Each author has his preferred approach. The Port-Royal author argues that only the pathologically risk-averse make choices based on the consequences without regard to the probability involved. The author of the New Theory argues that only the foolhardy make choices based on the probability of an outcome without regard to its consequences.
The author of the St. Petersburg paper was a Swiss mathematician named Daniel Bernoulli, who was then 38 years old.' Although Daniel Bernoulli's name is familiar only to scientists, his paper is one of the most profound documents ever written, not just on the subject of risk but on human behavior as well. Bernoulli's emphasis on the complex relationships between measurement and gut touches on almost every aspect of life.
Daniel Bernoulli was a member of a remarkable family. From the late 1600s to the late 1700s, eight Bernoullis had been recognized as celebrated mathematicians. Those men produced what the historian Eric Bell describes as "a swarm of descendants ... and of this posterity the majority achieved distinction-sometimes amounting to eminence-in the law, scholarship, literature, the learned professions, administration and the arts. None were failures."3
The founding father of this tribe was Nicolaus Bernoulli of Basel, a wealthy merchant whose Protestant forebears had fled from Catholicdominated Antwerp around 1585. Nicolaus lived a long life, from 1623 to 1708, and had three sons, Jacob, Nicolaus (known as Nicolaus I), and Johann. We shall meet Jacob again shortly, as the discoverer of the Law of Large Numbers in his book Ars Conjectandi (The Art of Conjecture). Jacob was both a great teacher who attracted students from all over Europe and an acclaimed genius in mathematics, engineering, and astron omy. The Victorian statistician Francis Galton describes him as having "a bilious and melancholic temperament ... sure but slow."4 His relationship with his father was so poor that he took as his motto Invito patre sidera verso-"I am among the stars in spite of my father."5
Galton did not limit his caustic observations to Jacob. Despite the evidence that the Bernoulli family provided in confirmation of Galton's theories of eugenics, he depicts them in his book, Hereditary Genius as "mostly quarrelsome and jealous."'
These traits seem to have run through the family. Jacob's younger brother and fellow-mathematician Johann, the father of Daniel, is described by James Newman, an anthologist of science, as "violent, abusive ... and, when necessary, dishonest." When Daniel won a prize from the French Academy of Sciences for his work on planetary orbits, his father, who coveted the prize for himself, threw him out of the house. Newman reports that Johann lived to be 80 years old, "retaining his powers and meanness to the end."
And then there was the son of the middle brother, Nicolaus I, who is known as Nicolaus II. When Nicolaus II's uncle Jacob died in 1705 after a long illness, leaving The Art of Conjecture all but complete, Nicolaus II was asked to edit the work for publication even though he was only 18 at the time. He took eight years to finish the task! In his introduction he confesses to the long delay and to frequent prodding by the publishers, but he offers as an excuse of "my absence on travels" and the fact that "I was too young and inexperienced to know how to complete it."'
Perhaps he deserves the benefit of the doubt: he spent those eight years seeking out the opinions of the leading mathematicians of his time, including Isaac Newton. In addition to conducting an active correspondence for the exchange of ideas, he traveled to London and Paris to consult with outstanding scholars in person. And he made a number of contributions to mathematics on his own, including an analysis of the use of conjecture and probability theory in applications of the law.
To complicate matters further, Daniel Bernoulli had a brother five years older than he, also named Nicolaus; by convention, this Nicolaus is known as Nicolaus III, his grandfather being numberless, his uncle being Nicolaus I, and his elder first cousin being Nicolaus II. It was Nicolaus III, a distinguished scholar himself, who started Daniel off in mathematics when Daniel was only eleven years old. As the oldest son, Nicolaus III had been encouraged by his father to become a mathematician. When he was only eight years old, he was able to speak four languages; he became Doctor of Philosophy at Basel at the age of nineteen; and he was appointed Professor of Mathematics at St. Petersburg in 1725 at the age of thirty. He died of some sort of fever just a year later.
Daniel Bernoulli received an appointment at St. Petersburg in the same year as Nicolaus III and remained there until 1733, when he returned to his hometown of Basel as Professor of Physics and Philosophy. He was among the first of many outstanding scholars whom Peter the Great would invite to Russia in the hope of establishing his new capital as a center of intellectual activity. According to Galton, Daniel was "physician, botanist, and anatomist, writer on hydrodynamics; very precocious."9 He was also a powerful mathematician and statistician, with a special interest in probability.
Bernoulli was very much a man of his times. The eighteenth century came to embrace rationality in reaction to the passion of the endless religious wars of the past century. As the bloody conflict finally wound down, order and appreciation of classical forms replaced the fervor of the Counter-Reformation and the emotional character of the baroque style in art. A sense of balance and respect for reason were hallmarks of the Enlightenment. It was in this setting that Bernoulli transformed the mysticism of the Fort-Royal Logic into a logical argument addressed to rational decision-makers.
Daniel Bernoulli's St. Petersburg paper begins with a paragraph that sets forth the thesis that he aims to attack:
Ever since mathematicians first began to study the measurement of risk, there has been general agreement on the following proposition: Expected values are computed by multiplying each possible gain by the num ber of ways in which it can occur, and then dividing the sum of these products by the total number of cases. *'0
Bernoulli finds this hypothesis flawed as a description of how people in real life go about making decisions, because it focuses only on the facts; it ignores the consequences of a probable outcome for a person who has to make a decision when the future is uncertain. Price-and probability-are not enough in determining what something is worth. Although the facts are the same for everyone, "the utility ... is dependent on the particular circumstances of the person making the estimate .... There is no reason to assume that ... the risks anticipated by each [individual] must be deemed equal in value." To each his own.
The concept of utility is experienced intuitively. It conveys the sense of usefulness, desirability, or satisfaction. The notion that arouses Bernoulli's impatience with mathematicians-"expected value"-is more technical. As Bernoulli points out, expected value equals the sum of the values of each of a number of outcomes multiplied by the probability of each outcome relative to all the other possibilities. On occasion, mathematicians still use the term "mathematical expectation" for expected value.
A coin has two sides, heads and tails, with a 50% chance of landing with one side or the other showing-a coin cannot come up showing both heads and tails at the same time. What is the expected value of a coin toss? We multiply 50% by one for heads and do the same for tails, take the sum-100%-and divide by two. The expected value of betting on a coin toss is 50%. You can expect either heads or tails, with equal likelihood.
What is the expected value of rolling two dice? If we add up the 11 numbers that mightcome up-2+3+4+5+6+7+8+9+10+ 11 + 12-the total works out to 77. The expected value of rolling two dice is 77/11, or exactly 7.
Yet these 11 numbers do not have an equal probability of coming up. As Cardano demonstrated, some outcomes are more likely than others when there are 36 different combinations that produce the 11 outcomes ranging from 2 to 12; two can be produced only by doubleone, but four can be produced in three ways, by 3 + 1, by 1 + 3, and by 2 + 2. Cardano's useful table (page 52) lists a number of combinations in which each of the 11 outcomes can occur:
The expected value, or the mathematical expectation, of rolling two dice is exactly 7, confirming our calculation of 77/11. Now we can see why a roll of 7 plays such
a critical role in the game of craps.
Bernoulli recognizes that such calculations are fine for games of chance but insists that everyday life is quite a different matter. Even when the probabilities are known (an oversimplification that later mathematicians would reject), rational decision-makers will try to maximize expected utility-usefulness or satisfaction-rather than expected value. Expected utility is calculated by the same method as that used to calculate expected value but with utility serving as the weighting factor.11
For example, Antoine Arnauld, the reputed author of the PortRoyal Logic, accused people frightened by thunderstorms of overestimating the small probability of being struck by lightning. He was wrong. It was he who was ignoring something. The facts are the same for everyone, and even people who are terrified at the first rumble of thun der are fully aware that it is highly unlikely that lightning will strike precisely where they are standing. Bernoulli saw the situation more clearly: people with a phobia about being struck by lightning place such a heavy weight on the consequences of that outcome that they tremble even though they know that the odds on being hit are tiny.
Gut rules the measurement. Ask passengers in an airplane during turbulent flying conditions whether each of them has an equal degree of anxiety. Most people know full well that flying in an airplane is far safer than driving in an automobile, but some passengers will keep the flight attendants busy while others will snooze happily regardless of the weather.
And that's a good thing. If everyone valued every risk in precisely the same way, many risky opportunities would be passed up. Venturesome people place high utility on the small probability of huge gains and low utility on the larger probability of loss. Others place little utility on the probability of gain because their paramount goal is to preserve their capital. Where one sees sunshine, the other sees a thunderstorm. Without the venturesome, the world would turn a lot more slowly. Think of what life would be like if everyone were phobic about lightning, flying in airplanes, or investing in start-up companies. We are indeed fortunate that human beings differ in their appetite for risk.
Once Bernoulli has established his basic thesis that people ascribe different values to risk, he introduces a pivotal idea: "[The] utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed." Then he observes, "Considering the nature of man, it seems to me that the foregoing hypothesis is apt to be valid for many people to whom this sort of comparison can be applied."
The hypothesis that utility is inversely related to the quantity of goods previously possessed is one of the great intellectual leaps in the history of ideas. In less than one full printed page, Bernoulli converts the process of calculating probabilities into a procedure for introducing subjective considerations into decisions that have uncertain outcomes.
The brilliance of Bernoulli's formulation lies in his recognition that, while the role of facts is to provide a single answer to expected value (the facts are the same for everyone), the subjective process will pro duce as many answers as there are human beings involved. But he goes even further than that: he suggests a systematic approach for determining how much each individual desires more over less: the desire is inversely proportionate to the quantity of goods possessed.
For the first time in history Bernoulli is applying measurement to something that cannot be counted. He has acted as go-between in the wedding of intuition and measurement. Cardano, Pascal, and Fermat provided a method for figuring the risks in each throw of the dice, but Bernoulli introduces us to the risk-taker-the player who chooses how much to bet or whether to bet at all. While probability theory sets up the choices, Bernoulli defines the motivations of the person who does the choosing. This is an entirely new area of study and body of theory. Bernoulli laid the intellectual groundwork for much of what was to follow, not just in economics, but in theories about how people make decisions and choices in every aspect of life.
Bernoulli offers in his paper a number of interesting applications to illustrate his theory. The most tantalizing, and the most famous, of them has come to be known as the Petersburg Paradox, which was originally suggested to him by his "most honorable cousin the celebrated Nicolaus Bernoulli"-the dilatory editor of The Art of Conjecture.
Nicolaus proposes a game to be played between Peter and Paul, in which Peter tosses a coin and continues to toss it until it comes up heads. Peter will pay Paul one ducat if heads comes up on the first toss, two ducats if heads comes up on the second toss, four ducats on the third, and so on. With each additional throw the number of ducats Peter must pay Paul is doubled.* How much should someone pay Paul-who stands to rake in a sizable sum of money-for the privilege of taking his place in this game?
The paradox arises because, according to Bernoulli, "The accepted method of calculation [expected value] does, indeed, value Paul's prospects at infinity [but] no one would be willing to purchase [those prospects] at a moderately high price .... [A]ny fairly reasonable man would sell his chance, with great pleasure, for twenty ducats.`
Bernoulli undertakes an extended mathematical analysis of the problem, based on his assumption that increases in wealth are inversely related to initial wealth. According to that assumption, the prize Paul might win on the two-hundredth throw would have only an infinitesimal amount of additional utility over what he would receive on the one-hundredth throw; even by the 51st throw, the number of ducats won would already have exceeded 1,000,000,000,000,000. (Measured in dollars, the total national debt of the U.S. government today is only four followed by twelve zeroes.)
Whether it be in ducats or dollars, the evaluation of Paul's expectation has long attracted the attention of leading scholars in mathematics, philosophy, and economics. An English history of mathematics by Isaac Todhunter, published in 1865, makes numerous references to the Petersburg Paradox and discusses some of the solutions that various mathematicians had proposed during the intervening years.12 Meanwhile, Bernoulli's paper remained in its original Latin until a German translation appeared in 1896. Even more sophisticated, complex mathematical treatments of the Paradox appeared after John Maynard Keynes made a brief reference to it in his Treatise on Probability, published in 1921. But it was not until 1954-216 years after its original publication-that the paper by Bernoulli finally appeared in an English translation.
The Petersburg Paradox is more than an academic exercise in the exponents and roots of tossing coins. Consider a great growth company whose prospects are so brilliant that they seem to extend into infinity. Even under the absurd assumption that we could make an accurate forecast of a company's earnings into infinity-we are lucky if we can make an accurate forecast of next quarter's earnings-what is a share of stock in that company worth? An infinite amount?*
There have been moments when real, live, hands-on professional investors have entertained dreams as wild as that-moments when the laws of probability are forgotten. In the late 1960s and early 1970s, major institutional portfolio managers became so enamored with the idea of growth in general, and with the so-called "Nifty-Fifty" growth stocks in particular, that they were willing to pay any price at all for the privilege of owning shares in companies like Xerox, Coca-Cola, IBM, and Polaroid. These investment managers defined the risk in the NiftyFifty, not as the risk of overpaying, but as the risk of not owning them: the growth prospects seemed so certain that the future level of earnings and dividends would, in God's good time, always justify whatever price they paid. They considered the risk of paying too much to be minuscule compared with the risk of buying shares, even at a low price, in companies like Union Carbide or General Motors, whose fortunes were uncertain because of their exposure to business cycles and competition.
This view reached such an extreme point that investors ended up by placing the same total market value on small companies like International Flavors and Fragrances, with sales of only $138 million, as they placed on a less glamorous business like U.S. Steel, with sales of $5 billion. In December 1972
, Polaroid was selling for 96 times its 1972 earnings, McDonald's was selling for 80 times, and IFF was selling for 73 times; the Standard & Poor's Index of 500 stocks was selling at an average of 19 times. The dividend yields on the Nifty-Fifty averaged less than half the average yield on the 500 stocks in the S&P Index.
The proof of this particular pudding was surely in the eating, and a bitter mouthful it was. The dazzling prospect of earnings rising up to the sky turned out to be worth a lot less than an infinite amount. By 1976, the price of IFF had fallen 40% but the price of U.S. Steel had more than doubled. Figuring dividends plus price change, the S&P 500 had surpassed its previous peak by the end of 1976, but the Nifty-Fifty did not surpass their 1972 bull-market peak until July 1980. Even worse, an equally weighted portfolio of the Nifty-Fifty lagged the performance of the S&P 500 from 1976 to 1990.
But where is infinity in the world of investing? Jeremy Siegel, a professor at the Wharton School of Business at the University of Pennsylvania, has calculated the performance of the Nifty-Fifty in detail from the end of 1970 to the end of 1993.13 The equally weighted portfolio of fifty stocks, even if purchased at its December 1972 peak, would have realized a total return by the end of 1993 that was less than one percentage point below the return on the S&P Index. If the same stocks had been bought just two years earlier, in December 1970, the portfolio would have outperformed the S&P by a percentage point per year. The negative gap between cost and market value at the bottom of the 1974 debacle would also have been smaller.
For truly patient individuals who felt most comfortable owning familiar, high-quality companies, most of whose products they encountered in their daily round of shopping, an investment in the Nifty-Fifty would have provided ample utility. The utility of the portfolio would have been much smaller to a less patient investor who had no taste for a fifty-stock portfolio in which five stocks actually lost money over twenty-one years, twenty earned less than could have been earned by rolling over ninety-day Treasury bills, and only eleven outperformed the S&P 500. But, as Bernoulli himself might have put it in a more informal moment, you pays your money and you takes your choice.
Against the Gods: The Remarkable Story of Risk Page 11