Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

His second trip to Africa when he was 27 left Galton “rather used up in health,” the result of a combination of physical exhaustion and bouts of depression that were to recur often though briefly throughout his life. He referred to himself on those occasions as someone with a “sprained brain.”

Whether Galton should be charged with responsibility for that evil outcome has been the subject of spirited debate. There is nothing about the man to suggest that he would have condoned such barbaric behavior.

In one experiment, for example, Quetelet took chest measurements on 5,738 Scottish soldiers. He concocted a normal distribution for the group and then compared the actual result with the theoretical result. The fit was almost perfect.

One of Quetelet’s studies, however, revealed a less than perfect fit with the normal distribution. His analysis of the heights of 100,000 French conscripts revealed that too many of them fell in the shortest class for the distribution to be normal. Since being too short was an excuse for exemption from service, Quetelet asserted that the measurements must have been distorted by fraud in order to accommodate draft-dodgers.

He found that only 36% of the sons of eminent men were themselves eminent; even worse, only 9% of their grandsons made the grade.

Note that the spread of diameters among the parents was wider than the dispersion among the offspring. The average diameter of the parents was 0.18 inches within a range of 0.15 to 0.21 inches, or 0.03 on either side of the mean.

“Reversion,” he wrote, “is the tendency of the ideal mean filial type to depart from the parental type, reverting to what may be roughly and perhaps fairly described as the average ancestral type.”

The mid-parents with heights of 68.5 inches and up all had children whose median heights were below the height of the mid-parents; the mid-parents who were shorter than 68.5 inches all had children who tended to be taller than they were.

Karl Pearson, Galton’s principal biographer

Francis Galton was a proud man, but he never suffered a fall. His many achievements were widely recognized. He ended a long, full life as a widower traveling and writing in the company of a much younger female relative.

It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as that of the native of one of our flat English counties, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be got rid of at once.41

Galton

There are three reasons why regression to the mean can be such a frustrating guide to decision-making. First, it sometimes proceeds at so slow a pace that a shock will disrupt the process. Second, the regression may be so strong that matters do not come to rest once they reach the mean. Rather, they fluctuate around the mean, with repeated, irregular deviations on either side. Finally, the mean itself may be unstable, so that yesterday’s normality may be supplanted today by a new normality that we know nothing about.

Reichenstein and Dorsett studied the S&P 500 from 1926 to 1993 and found that the variance of three-year returns was only 2.7 times the variance of annual returns; the variance of eight-year returns was only 5.6 times the variance of annual returns.

H < 0.5

The overreaction to new information that DeBondt and Thaler reported in the behavior of stock prices was the result of the human tendency to overweight recent evidence and to lose sight of the long run.

William Baumol published an enlightening study of long-run trends in productivity. His data came from 72 countries and reached back to 1870.8 The study focused on what Baumol calls the process of convergence. According to this process, the countries with the lowest levels of productivity in 1870 have had the highest rates of improvement over the years, while the most productive countries in 1870 have exhibited the slowest rates of improvement—the peapods at work again, in other words.

The Reichenstein-Dorsettt projections assume that the future will look like the past, but there is no natural law that says it always will. If global warming indeed lies ahead, a long string of hot years will not necessarily be followed by a long string of cold years.

When discontinuities threaten, it is perilous to base decisions on established trends that have always seemed to make perfect sense but suddenly do not.

Until 1959, that is. At that point, stock prices were soaring and bond prices were falling. This meant that the ratio of bond interest to bond prices was shooting up and the ratio of stock dividends to stock prices was declining. The old relationship between bonds and stocks vanished, opening up a gap so huge that ultimately bonds were yielding more than stocks by an even greater margin than when stocks had yielded more than bonds.

The Second World War and its aftermath changed all that. From 1941 to 1959, inflation averaged 4.0% a year, with the cost-of-living index rising every year but one. The relentlessly rising price level transformed bonds from a financial instrument that had appeared inviolate into an extremely risky investment.

Although the contours of this new world were visible well before 1959, the old relationships in the capital markets tended to persist as long as people with memories of the old days continued to be the main investors.

The trick is to be flexible enough to recognize that regression to the mean is only a tool; it is not a religion with immutable dogma and ceremonies.

Rational people process information objectively: whatever errors they make in forecasting the future are random errors rather than the result of a stubborn bias toward either optimism or pessimism.

Bernoulli’s choice of Latin may help explain why his accomplishments have received greater notice from mathematicians than from economists and students of human behavior.

Utility theory was rediscovered toward the end of the eighteenth century by Jeremy Bentham, a popular English philosopher who lived from 1748 to 1832.

the economists of the nineteenth century fastened onto utility as a tool for discovering how prices result from interactive decisions by buyers and sellers. That detour led directly to the law of supply and demand.

Alfred Marshall, the pre-eminent economist of the Victorian age, once remarked, “No one should have an occupation which tends to make him anything less than a gentleman.”3

As late as the mid-1920s Frank Ramsay, a brilliant young Cambridge mathematician, was exploring the possibility of creating a “psychogalvanometer.”

Some Victorians protested that the rush toward measurement smacked of materialism. In 1860, when Florence Nightingale, after consulting with Galton and others, offered to fund a chair in applied statistics at Oxford, her offer was flatly refused.

@sidin

Today, people in the world of finance use terms like financial engineering, neural networks, and genetic algorithms.

from a book published in the 1990s

Did this devastating experience cause Jevons to question whether the economic system might be inherently stable at optimal levels of output and employment, as Ricardo and his followers had promised? Not in the least. Instead, he came up with a theory of business cycles based on the influence of sunspots on weather, of weather on harvests, and of harvests on prices, wages, and the level of employment.

Our confidence in measurement often fails, and we reject it. “Last night they got the elephant.” Our favorite explanation for such an event is to ascribe it to luck, good or bad as the case may be.

When we say that someone has fallen on bad luck, we relieve that person of any responsibility for what has happened. When we say that someone has had good luck, we deny that person credit for the effort that might have led to the happy outcome. But how sure can we be? Was it fate or choice that decided the outcome?

We would be surprised if we drew the number 1,000 out of a bottle containing 1,000 numbers; yet the probability of drawing 457 is also only one in a thousand. “The more extraordinary the event,” Laplace concludes, “the greater the need of it being supported by strong proofs.”4

Bachelier’s thesis came to light only by accident more than fifty years after he wrote it. Young as he was at the time, the mathematics he developed to explain the pricing of options on French government bonds anticipated by five years Einstein’s discovery of the motion of electrons—which, in turn, provided the basis for the theory of the random walk in finance.

The world, he concluded, would be a better place if we could insure against every future possibility. Then people would be more willing to engage in risk-taking, without which economic progress is impossible.

Arrow

In practice, insurance is available only when the Law of Large Numbers is observed. The law requires that the risks insured must be both large in number and independent of one another, like successive deals in a game of poker.

Keynes once asked, “[Why] should anyone outside a lunatic asylum wish to hold money as a store of wealth?”

he switched to economics after a professor declared, “Stop talking so much, or leave the philosophy department!”

In his presidential address to the American Economic Association in 1950, he likened the pope to Hitler and Stalin.

Frank knight

Noting a quotation by Lord Kelvin chiseled in stone on the social science building at Chicago—”[W]hen you cannot measure it . . . your knowledge is of a meager and unsatisfactory kind”—Knight sarcastically interpreted it to mean, “Oh, well, if you cannot measure, measure anyhow.”6

14 In an essay written in 1938, tided “My Early Beliefs,” Keynes condemns as “flimsily based [and] disastrously mistaken” the assumption of classical economists that human nature is reasonable.15

Ideas came to Keynes in such a rush and in such volume that he often found himself at odds with something he had said or written earlier. That did not disturb him. “When somebody persuades me that I am wrong,” he wrote, “I change my mind. What do you do?”19

His biographer Norman Macrae describes him as “excessively polite to everybody except . . . two long-suffering wives,” one of whom once remarked, “He can count everything except calories.”2

Von Neumann

Von Neumann was born in Budapest to a well-to-do, cultured, jolly family. Budapest “at the time was the sixth-largest city in Europe, prosperous and growing, with the world’s first underground subway. Its literacy rate was over 90%. More than 25% of the population was Jewish, including the von Neumanns, although John von Neumann paid little attention to his Jewishness except as a source of jokes.

They completed the 650 pages of their book in 1944, but the wartime paper shortage made Princeton University Press hesitant to publish it. At last a member of the Rockefeller family personally subsidized the publication of the book in 1953.

The Theory of Games and Economic Behavior loses no time in advocating the use of the mathematics in economic decision-making. Von Neumann and Morgenstern dismiss as “utterly mistaken” the view that the human and psychological elements of economics stand in the way of mathematical analysis.

Throughout most of the history of stock markets—about 200 years in the United States and even longer in some European countries—it never occurred to anyone to define risk with a number. Stocks were risky and some were riskier than others, and people let it go at that.

He had no interest in the foolishness that characterized most stock-market literature, such as lessons from a ballet dancer on how to become a millionaire without really trying, or how to be recognized as a guru among market forecasters.

Markowitz

Markowitz is parsimonious in providing footnotes and bibliography: he makes only three references to other writers in a setting where many academics measured accomplishment by the number of footnotes an author could manage to compile.

This failure to credit his intellectual forebears is curious: Markowitz’s methodology is a synthesis of the ideas of Pascal, de Moivre, Bayes, Laplace, Gauss, Galton, Daniel Bernoulli, Jevons, and von Neumann and Morgenstern. It draws on probability theory, on sampling, on the bell curve and dispersion around the mean, on regression to the mean, and on utility theory.