Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

  One other aspect ofJevons's work as an economist deserves mention. As a man trained in the natural sciences, he could not avoid taking note of what was right in front of his face-the economy did fluctuate. In 1873, just two years after the publication of The Theory of Political Economy, a great economic boom that had lasted for over twenty years in Europe and the United States came to an end. Business activity fell steadily for three years, and recovery was slow to come. Industrial production in the United States in 1878 was only 6% higher than it had been in 1872. Over the next 23 years, the prices of U.S. goods and services fell almost uninterruptedly by some 40%, creating much hardship throughout western Europe and North America.

  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. For Jevons, the trouble with the economy was in heaven and earth, not in its philosophy.

  Theories of how people make decisions and choices seem to have become detached from everyday life in the real world. Yet those theo ries prevailed for nearly a hundred years. Even well into the Great Depression, the notion persisted that economic fluctuations were accidents of some kind rather than events inherent in an economic system driven by risk-taking. Hoover's promise in 1930 that prosperity was just around the corner reflected his belief that the Great Crash had been caused by a passing aberration rather than by some structural fault. In 1931, Keynes himself still exhibited the optimism of his Victorian upbringing when he expressed his "... profound conviction that the Economic Problem ... is nothing but a frightful muddle, a transitory and an unnecessary muddle."6 The italics are his.

  ur 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.

  If everything is a matter of luck, risk management is a meaningless exercise. Invoking luck obscures truth, because it separates an event from its cause.

  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?

  Until we can distinguish between an event that is truly random and an event that is the result of cause and effect, we will never know whether what we see is what we'll get, nor how we got what we got. When we take a risk, we are betting on an outcome that will result from a decision we have made, though we do not know for certain what the outcome will be. The essence of risk management lies in maximizing the areas where we have some control over the outcome while minimizing the areas where we have absolutely no control over the outcome and the linkage between effect and cause is hidden from us.

  Just what do we mean by luck? Laplace was convinced that there is no such thing as luck-or hazard as he called it. In his Essai philosophique sur les probabilite's, he declared:

  Present events are connected with preceding ones by a tie based upon the evident principle that a thing cannot occur without a cause that produces it.... All events, even those which on account of their insignificance do not seem to follow the great laws of nature, are a result of it just as necessarily as the revolutions of the sun.'

  This statement echoes an observation by Jacob Bernoulli that if all events throughout eternity could be repeated, we would find that every one of them occurred in response to "definite causes" and that even the events that seemed most fortuitous were the result of "a certain necessity, or, so to say, FATE." We can also hear de Moivre, submitting to the power of ORIGINAL DESIGN. Laplace, surmising that there was a "vast intelligence" capable of understanding all causes and effects, obliterated the very idea of uncertainty. In the spirit of his time, he predicted that human beings would achieve that same level of intelligence, citing the advances already made in astronomy, mechanics, geometry, and gravity. He ascribed those advances to "the tendency, peculiar to the human race [that] renders it superior to animals; and their progress in this respect distinguishes nations and ages and constitutes their true glory. "2

  Laplace admitted that it is sometimes hard to find a cause where there seems to be none, but he also warns against the tendency to assign a particular cause to an outcome when in fact only the laws of probability are at work. He offers this example: "On a table, we see the letters arranged in this order, CONSTANTINOPLE, and we judge that this arrangement is not the result of chance. [Yet] if this word were not employed in any language we should not suspect it came from any particular cause."3 If the letters happened to be BZUXRQVICPRGAB, we would not give the sequence of letters a second thought, even though the odds on BZUXRQVICPRGAB's showing up in a random drawing are precisely the same as the odds on CONSTANTINOPLE's showing up. 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

  In the month of October 1987, the stock market fell by more than 20%. That was only the fourth time since 1926 that the market had dropped by more than 20% in a single month. But the 1987 crash came out of nowhere. There is no agreement on what caused it, though theories abound. It could not have occurred without a cause, and yet that cause is obscure. Despite its extraordinary character, no one could come up with "strong proofs" of its origins.

  Another French mathematician, born about a century after Laplace, gave further emphasis to the concept of cause and effect and to the importance of information in decision-making. Jules-Henri Poincare, (1854-1912) was, according to James Newman,

  ... a French savant who looked alarmingly like a French savant. He was short and plump, carried an enormous head set off by a thick spade beard and splendid mustache, was myopic, stooped, distraught in speech, absent-minded and wore pince-nez glasses attached to a black silk ribbon.5

  Poincare was another mathematician in the long line of child prodigies that we have met along the way. He grew up to be the leading French mathematician of his time.

  Nevertheless, Poincare made the great mistake of underestimating the accomplishments of a student named Louis Bachelier, who earned a degree in 1900 at the Sorbonne with a dissertation titled "The Theory of Speculation."6 Poincare, in his review of the thesis, observed that "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." The thesis was awarded "mention honorable," rather than the highest award of "mention tres honorable," which was essential for anyone hoping to find a decent job in the academic community. Bachelier never found such a job.

  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. Moreover, his description of the process of speculation anticipated many of the theories observed in financial markets today. "Mention honorable"!

  The central idea of Bachelier's thesis was this: "The mathematical expectation of the speculator is zero." The ideas that flowed from that startling statement are now evident in everything from trading strategies and the use of derivative instruments to the most sophisticated techniques of portfolio management. Bachelier knew that he was onto something big, despite the indifference he was accorded. "It is evident," he wrote, "that the present theory solves the majority of problems in the study of speculation by the calculus of probability."

  B
ut we must return to Poincare, Bachelier's nemesis. Like Laplace, Poincare believed that everything has a cause, though mere mortals are incapable of divining all the causes of all the events that occur. "A mind infinitely powerful, infinitely well-informed about the laws of nature, could have foreseen [all events] from the beginning of the centuries. If such a mind existed, we could not play with it at any game of chance, for we would lose."7

  To dramatize the power of cause-and-effect, Poincare suggests what the world would be like without it. He cites a fantasy imagined by Camile Flammarion, a contemporary French astronomer, in which an observer travels into space at a velocity greater than the speed of light:

  [F] or him time would have changed sign [from positive to negative]. History would be turned about, and Waterloo would precede Austerlitz.... [A]ll would seem to him to come out of a sort of chaos in unstable equilibrium. All nature would appear to him delivered over to chance.8

  But in a cause-and-effect world, if we know the causes we can predict the effects. So "what is chance for the ignorant is not chance for the scientist. Chance is only the measure of our ignorance."9

  But then Poincare asks whether that definition of chance is totally satisfactory. After all, we can invoke the laws of probability to make predictions. We never know which team is going to win the World Series, but Pascal's Triangle demonstrates that a team that loses the first game has a probability of 22/64 of winning four games before their opponents have won three more. There is one chance in six that the roll of a single die will come up 3. The weatherman predicts today that the probability of rain tomorrow is 30%. Bachelier demonstrates that the odds that the price of a stock will move up on the next trade are precisely 50%. Poincare points out that the director of a life insurance company is ignorant of the time when each of his policyholders will die, but "he relies upon the calculus of probabilities and on the law of great numbers, and he is not deceived, since he distributes dividends to his stockholders."10

  Poincare also points out that some events that appear to be fortuitous are not; instead, their causes stem from minute disturbances. A cone perfectly balanced on its apex will topple over if there is the least defect in symmetry; and even if there is no defect, the cone will topple in response to "a very slight tremor, a breath of air." That is why, Poincare explained, meteorologists have such limited success in predicting the weather:

  Many persons find it quite natural to pray for rain or shine when they would think it ridiculous to pray for an eclipse.... [O)ne-tenth of a degree at any point, and the cyclone bursts here and not there, and spreads its ravages over countries it would have spared. This we could have foreseen if we had known that tenth of a degree, but ... all seems due to the agency of chance.11

  Even spins of a roulette wheel and throws of dice will vary in response to slight differences in the energy that puts them in motion. Unable to observe such tiny differences, we assume that the outcomes they produce are random, unpredictable. As Poincare observes about roulette, "This is why my heart throbs and I hope everything from luck."12

  Chaos theory, a more recent development, is based on a similar premise. According to this theory, much of what looks like chaos is in truth the product of an underlying order, in which insignificant perturbations are often the cause of predestined crashes and long-lived bull markets. The New York Times of July 10, 1994, reported a fanciful application of chaos theory by a Berkeley computer scientist named James Crutchfield, who "estimated that the gravitational pull of an electron, randomly shifting position at the edge of the Milky Way, can change the outcome of a billiard game on Earth."

  Laplace and Poincare recognized that we sometimes have too little information to apply the laws of probability. Once, at a professional investment conference, a friend passed me a note that read as follows:

  We can assemble big pieces of information and little pieces, but we can never get all the pieces together. We never know for sure how good our sample is. That uncertainty is what makes arriving at judgments so difficult and acting on them so risky. We cannot even be 100% certain that the sun will rise tomorrow morning: the ancients who predicted that event were themselves working with a limited sample of the history of the universe.

  When information is lacking, we have to fall back on inductive reasoning and try to guess the odds. John Maynard Keynes, in a treatise on probability, concluded that in the end statistical concepts are often useless: "There is a relation between the evidence and the event considered, but it is not necessarily measurable."13

  Inductive reasoning leads us to some curious conclusions as we try to cope with the uncertainties we face and the risks we take. Some of the most impressive research on this phenomenon has been done by Nobel Laureate Kenneth Arrow. Arrow was born at the end of the First World War and grew up in New York City at a time when the city was the scene of spirited intellectual activity and controversy. He attended public school and City College and went on to teach at Harvard and Stanford. He now occupies two emeritus professorships at Stanford, one in operations research and one in economics.

  Early on, Arrow became convinced that most people overestimate the amount of information that is available to them. The failure of economists to comprehend the causes of the Great Depression at the time demonstrated to him that their knowledge of the economy was "very limited." His experience as an Air Force weather forecaster during the Second World War "added the news that the natural world was also unpredictable." 14 Here is a more extended version of the passage from which I quoted in the Introduction:

  To me our knowledge of the way things work, in society or in nature, comes trailing clouds of vagueness. Vast ills have followed a belief in certainty, whether historical inevitability, grand diplomatic designs, or extreme views on economic policy. When developing policy with wide effects for an individual or society, caution is needed because we cannot predict the consequences."15

  One incident that occurred while Arrow was forecasting the weather illustrates both uncertainty and the human unwillingness to accept it. Some officers had been assigned the task of forecasting the weather a month ahead, but Arrow and his statisticians found that their long-range forecasts were no better than numbers pulled out of a hat. The forecasters agreed and asked their superiors to be relieved of this duty. The reply was: "The Commanding General is well aware that the forecasts are no good. However, he needs them for planning purposes."16

  In an essay on risk, Arrow asks why most of us gamble now and then and why we regularly pay premiums to an insurance company. The mathematical probabilities indicate that we will lose money in both instances. In the case of gambling, it is statistically impossible to expect-though possible to achieve-more than a break-even, because the house edge tilts the odds against us. In the case of insurance, the premiums we pay exceed the statistical odds that our house will burn down or that our jewelry will be stolen.

  Why do we enter into these losing propositions? We gamble because we are willing to accept the large probability of a small loss in the hope that the small probability of scoring a large gain will work in our favor; for most people, in any case, gambling is more entertainment than risk. We buy insurance because we cannot afford to take the risk of losing our home to fire-or our life before our time. That is, we prefer a gamble that has 100% odds on a small loss (the premium we must pay) but a small chance of a large gain (if catastrophe strikes) to a gam ble with a certain small gain (saving the cost of the insurance premium) but with uncertain but potentially ruinous consequences for us or our family.

  Arrow won his Nobel Prize in part as a result of his speculations about an imaginary insurance company or other risk-sharing institution that would insure against any loss of any kind and of any magnitude, in what he describes as a "complete market." 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.

  Often we are unable to conduct enough tr
ials or take enough samples to employ the laws of probability in making decisions. We decide on the basis of ten tosses of the coin instead of a hundred. Consequently, in the absence of insurance, just about any outcome seems to be a matter of luck. Insurance, by combining the risks of many people, enables each individual to enjoy the advantages provided by the Law of Large Numbers.

  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.

  "Independent" means several things: it means that the cause of a fire, for example, must be independent of the actions of the policyholder. It also means that the risks insured must not be interrelated, like the probable movement of any one stock at a time when the whole stock market is taking a nose dive, or the destruction caused by a war. Finally, it means that insurance will be available only when there is a rational way to calculate the odds of loss, a restriction that rules out insurance that a new dress style will be a smashing success or that the nation will be at war at some point in the next ten years.

  Consequently, the number of risks that can be insured against is far smaller than the number of risks we take in the course of a lifetime. We often face the possibility that we will make the wrong choice and end up regretting it. The premium we pay the insurance company is only one of many certain costs we incur in order to avoid the possibility of a larger, uncertain loss, and we go to great lengths to protect ourselves from the consequences of being wrong. Keynes once asked, "[Why] should anyone outside a lunatic asylum wish to hold money as a store of wealth?" His answer: "The possession of actual money lulls our disquietude; and the premium we require to make us part with money is the measure of our disquietude."17