve seen that effective forecasting requires self-critical questioning and doubt.  How can a leader be both a forecaster and a leader?  Tetlock:

Fortunately, the contradiction between being a superforecaster and a superleader is more apparent than real.  In fact, the superforecaster model can help make good leaders superb and the organizations they lead smart, adaptable, and effective.  The key is an approach to leadership and organization first articulated by a nineteenth-century Prussian general, perfected by the German army of World War II, made foundational doctrine by the modern American military, and deployed by many successful corporations today.  You might even find it at your neighborhood Walmart.

Helmuth von Moltke the Elder (1800-1891). Photo by Georgios Kollidas.

Helmuth von Moltke was a famous nineteenth-century Prussian general.  One of Moltke’s axioms is: “In war, everything is uncertain.”  While having a plan is important, you can never entirely trust your plan.  Moltke: “No plan of operations extends with certainty beyond the first encounter with the enemy’s main strength.”  Moltke also wrote that, “It is impossible to lay down binding rules” that apply in all circumstances.  “Two cases will never be exactly the same” in war.  Improvisation is essential.  Tetlock:

Moltke trusted that his officers were up to the task… In Germany’s war academies, scenarios were laid out and students were invited to suggest solutions and discuss them collectively.  Disagreement was not only permitted, it was expected, and even the instructor’s views could be challenged… Even the views of generals were subject to scrutiny.

Tetlock continues:

What ties all of this together… is the command principle of Auftragstaktik.  Usually translated today as “mission command,” the basic idea is simple. “War cannot be conducted from the green table,” Moltke wrote, using an expression that referred to top commanders at headquarters.  “Frequent and rapid decisions can be shaped only on the spot according to estimates of local conditions.”  Decision-making power must be pushed down the hierarchy so that those on the ground—the first to encounter surprises on the evolving battlefield—can respond quickly.  Of course those on the ground don’t see the bigger picture.  If they made strategic decisions the army would lose coherence and become a collection of tiny units, each seeking its own ends.  Auftragstaktik blended strategic coherence and decentralized decision making with a simple principle: commanders were to tell subordinates what their goal is but not how to achieve it.



Are the superforecasters super primarily due to innate abilities or primarily due to specific habits they’ve developed?   Tetlock answers:

They score higher than average on measures of intelligence and open-mindedness, although they are not off the charts.  What makes them so good is less what they are than what they do—the hard work of research, the careful thought and self-criticism, the gathering and synthesizing of other perspectives, the granular judgments and relentless updating.

Tetlock again:

My sense is that some superforecasters are so well practiced in System 2 corrections—such as stepping back to take the outside view—that these techniques have become habitual.  In effect, they are now part of their System 1… No matter how physically or cognitively demanding a task may be—cooking, sailing, surgery, operatic singing, flying fighter jets—deliberative practice can make it second nature.

Very little is predictable five or more years into the future.  Tetlock confirmed this fact in his EPJ research: Expert predictions declined towards chance five years out.  And yet governments need to make plans that extend five or more years into the future.  Tetlock comments:

Probability judgments should be explicit so we can consider whether they are as accurate as they can be.  And if they are nothing but a guess, because that’s the best we can do, we should say so.  Knowing what we don’t know is better than thinking we know what we don’t.

Tetlock adds:

Kahneman and other pioneers of modern psychology have revealed that our minds crave certainty and when they don’t find it, they impose it.  In forecasting, hindsight bias is the cardinal sin.  Recall how experts stunned by the Gorbachev surprise quickly became convinced it was perfectly explicable, even predictable, although they hadn’t predicted it.

Brushing off surprises makes the past look more predictable than it was—and this encourages the belief that the future is much more predictable that it is.

Tetlock concludes:

Savoring how history could have generated an infinite array of alternative outcomes and could now generate a similar array of alternative futures, is like contemplating the one hundred billion known stars in our galaxy and the one hundred billion known galaxies.  It instills profound humility.

…But I also believe that humility should not obscure the fact that people can, with considerable effort, make accurate forecasts about at least some developments that really do matter.



Tetlock hopes that the forecasting lessons discussed in his book will be widely adopted:

Consumer of forecasts will stop being gulled by pundits with good stories and start asking pundits how their past predictions fared—and reject answers that consist of nothing but anecdotes and credentials.  Just as we now expect a pill to have been tested in peer-reviewed experiments before we swallow it, we will expect forecasters to establish the accuracy of their forecasting with rigorous testing before we heed their advice.  And forecasters themselves will realize… that these higher expectations will ultimately benefit them, because it is only with the clear feedback that comes from rigorous testing that they can improve their foresight.  It could be huge—an “evidence-based forecasting” revolution similar to the “evidence-based medicine” revolution, with consequences every bit as significant.

Or nothing may change.  Revolutionaries aren’t supposed to say failure is possible, but let’s think like superforecasters here and acknowledge that things may go either way.


Tetlock writes about a Boston doctor named Ernest Amory Codman who proposed an idea he called the End Result System.

Hospitals should record what ailments incoming patients had, how they were treated, and—most important—the end result of each case.  These records should be compiled and statistics released so consumers could choose hospitals on the basis of good evidence.  Hospitals would respond to consumer pressure by hiring and promoting doctors on the same basis.  Medicine would improve, to the benefit of all.

Today, hospitals do much of what Codman suggested, and physicians have embraced such measurement.  But when Codman first put forth his idea, the medical establishment rejected it.  Hospitals didn’t want to pay for record keepers.  And doctors were already respected.  Keeping score could only hurt their reputations.   Codman was fired from Massachusetts General Hospital.  And Codman lost his teaching post at Harvard.  Eventually, however, his core idea was accepted.

Other areas of society are following the example of evidence-based medicine.  There’s evidence-based government policy.  There’s evidence-based philanthropy, led by the Gates Foundation.  There’s evidence-based management of sports teams.

One possible criticism of superforecasting is that it doesn’t deal with big enough questions.  Tetlock responds that a lot of smaller questions can ultimately shed light on bigger questions.  He calls this Bayesian question clustering.  For instance, in considering a big question like whether there will be another Korean war, you can focus on smaller questions related to missile launches, nuclear tests, cyber attacks, and artillery shelling.  The answers are cumulative.  The more yeses, the more likely the overall situation will end badly.

It’s interesting that hedgehog-minded experts are often good at coming up with big questions even though they’re usually not good at making forecasts.  Tetlock writes:

While we may assume that a superforecaster would also be a superquestioner, we don’t actually know that.  Indeed, my best scientific guess is that they often are not.  The psychological recipe for the ideal superforecaster may prove to be quite different from that for the ideal superquestioner, as superb question generation often seems to accompany a hedgehog-like incisiveness and confidence that one has a Big Idea grasp of the deep drivers of an event.  That’s quite a different mindset from the foxy eclecticism and sensitivity to uncertainty that characterizes superb forecasting.

Tetlock continues:

Superforecasters and superquestioners need to acknowledge each other’s complementary strengths, not dwell on each other’s alleged weaknesses.  Friedman poses provocative questions that superforecasters should use to sharpen their foresight; superforecasters generate well-calibrated answers that superquestioners should use to fine-tune and occasionally overhaul their mental models of reality…

…But there’s a much bigger collaboration I’d like to see.  It would be the Holy Grail of my research program: using forecasting tournaments to depolarize unnecessarily polarized policy debates and make us collectively smarter.




Focus on questions where your hard work is likely to pay off.  Don’t waste time either on easy “clocklike” questions (where simple rules of thumb can get you close to the right answer) or on impenetrable “cloud-like” questions (where even fancy statistical models can’t beat the dart-throwing chimp).  Concentrate on questions in the Goldilocks zone of difficulty, where effort pays off the most.

Break seemingly intractable problems into tractable sub-problems.

Decompose the problem into its knowable and unknowable parts.  Flush ignorance into the open.  Expose and examine your assumptions.  Dare to be wrong by making your best guesses.  Better to discover errors quickly than to hide them behind vague verbiage… The surprise is now often remarkably good probability estimates arise from a remarkably crude series of assumptions and guesstimates.

Strike the right balance between inside and outside views.

Superforecasters know that there is nothing new under the sun.  Nothing is 100% “unique.”… Superforecasters are in the habit of posing the outside-view question: How often do things of this sort happen in situations of this sort?

Strike the right balance between under- and overreacting to evidence.

The best forecasters tend to be incremental belief updaters, often moving from probabilities of, say, 0.4 to 0.35 or from 0.6 to 0.65, distinctions too subtle to capture with vague verbiage, like “might” or “maybe,” but distinctions that, in the long run, define the difference between good and great forecasters.

Yet superforecasters also know how to jump, to move their probability estimates fast in response to diagnostic signals.  Superforecasters are not perfect Bayesian updaters but they are better than most of us.  And that is largely because they value this skill and work hard at cultivating it.

Look for the clashing causal forces at work in each problem.

For every good policy argument, there is typically a counterargument that is at least worth acknowledging.  For instance, if you are a devout dove who believes that threatening military action never brings peace, be open to the possibility that you might be wrong about Iran.  And the same advice applies if you are a devout hawk who believes that soft “appeasement” policies never pay off.  Each side should list, in advance, the signs that would nudge them toward the other.

Now here comes the really hard part.  In classical dialectics, thesis meets antithesis, producing synthesis.  In dragonfly eye, one view meets another and another and another—all of which must be synthesized into a single image.  There are no paint-by-number rules here.  Synthesis is an art that requires reconciling irreducibly subjective judgments.  If you do it well, engaging in this process of synthesizing should transform you from a cookie-cutter dove or hawk into an odd hybrid creature, a dove-hawk, with a nuanced view of when tougher or softer policies are likelier to work.

Strive to distinguish as many degrees of doubt as the problem permits but no more.

Few things are either certain or impossible.  And “maybe” isn’t all that informative.  So your uncertainty dial needs more than three settings.  Nuance matters.  The more degrees of uncertainty you can distinguish, the better a forecaster you are likely to be.  As in poker, you have an advantage if you are better than your competitors at separating 60/40 bets from 40/60—or 55/45 from 45/55.  Translating vague-verbiage hunches into numeric probabilities feels unnatural at first but it can be done.  It just requires patience and practice.  The superforecasters have shown what is possible.

Strike the right balance between under- and overconfidence, between prudence and decisiveness.

Superforecasters understand the risks both of rushing to judgment and of dawdling too long near “maybe.”…They realize that long-term accuracy requires getting good scores on both calibration and resolution… It is not enough just to avoid the most recent mistake.  They have to find creative ways to tamp down both types of forecasting errors—misses and false alarms—to the degree a fickle world permits such uncontroversial improvements in accuracy.

Look for the errors behind your mistakes but beware of rearview-mirror hindsight biases.

Don’t try to justify or excuse your failures.  Own them!  Conduct unflinching postmortems: Where exactly did I go wrong?  And remember that although the more common error is to learn too little from failure and to overlook flaws in your basic assumptions, it is also possible to learn too much (you may have been basically on the right track but made a minor technical mistake that had big ramifications).  Also don’t forget to do postmortems on your successes too.  Not all successes imply that your reasoning was right.  You may have just lucked out by making offsetting errors.  And if you keep confidently reasoning along the same lines, you are setting yourself up for a nasty surprise.

Bring out the best in others and let others bring out the best in you.

Master the fine arts of team management, especially perspective taking (understanding the arguments of the other side so well that you can reproduce them to the other’s satisfaction), precision questioning (helping others to clarify their arguments so they are not misunderstood), and constructive confrontation (learning to disagree without being disagreeable).  Wise leaders know how fine the line can be between a helpful suggestion and micromanagerial meddling or between a rigid group and a decisive one or between a scatterbrained group and an open-minded one.

Master the error-balancing bicycle.

Implementing each commandment requires balancing opposing errors… Learning requires doing, with good feedback that leaves no ambiguity about whether you are succeeding… or whether you are failing… Also remember that practice is not just going through the motions of making forecasts, or casually reading the news and tossing out probabilities.  Like all other known forms of expertise, superforecasting is the product of deep, deliberative practice.

Don’t treat commandments as commandments.

Guidelines are the best we can do in a world where nothing is certain or exactly repeatable.  Superforecasting requires constant mindfulness, even when—perhaps especially when—you are dutifully trying to follow these commandments.



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:

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.

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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.

Fooled by Randomness

(Image:  Zen Buddha Silence by Marilyn Barbone.)

October 20, 2019

Nassim Nicholas Taleb’s Fooled by Randomness: The Hidden Role of Chance in the Markets and in Life, is an excellent book.  Below I summarize the main points.

Here’s the outline:

    • Prologue

Part I: Solon’s Warning—Skewness, Asymmetry, and Induction

    • One: If You’re So Rich, Why Aren’t You So Smart?
    • Two: A Bizarre Accounting Method
    • Three: A Mathematical Meditation on History
    • Four: Randomness, Nonsense, and the Scientific Intellectual
    • Five: Survival of the Least Fit—Can Evolution Be Fooled By Randomness?
    • Six: Skewness and Asymmetry
    • Seven: The Problem of Induction

Part II: Monkeys on Typewriters—Survivorship and Other Biases

    • Eight: Too Many Millionaires Next Door
    • Nine: It Is Easier to Buy and Sell Than Fry an Egg
    • Ten: Loser Takes All—On the Nonlinearities of Life
    • Eleven: Randomness and Our Brain—We Are Probability Blind

Part III: Wax in my Ears—Living With Randomitis

    • Twelve: Gamblers’ Ticks and Pigeons in a Box
    • Thirteen: Carneades Comes to Rome—On Probability and Skepticism
    • Fourteen: Bacchus Abandons Antony

(Albrecht Durer’s Wheel of Fortune from Sebastien Brant’s Ship of Fools (1494) via Wikimedia Commons)



Taleb presents Table P.1 Table of Confusion, listing the central distinctions used in the book.


Luck Skills
Randomness Determinism
Probability Certainty
Belief, conjecture Knowledge, certitude
Theory Reality
Anecdote, coincidence Causality, law
Forecast Prophecy


Lucky idiot Skilled investor
Survivorship bias Market outperformance


Volatility Return (or drift)
Stochastic variable Deterministic variable


Noise Signal


None Symbol


Epistemic probability Physical probability
Induction Deduction
Synthetic proposition Analytic proposition



Taleb introduces an options trader Nero Tulip.  He became convinced that being an options trader was even more interesting that being a pirate would be.

Nero is highly educated (like Taleb himself), with an undergraduate degree in ancient literature and mathematics from Cambridge University, a PhD. in philosophy from the University of Chicago, and a PhD. in mathematical statistics.  His thesis for the PhD. in philosophy had to do with the methodology of statistical inference in its application to the social sciences.  Taleb comments:

In fact, his thesis was indistinguishable from a thesis in mathematical statistics—it was just a bit more thoughtful (and twice as long).

Nero left philosophy because he became bored with academic debates, particularly over minor points.  Nero wanted action.

(Photo by Neil Lockhart)

Nero became a proprietary trader.  The firm provided the capital.  As long as Nero generated good results, he was free to work whenever he wanted.  Generally he was allowed to keep between 7% and 12% of his profits.

It is paradise for an intellectual like Nero who dislikes manual work and values unscheduled meditation.

Nero was an extremely conservative options trader.  Over his first decade, he had almost no bad years and his after-tax income averaged $500,000.  Due to his extreme risk aversion, Nero’s goal is not to maximize profits as much as it is to avoid having such a bad year that his “entertaining money machine called trading” would be taken away from him.  In other words, Nero’s goal was to avoid blowing up, or having such a bad year that he would have to leave the business.

Nero likes taking small losses as long as his profits are large.  Whereas most traders make money most of the time during a bull market and lose money during market panics or crashes, Nero would lose small amounts most of the time during a bull market and then make large profits during a market panic or crash.

Nero does not do as well as some other traders.  One reason is that his extreme risk aversion leads him to invest his own money in treasury bonds.  So he missed most of the bull market from 1982 to 2000.

Note: From a value investing point of view, Nero should at least have invested in undervalued stocks, since such a strategy will almost certainly do well after 10+ years.  But Nero wasn’t trained in value investing, and he was acutely aware of what can happen during market panics or crashes.

Also Note:  For a value investor, a market panic or crash is an opportunity to buy more stock at very cheap prices.  Thus bear markets benefit the value investor who can add to his or her positions.

Nero and his wife live across the street from John the High-Yield Trader and his wife.  John was doing much better than Nero.  John’s strategy was to maximize profits for as long as the bull market lasted.  Nero’s wife and even Nero himself would occasionally feel jealous when looking at the much larger house in which John and his wife lived.  However, one day there was a market panic and John blew up, losing virtually everything including his house.

Taleb writes:

…Nero’s merriment did not come from the fact that John went back to his place in life, so much as it was from the fact that Nero’s methods, beliefs, and track record had suddenly gained in credibility.  Nero would be able to raise public money on his track record precisely because such a thing could not possibly happen to him.  A repetition of such an event would pay off massively for him.  Part of Nero’s elation also came from the fact that he felt proud of his sticking to his strategy for so long, in spite of the pressure to be the alpha male.  It was also because he would no longer question his trading style when others were getting rich because they misunderstood the structure of randomness and market cycles.

Taleb then comments that lucky fools never have the slightest suspicion that they are lucky fools.  As long as they’re winning, they get puffed up from the release of the neurotransmitter serotonin into their systems.  Taleb notes that our hormonal system can’t distinguish between winning based on luck and winning based on skill.

(A lucky seven.  Photo by Eagleflying)

Furthermore, when serotonin is released into our system based on some success, we act like we deserve the success, regardless of whether it was based on luck or skill.  Our new behavior will often lead to a virtuous cycle during which, if we continue to win, we will rise in the pecking order.  Similarly, when we lose, whether that loss is due to bad luck or poor skill, our resulting behavior will often lead to a vicious cycle during which, if we continue to lose, we will fall in the pecking order.  Taleb points out that these virtuous and vicious cycles are exactly what happens with monkeys who have been injected with serotonin.

Taleb adds that you can always tell whether some trader has had a winning day or a losing day.  You just have to observe his or her gesture or gait.  It’s easy to tell whether the trader is full of serotonin or not.

Photo by Antoniodiaz



Taleb introduces the concept of alternative histories.  This concept applies to many areas of human life, including many different professions (war, politics, medicine, investments).  The main idea is that you cannot judge the quality of a decision based only on its outcome.  Rather, the quality of a decision can only be judged by considering all possible scenarios (outcomes) and their associated probabilities.

Once again, our brains deceive us unless we develop the habit of thinking probabilistically, in terms of alternative histories.  Without this habit, if a decision is successful, we get puffed up with serotonin and believe that the successful outcome is based on our skill.  By nature, we cannot account for luck or randomness.

Taleb offers Russian roulette as an analogy.  If you are offered $10 million to play Russian roulette, and if you play and you survive, then you were lucky even though you will get puffed up with serotonin.

Photo by Banjong Khanyai

Taleb argues that many (if not most) business successes have a large component of luck or randomness.  Again, though, successful businesspeople in general will be puffed up with serotonin and they will attribute their success primarily to skill.  Taleb:

…the public observes the external signs of wealth without even having a glimpse at the source (we call such source the generator).

Now, if the lucky Russian roulette player continues to play the game, eventually the bad histories will catch up with him or her.  Here’s an important point:  If you start out with thousands of people playing Russian roulette, then after the first round roughly 83.3% will be successful.  After the second round, roughly 83.3% of the survivors of round one will be successful.  After the third round, roughly 83.3% of the survivors of round two will be successful.  And on it goes…  After twenty rounds, there will be a small handful of extremely successful and wealthy Russian roulette players.  However, these cases of extreme success are due entirely to luck.

In the business world, of course, there are many cases where skill plays a large role.  The point is that our brains by nature are unable to see when luck has played a role in some successful outcome.  And luck almost always plays an important role in most areas of life.

Taleb points out that there are some areas where success is due mostly to skill and not luck.  Taleb likes to give the example of dentistry.  The success of a dentist will typically be due mostly to skill.

Taleb attributes some of his attitude towards risk to the fact that at one point he had a boss who forced him to consider every possible scenario, no matter how remote.

Interestingly, Taleb understands Homer’s The Iliad as presenting the following idea: heroes are heroes based on heroic behavior and not based on whether they won or lost.  Homer seems to have understood the role of chance (luck).



A Monte Carlo generator creates many alternative random sample paths.  Note that a sample path can be deterministic, but our concern here is with random sample paths.  Also note that some random sample paths can have higher probabilities than other random sample paths.  Each sample path represents just one sequence of events out of many possible sequences, ergo the word “sample”.

Taleb offers a few examples of random sample paths.  Consider the price of your favorite technology stock, he says.  It may start at $100, hit $220 along the way, and end up at $20.  Or it may start at $100 and reach $145, but only after touching $10.  Another example might be your wealth during at a night at the casino.  Say you begin with $1,000 in your pocket.  One possibility is that you end up with $2,200, while another possibility is that you end up with only $20.

Photo by Emily2k

Taleb says:

My Monta Carlo engine took me on a few interesting adventures.  While my colleagues were immersed in news stories, central bank announcements, earnings reports, economic forecasts, sports results and, not least, office politics, I started toying with it in fields bordering my home base of financial probability.  A natural field of expansion for the amateur is evolutionary biology… I started simulating populations of fast mutating animals called Zorglubs under climactic changes and witnessing the most unexpected of conclusions… My aim, as a pure amateur fleeing the boredom of business life, was merely to develop intuitions for these events… I also toyed with molecular biology, generating randomly occurring cancer cells and witnessing some surprising aspects to their evolution.

Taleb continues:

Naturally the analogue to fabricating populations of Zorglubs was to simulate a population of “idiotic bull”, “impetuous bear”, and “cautious” traders under different market regimes, say booms and busts, and to examine their short-term and long-term survival… My models showed almost nobody to really ultimately make money; bears dropped out like flies in the rally and bulls got ultimately slaughtered, as paper profits vanished when the music stopped.  But there was one exception; some of those who traded options (I called them option buyers) had remarkable staying power and I wanted to be one of those.  How?  Because they could buy insurance against the blowup; they could get anxiety-free sleep at night, thanks to the knowledge that if their careers were threatened, it would not be owing to the outcome of a single day.

Note from a value investing point of view

A value investor seeks to pay low prices for stock in individual businesses.  Stock prices can jump around in the short term.  But over time, if the business you invest in succeeds, then the stock will follow, assuming you bought the stock at relatively low prices.  Again, if there’s a bear market or a market crash, and if the stock prices of the businesses in which you’ve invested decline, then that presents a wonderful opportunity to buy more stock at attractively low prices.  Over time, the U.S. and global economy will grow, regardless of the occasional market panic or crash.  Because of this growth, one of the lowest risk ways to build wealth is to invest in businesses, either on an individual basis if you’re a value investor or via index funds.

Taleb’s methods of trying to make money during a market panic or crash will almost certainly do less well over the long term than simple index funds.

Taleb makes a further point: The vast majority of people learn only from their own mistakes, and rarely from the mistakes of others.  Children only learn that the stove is hot by getting burned.  Adults are largely the same way: We only learn from our own mistakes.  Rarely do we learn from the mistakes of others.  And rarely do we heed the warnings of others.  Taleb:

All of my colleagues whom I have known to denigrate history blew up spectacularly—and I have yet to encounter some such person who has not blown up.

Keep in mind that Taleb is talking about traders here.  For a regular investor who dollar cost averages into index funds and/or who uses value investing, Taleb’s warning does not apply.  As a long-term investor in index funds and/or in value investing techniques, you do have to be ready for a 50% decline at some point.  But if you buy more after such a decline, your long-term results will actually be helped, not hurt, by a 50% decline.

Taleb points out that aged traders and investors are likely better to use as role models precisely because they have been exposed to markets longer.  Taleb:

I toyed with Monte Carlo simulations of heterogeneous populations of traders under a variety of regimes (closely resembling historical ones), and found a significant advantage in selecting aged traders, using, as a selection criterion their cumulative years of experience rather than their absolute success (conditional on their having survived without blowing up).

Taleb also observes that there is a similar phenomenon in mate selection.  All else equal, women prefer to mate with healthy older men over healthy younger ones.  Healthy older men, by having survived longer, show some evidence of better genes.



Using a random generator of words, it’s possible to create rhetoric, but it’s not possible to generate genuine scientific knowledge.



Taleb writes about Carlos “the emerging markets wizard.”  After excelling as an undergraduate, Carlos went for a PhD. in economics from Harvard.  Unable to find a decent thesis topic for his dissertation, he settled for a master’s degree and a career on Wall Street.

Carlos did well investing in emerging markets bonds.  One important reason for his success, beyond the fact that he bought emerging markets bonds that later went up in value, was that he bought the dips.  Whenever there was a momentary panic and emerging markets bonds dropped in value, Carlos bought more.  This dip buying improved his performance.  Taleb:

It was the summer of 1998 that undid Carlos—that last dip did not translate into a rally.  His track record today includes just one bad quarter—but bad it was.  He had earned close to $80 million cumulatively in his previous years.  He lost $300 million in just one summer.

When the market first started dipping, Carlos learned that a New Jersey hedge fund was liquidating, including its position in Russian bonds.  So when Russian bonds dropped to $52, Carlos was buying.  To those who questioned his buying, he yelled: “Read my lips: it’s li-qui-da-tion!”

Taleb continues:

By the end of June, his trading revenues for 1998 had dropped from up $60 million to up $20 million.  That made him angry.  But he calculated that should the market rise back to the pre-New Jersey selloff, then he would be up $100 million.  That was unavoidable, he asserted.  These bonds, he said, would never, ever trade below $48.  He was risking so little, to possibly make so much.

Then came July.  The market dropped a bit more.  The benchmark Russian bond was now $43.  His positions were under water, but he increased his stakes.  By now he was down $30 million for the year.  His bosses were starting to become nervous, but he kept telling them that, after all, Russia would not go under.  He repeated the cliche that it was too big to fail.  He estimated that bailing them out would cost so little and would benefit the world economy so much that it did not make sense to liquidate his inventory now.

Carlos asserted that the Russian bonds were trading near default value.  If Russia were to default, then Russian bonds would stay at the same prices they were at currently.  Carlos took the further step of investing half of his net worth, then $5,000,000, into Russian bonds.

Russian bond prices then dropped into the 30s, and then into the 20s.  Since Carlos thought the bonds could not be less than the default values he had calculated, and were probably worth much more, he was not alarmed.  He maintained that anyone who invested in Russian bonds at these levels would realize wonderful returns.  He claimed that stop losses “are for schmucks!  I am not going to buy high and sell low!”  He pointed out that in October 1997 they were way down, but that buying the dip ended up yielding excellent profits for 1997.  Furthermore, Carlos pointed out that other banks were showing even larger losses on their Russian bond positions.  Taleb:

Towards the end of August, the bellwether Russian Principal Bonds were trading below $10.  Carlos’s net worth was reduced by almost half.  He was dismissed.  So was his boss, the head of trading.  The president of the bank was demoted to a “newly created position”.  Board members could not understand why the bank had so much exposure to a government that was not paying its own employees—which, disturbingly, included armed soldiers.  This was one of the small points that emerging market economists around the globe, from talking to each other so much, forgot to take into account.

Taleb adds:

Louie, a veteran trader on the neighboring desk who suffered much humiliation by these rich emerging market traders, was there, vindicated.  Louie was then a 52-year-old Brooklyn-born-and-raised trader who over three decades survived every single conceivable market cycle.

Taleb concludes that Carlos is a gentleman, but a bad trader:

He has all of the traits of a thoughtful gentleman, and would be an ideal son-in-law.  But he has most of the attributes of the bad trader.  And, at any point in time, the richest traders are often the worst traders.  This, I will call the cross-sectional problem: at a given time in the market, the most profitable traders are likely to be those that are best fit to the latest cycle.

Taleb discusses John the high-yield trader, who was mentioned near the beginning of the book, as another bad trader.  What traits do bad traders, who may be lucky idiots for awhile, share?  Taleb:

    • An overestimation of the accuracy of their beliefs in some measure, either economic (Carlos) or statistical (John).  They don’t consider that what they view as economic or statistical truth may have been fit to past events and may no longer be true.
    • A tendency to get married to positions.
    • The tendency to change their story.
    • No precise game plan ahead of time as to what to do in the event of losses.
    • Absence of critical thinking expressed in absence of revision of their stance with “stop losses”.
    • Denial.



Taleb presents the following Table:

Event Probability Outcome Expectation
A 999/1000 $1 $.999
B 1/1000 -$10,000 -$10.00
Total -$9.001

The point is that the frequency of losing cannot be considered apart from the magnitude of the outcome.  If you play the game, you’re extremely likely to make $1.  But it’s not a good idea to play.  If you play this game millions of times, you’re virtually guaranteed to lose money.

Taleb comments that even professional investors misunderstand this bet:

How could people miss such a point?  Why do they confuse probability and expectation, that is, probability and probability times the payoff?  Mainly because much of people’s schooling comes from examples in symmetric environments, like a coin-toss, where such a difference does not matter.  In fact the so-called “Bell Curve” that seems to have found universal use in society is entirely symmetric.

(Coin toss.  Photo by Christian Delbert)

Taleb gives an example where he is shorting the S&P 500 Index.  He thought the market had a 70% chance of going up and a 30% chance of going down.  But he thought that if the market went down, it could go down a lot.  Therefore, it was profitable over time (by repeating the bet) to be short the S&P 500.

Note: From a value investing point of view, no one can predict what the market will do.  But you can predict what some individual businesses are likely to do.  The key is to invest in businesses when the price (stock) is low.

Rare Events

Taleb explains his trading strategy:

The best description of my lifelong business in the market is “skewed bets”, that is, I try to benefit from rare events, events that do not tend to repeat themselves frequently, but, accordingly, present a large payoff when they occur.  I try to make money infrequently, as infrequently as possible, simply because I believe that rare events are not fairly valued, and that the rarer the event, the more undervalued it will be in price.

Illustration by lqoncept

Taleb gives an example where his strategy paid off:

One such rare event is the stock market crash of 1987, which made me as a trader and allowed me the luxury of becoming involved in all manner of scholarship.

Taleb notes that in most areas of science, it is common practice to discard outliers when computing the average.  For instance, a professor calculating the average grade in his or her class might discard the highest and the lowest values.  In finance, however, it is often wrong to discard the extreme outcomes because, as Taleb has shown, the magnitude of an extreme outcome can matter.

Taleb advises studying market history.  But then again, you have to be careful, as Taleb explains:

Sometimes market data becomes a simple trap; it shows you the opposite of its nature, simply to get you to invest in the security or mismanage your risks.  Currencies that exhibit the largest historical stability, for example, are the most prone to crashes…

Taleb notes the following:

In other words history teaches us that things that never happened before do happen.

History does not always repeat.  Sometimes things change.  For instance, today the U.S. stock market seems high.  The S&P 500 Index is over 3,000.  Based on history, one might expect a bear market and/or a recession.  There hasn’t been a recession in the U.S. since 2009.

However, with interest rates low, and with the profit margins on many technology companies high, it’s possible that stocks will not decline much, even if there’s a recession.  It’s also possible that any recession could be delayed, partly because the Fed and other central banks remain very accommodative.  It’s possible that the business cycle itself may be less volatile because the fiscal and monetary authorities have gotten better at delaying recessions or at making recessions shallower than before.

Ironically, to the extent that Taleb seeks to profit from a market panic or crash, for the reasons just mentioned, Taleb’s strategy may not work as well going forward.

Taleb introduces the problem of stationarity.  To illustrate the problem, think of an urn with red balls and black balls in it.  Taleb:

Think of an urn that is hollow at the bottom.  As I am sampling from it, and without my being aware of it, some mischievous child is adding balls of one color or another.  My inference thus becomes insignificant.  I may infer that the red balls represent 50% of the urn while the mischievous child, hearing me, would swiftly replace all the red balls with black ones.  This makes much of our knowledge derived through statistics quite shaky.

The very same effect takes place in the market.  We take past history as a single homogeneous sample and believe that we have considerably increased our knowledge of the future from the observation of the sample of the past.  What if vicious children were changing the composition of the urn?  In other words, what if things have changed?

Taleb notes that there are many techniques that use past history in order to measure risks going forward.  But to the extent that past data are not stationary, depending upon these risk measurement techniques can be a serious mistake.  All of this leads to a more fundamental issue: the problem of induction.



Taleb quotes the Scottish philosopher David Hume:

No amount of observations of white swans can allow the inference that all swans are white, but the observation of a single black swan is sufficient to refute that conclusion.

(Black swan.  Photo by Damithri)

Taleb came to believe that Sir Karl Popper had an important answer to the problem of induction.  According to Popper, there are only two types of scientific theories:

    • Theories that are known to be wrong, as they were tested and adequately rejected (i.e., falsified).
    • Theories that have not yet been known to be wrong, not falsified yet, but are exposed to be proved wrong.

It also follows that we should not always rely on statistics.  Taleb:

More practically to me, Popper had many problems with statistics and statisticians.  He refused to blindly accept the notion that knowledge can always increase with incremental information—which is the foundation for statistical inference.  It may in some instances, but we do not know which ones.  Many insightful people, such as John Maynard Keynes, independently reached the same conclusions.  Sir Karl’s detractors believe that favorably repeating the same experiment again and again should lead to an increased comfort with the notion that “it works”.

Taleb explains the concept of an open society:

Popper’s falsificationism is intimately connected to the notion of an open society.  An open society is one in which no permanent truth is held to exist; this would allow counterideas to emerge.

For Taleb, a successful trader or investor must have an open mind in which no permanent truth is held to exist.

Taleb concludes the chapter by applying the logic of Pascal’s wager to trading and investing:

…I will use statistics and inductive methods to make aggressive bets, but I will not use them to manage my risks and exposure.  Surprisingly, all the surviving traders I know seem to have done the same.  They trade on ideas based on some observation (that includes past history) but, like the Popperian scientists, they make sure that the costs of being wrong are limited (and their probability is not derived from past data).  Unlike Carlos and John, they know before getting involved in the trading strategy which events would prove their conjecture wrong and allow for it (recall the Carlos and John used past history both to make their bets and measure their risk).



If you put an infinite number of monkeys in front of typewriters, it is certain that one of them will type an exact version of Homer’s The Iliad.  Taleb asks:

Now that we have found that hero among monkeys, would any reader invest his life’s savings on a bet that the monkey would write The Odyssey next?

Infinite number of monkeys on typewriters.  Illustration by Robert Adrian Hillman.



Taleb begins the chapter by describing a lawyer named Marc.  Marc makes $500,000 a year.  He attended Harvard as an undergraduate and then Yale Law School.  The problem is that some of Marc’s neighbors are much wealthier.  Taleb discusses Marc’s wife, Janet:

Every month or so, Janet has a crisis… Why isn’t her husband so successful?  Isn’t he smart and hard working?  Didn’t he get close to 1600 on the SAT?  Why is Ronald Something whose wife never even nods to Janet worth hundred of millions when her husband went to Harvard and Yale and has such a high I.Q., and has hardly any substantial savings?

Note: Warren Buffett and Charlie Munger have long made the point that envy is a massively stupid sin because, unlike other sins (e.g., gluttony), you can’t have any fun with it.  Granted, envy is a very human emotion.  But we can and must train ourselves not to fall into it.

Daniel Kahneman and others have demonstrated that the average person would rather make $70,000 as long as his neighbor makes $60,000 than make $80,000 if his neighbor makes $90,000.  How stupid to compare ourselves to people who happen to be doing better!  There will always be someone doing better.

Taleb mentions the book, The Millionaire Next Door.  One idea from the book is that the wealthy often do not look wealthy because they’re focused on saving and investing, rather than on spending.  However, Taleb finds two problems with the book.  First, the book does not adjust for survivorship bias.  In other words, for at least some of the wealthy, there is some luck involved.  Second, there’s the problem of induction.  If you measure someone’s wealth in the year 2000 (Taleb was writing in 2001), at the end of one of the biggest bull markets in modern history (from 1982 to 2000), then in many cases a large degree of that wealth came as a result of the prolonged bull market.  By contrast, if you measure people’s wealth in 1982, there would be fewer people who are millionaires, even after adjusting for inflation.



Taleb writes about going to the dentist and being confident that his dentist knows something about teeth.  Later, Taleb goes to Carnegie Hall.  Before the pianist begins her performance, Taleb has zero doubt that she knows how to play the piano and is not about to produce cacophony.  Later still, Taleb is in London and ends up looking at some of his favorite marble statues.  Once again, he knows they weren’t produced by luck.

However, in many areas of business and even more so when it comes to investing, luck does tend to play a large role.  Taleb is supposed to meet with a fund manager who has a good track record and who is looking for investors.  Taleb comments that buying and selling, which is what the fund manager does, is easier than frying an egg.  The problem is that luck plays such a large role in almost any good investment track record.

Photo by Alhovik

In order to study the role luck plays for investors, Taleb suggests a hypothetical game.  There are 10,000 investors at the beginning.  In the first round, a fair coin is tossed for each investor.  Heads, and the investor makes $10,000, tails, and the investor loses $10,000.  (Any investor who has a losing year is not allowed to continue to play the game.)  After the first round, there will be about 5,000 successful investors.  In the second round, a fair coin is again tossed.  After the second round, there will be 2,500 successful investors.  Another round, and 1,250 will remain.  A fourth round, and 625 successful investors will remain.  A fifth round, and 313 successful investors will remain.  Based on luck alone, after five years there will be approximately 313 investors with winning track records.  No doubt these 313 winners will be puffed up with serotonin.

Taleb then observes that you can play the same hypothetical game with bad investors.  You assume each year that there’s a 45% chance of winning and a 55% chance of losing.  After one year, 4,500 successful (but bad) investors will remain.  After two years, 2,025.  After three years, 911.  After four years, 410.  After five years, there will be 184 bad investors who have successful track records.

Taleb makes two counterintuitive points:

    • First, even starting with only bad investors, you will end up with a small number of great track records.
    • Second, how many great track records you end up with depends more on the size of the initial sample—how many investors you started with—than it does on the individual odds per investor.  Applied to the real world, this means that if there are more investors who start in 1997 than in 1993, then you will see a greater number of successful track records in 2002 than you will see in 1998.

Taleb concludes:

Recall that the survivorship bias depends on the size of the initial population.  The information that a person made money in the past, just by itself, is neither meaningful nor relevant.  We need to know that size of the population from which he came.  In other words, without knowing how many managers out there have tried and failed, we will not be able to assess the validity of the track record.  If the initial population includes ten managers, then I would give the performer half my savings without a blink.  If the initial population is composed of 10,000 managers, I would ignore the results.

The mysterious letter

Taleb tells a story.  You get a letter on Jan. 2 informing you that the market will go up during the month.  It does.  Then you get a letter on Feb. 1 saying the market will go down during the month.  It does.  You get another letter on Mar. 1.  Same story.  Again for April and for May.  You’ve now gotten five letters in a row predicting what the market would do during the ensuing month, and all five letters were correct.  Next you are asked to invest in a special fund.  The fund blows up.  What happened?

The trick is as follows.  The con operator gets 10,000 random names.  On Jan. 2, he mails 5,000 letters predicting that the market will go up and 5,000 letters predicting that the market will go down.  The next month, he focuses only on the 5,000 names who were just mailed a correct prediction.  He sends 2,500 letters predicting that the market will go up and 2,500 letters predicting that the market will go down.  Of course, next he focuses on the 2,500 letters which gave correct predictions.  He mails 1,250 letters predicting a market rise and 1,250 predicting a market fall.  After five months of this, there will be approximately 200 people who received five straight correct predictions.

Taleb suggests the birthday paradox as an intuitive way to explain the data mining problem.  If you encounter a random person, there is a one in 365.25 chance that you have the same birthday.  But if you have 23 random people in a room, the odds are close to 50 percent that you can find two people who share a birthday.

Similarly, what are the odds that you’ll run into someone you know in a totally random place?  The odds are quite high because you are testing for any encounter, with any person you know, in any place you will visit.

Taleb continues:

What is your probability of winning the New Jersey lottery twice?  One in 17 trillion.  Yet it happened to Evelyn Adams, whom the reader might guess should feel particularly chosen by destiny.  Using the method we developed above, Harvard’s Percy Diaconis and Frederick Mosteller estimated at 30 to 1 the probability the someone, somewhere, in a totally unspecified way, gets so lucky!

What is data snooping?  It’s looking  at historical data to determine the hypothetical performance of a large number of trading rules.  The more trading rules you examine, the more likely you are to find trading rules that would have worked in the past and that one might expect to work in the future.  However, many such trading rules would have worked in the past based on luck alone.

Taleb next writes about companies that increase their earnings.  The same logic can be applied.  If you start out with 10,000 companies, then by luck 5,000 will increase their profits after the first year.  After three years, there will be 1,250 “stars” that increased their profits for three years in a row.  Analysts will rate these companies a “strong buy”.  The point is not that profit increases are entirely due to luck.  The poin, rather, is that luck often plays a significant role in business results, usually far more than is commonly supposed.



Taleb writes:

This chapter is about how a small advantage in life can translate into a highly disproportionate payoff, or, more viciously, how no advantage at all, but a very, very small help from randomness, can lead to a bonanza.

Nonlinearity is when a small input can lead to a disproportionate response.  Consider a sandpile.  You can add many grains of sand with nothing happening.  Then suddenly one grain of sand causes an avalanche.

(Photo by Maocheng)

Taleb mentions actors auditioning for parts.  A handful of actors get certain parts, and a few of them become famous.  The most famous actors are not always the best actors (although they often are).  Rather, there could have been random (lucky) reasons why a handful of actors got certain parts and why a few of them became famous.

The QWERTY keyboard is not optimal.  But so many people were trained on it, and so many QWERTY keyboards were manufactured, that it has come to dominate.  This is called a path dependent outcome.  Taleb comments:

Such ideas go against classical economic models, in which results either come from a precise reason (there is no account for uncertainty) or the good guy wins (the good guy is the one who is more skilled and has some technical superiority)… Brian Arthur, an economist concerned with nonlinearities at the Santa Fe Institute, wrote that chance events coupled with positive feedback rather than technological superiority will determine economic superiority—not some abstrusely defined edge in a given area of expertise.  While early economic models excluded randomness, Arthur explained how “unexpected orders, chance meetings with lawyers, managerial whims… would help determine which ones achieved early sales and, over time, which firms dominated”.

Taleb continues by noting that Arthur suggests a mathematical model called the Polya process:

The Polya process can be presented as follows: assume an urn initially containing equal quantities of black and red balls.  You are to guess each time which color you will pull out before you make the draw.  Here the game is rigged.  Unlike a conventional urn, the probability of guessing correctly depends on past success, as you get better or worse at guessing depending on past performance.  Thus the probability of winning increases after past wins, that of losing increases after past losses.  Simulating such a process, one can see a huge variance of outcomes, with astonishing successes and a large number of failures (what we called skewness).



Our genes have not yet evolved to the point where our brains can naturally compute probabilities.  Computing probabilities is not something we even needed to do until very recently.

Here’s a diagram of how to compute the probability of A, conditional on B having happened:

(Diagram by Oleg Alexandrov, via Wikimedia Commons)


We are capable of sending a spacecraft to Mars, but we are incapable of having criminal trials managed by the basic laws of probability—yet evidence is clearly a probabilistic notion…

People who are as close to being criminal as probability laws can allow us to infer (that is with a confidence that exceeds the shadow of a doubt) are walking free because of our misunderstanding of basic concepts of the odds… I was in a dealing room with a TV set turned on when I saw one of the lawyers arguing that there were at least four people in Los Angeles capable of carrying O.J. Simpson’s DNA characteristics (thus ignoring the joint set of events…).  I then switched off the television set in disgust, causing an uproar among the traders.  I was under the impression until then that sophistry had been eliminated from legal cases thanks to the high standards of republican Rome.  Worse, one Harvard lawyer used the specious argument that only 10% of men who brutalize their wives go on to murder them, which is a probability unconditional on the murder… Isn’t the law devoted to the truth?  The correct way to look at it is to determine the percentage of murder cases where women were killed by their husband and had previously been battered by him (that is, 50%)—for we are dealing with what is called conditional probabilities; the probability that O.J. killed his wife conditional on the information of her having been killed, rather than the unconditional probability of O.J. killing his wife.  How can we expect the untrained person to understand randomness when a Harvard professor who deals and teaches the concept of probabilistic evidence can make such an incorrect statement?

Speaking of people misunderstanding probabilities, Daniel Kahneman and Amos Tversky have asked groups to answer the following question:

Linda is 31 years old, single, outspoken, and very bright.  She majored in philosophy.  As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more probable?

    1. Linda is a bank teller.
    2. Linda is a bank teller and is active in the feminist movement.

The majority of people believe that 2. is more probable the 1.  But that’s an obvious fallacy.  Bank tellers who are also feminists is a subset of all bank tellers, therefore 1. is more probable than 2.  To see why, consider the following diagram:

(By svjo, via Wikimedia Commons)

B represents ALL bank tellers.  Out of ALL bank tellers, some are feminists and some are not.  Those bank tellers that are also feminists is represented by A.

Here’s a probability question that was presented to doctors:

A test of a disease presents a rate of 5% false positives.  The disease strikes 1/1,000 of the population.  People are tested at random, regardless of whether they are suspected of having the disease.  A patient’s test is positive.  What is the probability of the patient being stricken with the disease?

Many doctors answer 95%, which is wildly incorrect.  The answer is close to 2%.  Less than one in five doctors get the question right.

To see the right answer, assume that there are no false negatives.  Out of 1,000 patients, one will have the disease.  Consider the remaining 999.  50 of them will test positive.  The probability of being afflicted with the disease for someone selected at random who tested positive is the following ratio:

Number of afflicted persons  /  Number of true and false positives

So the answer is 1/51, about 2%.

Another example where people misunderstand probabilities is when it comes to valuing options.  (Recall that Taleb is an options trader.)  Taleb gives an example.  Say that the stock price is $100 today.  You can buy a call option for $1 that gives you the right to buy the stock at $110 any time during the next month.  Note that the option is out-of-the-money because you would not gain if you exercised your right to buy now, given that the stock is $100, below the exercise price of $110.

Now, what is the expected value of the option?  About 90 percent of out-of-the-money options expire worthless, that is, they end up being worth $0.  But the expected value is not $0 because there is a 10 percent chance that the option could be worth, say $10, because the stock went to $120.  So even though it is 90 percent likely that the option will end up being worth $0, the expected value is not $0.  The actual expected value in this example is:

(90% x $0) + (10% x $10) = $0 + $1 = $1

The expected value of the option is $1, which means you would have paid a fair price if you had bought it for $1.  Taleb notes:

I discovered very few people who accepted losing $1 for most expirations and making $10 once in a while, even if the game were fair (i.e., they made the $10 more than 10% of the time).

“Fair” is not the right term here.  If you make $10 more than 10% of the time, then the game has a positive expected value.  That means if you play the game repeatedly, then eventually over time you will make money.  Taleb’s point is that even if the game has a positive expected value, very few people would like to play it because on your way to making money, you have to accept small losses most of the time.

Taleb distinguishes between premium sellers, who sell options, and premium buyers, who buy options.  Following the same logic as above, premium sellers make small amounts of money roughly 90% of the time, and then take a big loss roughly 10% of the time.  Premium buyers lose small amounts about 90% of the time, and then have a big gain about 10% of the time.

Is it better to be an option seller or an option buyer?  It depends on whether you can find favorable odds.  It also depends on your temperament.  Most people do not like taking small losses most of the time.  Taleb:

Alas, most option traders I encountered in my career are premium sellers—when they blow up it is generally other people’s money.



Taleb writes that when Odysseus and his crew encountered the sirens, Odysseus had his crew put wax in their ears.  He also instructed his crew to tie him to the mast.  With these steps, Odysseus and crew managed to survive the sirens’ songs.  Taleb notes that he would be not Odysseus, but one of the sailors who needed to have wax in his ears.

(Odysseus and crew at the sirens.  Illustration by Mr1805)

Taleb admits that he is dominated by his emotions:

The epiphany I had in my career in randomness came when I understood that I was not intelligent enough, nor strong enough, to even try to fight my emotions.  Besides, I believe that I need my emotions to formulate my ideas and get the energy to execute them.

I am just intelligent enough to understand that I have a predisposition to be fooled by randomness—and to accept the fact that I am rather emotional.  I am dominated by my emotions—but as an aesthete, I am happy about that fact.  I am just like every single character whom I ridiculed in this book… The difference between myself and those I ridicule is that I try to be aware of it.  No matter how long I study and try to understand probability, my emotions will respond to a different set of calculations, those that my unintelligent genes want me to handle.

Taleb says he has developed tricks in order to handle his emotions.  For instance, if he has financial news playing on the television, he keeps the volume off.  Without volume, a babbling person looks ridiculous.  This trick helps Taleb stay free of news that is not rationally presented.



Early in his career as a trader, Taleb says he had a particularly profitable day.  It just so happens that the morning of this day, Taleb’s cab driver dropped him off in the wrong location.  Taleb admits that he was superstitious.  So the next day, he not only wore the same tie, but he had his cab driver drop him off in the same wrong location.

(Skinner boxes.  Photo by Luis Dantas, via Wikimedia Commons)

B.F. Skinner did an experiment with famished pigeons.  There was a mechanism that would deliver food to the box in which the hungry pigeon was kept.  But Skinner programmed the mechanism to deliver the food randomly.  Taleb:

He saw quite astonishing behavior on the part of the birds; they developed an extremely sophisticated rain-dance type of behavior in response to their ingrained statistical machinery.  One bird swung its head rhythmically against a specific corner of the box, others spun their heads anti-clockwise; literally all of the birds developed a specific ritual that progressively became hard-wired into their mind as linked to their feeding.

Taleb observes that whenever we experience two events, A and B, our mind automatically looks for a causal link even though there often is none.  Note: Even if B always follows A, that doesn’t prove a causal link, as Hume pointed out.

Taleb again admits that after he has calculated the probabilities in some situation, he finds it hard to modify his own conduct accordingly.  He gives an example of trading.  Taleb says if he is up $100,000, there is a 98% chance that it’s just noise.  But if he is up $1,000,000, there is a 1% chance that it’s noise and a 99% chance that his strategy is profitable.  Taleb:

A rational person would act accordingly in the selection of strategies, and set his emotions in accordance with his results.  Yet I have experienced leaps of joy over results that I knew were mere noise, and bouts of unhappiness over results that did not carry the slightest degree of statistical significance.  I cannot help it…

Taleb uses another trick to deal with this.  He denies himself access to his performance report unless it hits a predetermined threshold.



Taleb writes:

Carneades was not merely a skeptic; he was a dialectician, someone who never committed himself to any of the premises from which he argued, or to any of the conclusions he drew from them.  He stood all his life against arrogant dogma and belief in one sole truth.  Few credible thinkers rival Carneades in their rigorous skepticism (a class that would include the medieval Arab philosopher Al Gazali, Hume, and Kant—but only Popper came to elevate his skepticism to an all-encompassing scientific methodology).  As the skeptics’ main teaching was that nothing could be accepted with certainty, conclusions of various degrees of probability could be formed, and these supplied a guide to conduct.

Taleb holds that Cicero engaged in probabilistic reasoning:

He preferred to be guided by probability than allege with certainty—very handy, some said, because it allowed him to contradict himself.  This may be a reason for us, who have learned from Popper how to remain self critical, to respect him more, as he did not hew stubbornly to an opinion for the mere fact that he had voiced it in the past.

Taleb asserts that the speculator George Soros has a wonderful ability to change his opinions rather quickly.  In fact, without this ability, Soros could not have become so successful as a speculator.  There are many stories about Soros holding one view strongly, only to abandon it very quickly and take the opposite view, leading to a large profit where there otherwise would have been a large loss.

Most of us tend to become married to our favorite ideas.  Most of us are not like George Soros.  Especially after we have invested time and energy into developing some idea.

At the extreme, just imagine a scientist who spent years developing some idea.  Many scientists in that situation have a hard time abandoning their idea, even after there is good evidence that they’re wrong.  That’s why it is said that science evolves from funeral to funeral.



Taleb refers to C.P. Cavafy’s poem, Apoleipein o Theos Antonion (The God Abandons Antony).  The poem addresses Antony after he has been defeated.  Taleb comments:

There is nothing wrong and undignified with emotions—we are cut to have them.  What is wrong is not following the heroic, or at least, the dignified path.  That is what stoicism means.  It is the attempt by man to get even with probability.

Seneca 4 BC-65 AD Roman stoic philosopher, statesman, and tutor to the future Emperor Nero.  Photo by Bashta.

Taleb concludes with some advice (stoicism):

Dress at your best on your execution day (shave carefully); try to leave a good impression on the death squad by standing erect and proud.  Try not to play victim when diagnosed with cancer (hide it from others and only share the information with the doctor—it will avert the platitudes and nobody will treat you like  a victim worthy of their pity; in addition the dignified attitude will make both defeat and victory feel equally heroic).  Be extremely courteous to your assistant when you lose money (instead of taking it out on him as many of the traders whom I scorn routinely do).  Try not to blame others for your fate, even if they deserve blame.  Never exhibit any self pity, even if your significant other bolts with the handsome ski instructor or the younger aspiring model.  Do not complain… The only article Lady Fortuna has no control over is your behavior.



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:

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.

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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.