# Book Review: Against the Gods

Peter Bernstein's latest book, *Against the Gods The Remarkable Story of Risk,* establishes his standing as America's preeminent scholar-practitioner in the field of finance. Together with his highly successful predecessor, *Capital Ideas The Improbable Origins of Modern Wall Street *(The Free Press, New York, 1992), Bernstein has laid out a remarkably insightful and entertaining history of the "science" of finance, including fascinating descriptions of the people who drove the development of modern financial economics.

Bernstein's earlier work,* Ideas,* emphasized the advances emerging from the notion that investors are rational and security markets are efficient It focused on modern portfolio theory and its implications for security prices, and on the development of new instruments, like equity options. *Gods* serves as a prequel. It searches out the roots of modern finance and provides an interpretation of the history of that "science." Throughout, Bernstein's subtext is an investigation of the meaning of rationality and of the limits of rationality in explaining our choices.

*Gods* begins with a sweeping proposition Modern thinking began when man abandoned the belief that events are due to the whim of the gods and embraced the notion that we are active, independent agents who can manage risks. Thus in the worldview of ancient Greece, a man's destiny swayed with the whim of the gods, logic prevailed over experimentation, and the use of letters for numbers inhibited man's ability to calculate. But by the thirteenth century, new mental tools were in place the Hindu-Arabic numbering system, algebra, accounting, and other necessary equipment for the first insights into the laws of chance.

Those insights came in the seventeenth century, in the analysis by Blaise Pascal, a dissolute who became a religious zealot, and Pierre de Fermat, a lawyer whose genius was in mathematics, of a gambling problem first proposed in the fifteenth century. The Pascal-Fermat contribution to probability theory, which helps us to analyze risk, was mixed in the next century with insights into the role risk plays in our choices arising from the work of Daniel Bernoulli, a Swiss mathematician whose father and uncles were confirmed eighteenth-century geniuses. The foundation for modern decision theory was laid. From that foundation, Bernstein sets off on a whirlwind tour of the development of modern decision theory.

In the last quarter of *Gods,* Bernstein focuses on the assumption that human choices are "rational," meaning that they are derived logically from a few axioms. In modern economics, one model of rational decision making is the expected utility hypothesis Decisions are made with the goal of maximizing one's expected satisfaction (back to Bernoulli again). Bernstein acquaints us with the many new ways of interpreting and measuring risk, and with the emerging field of behavioral finance, which recognizes and attempts to explain anomalies in finance, examples in which rational explanations fail. In this part of *Gods* Bernstein tips his hand, telling us that although the assumption of rational behavior is a useful starting point, it describes the real world only up to a point.

The most interesting part of this discussion is Bernstein's presentation of the path-breaking work of Daniel Kahneman and the late Amos Tversky; they were experimental psychologists whose work, called "prospect theory," is often used by students of behavioral finance to explain a variety of financial anomalies.

Among the human tendencies documented by Kahneman and Tversky are extrapolation from small and unreliable samples (I had a car accident at that intersection, therefore that intersection is more dangerous than others), giving greater weight to catastrophic outcomes than their low probabilities warrant (the Three Mile Island effect), loss aversion, and mental accounting. Loss aversion refers to our tendency, when faced with a choice between a sure loss and an uncertain gamble, to gamble unless the odds are strongly against us; embezzlers will recognize this, as will many investors who avoid selling at a loss in the hope that continuing the gamble will extricate them. Mental accounting refers to the tendency to sort decisions into compartments rather than to consider the overall position. Examples of mental accounting are Christmas saving clubs and other ways of segregating assets by intent; as Bernstein argues, this includes the practice of buying dividend-paying stocks so that one can avoid "dipping into capital" -- selling stock -- to pay for life's necessities.

By the end of the book, Bernstein has shown us how interpretations of rational behavior in the presence of risk have changed as the tools to understand decision making have changed. He almost gets us to move into the castle-in-the-air, the notion that we are, in fact, rational. But his good sense and long experience with security markets, supported by the work of behavioral economics, keeps him from entering the front door. Bernstein believes that "we are rational as far as it goes." To know what he means, read the book. That would be rational!

* Peter Fortune rationally economizes for the Boston Fed. *

**Peter Bernstein **

**Against The Gods: the Remarkable Story of Risk **

**New York: John Wiley , Sons, Inc., 1996. $27.95.**

**THE PASCAL-FERMAT ANSWER TO THE PROBLEM OF POINTS: AN EXAMPLE**

Mr. H and Mr. T each pay $50 to play a game of points a fair coin will be flipped 15 times, and the $100 stake will go to Mr. H if more heads than tails result, to Mr. T otherwise. However, the game stops prematurely, after 6 flips have resulted in four heads. How should the $100 be split?

Pascal and Fermat argue that the $100 should be split according to the probability that each player would have won if the game had continued through the agreed-upon 15 flips. Because the game involves Bernoulli trials (each flip has the same probability of heads, and the results on one flip are independent of the results on previous flips), the binomial distribution (based on Pascal's Triangle) gives the answer. The probability that Mr. H would have won 4 or more of the remaining 9 flips, bringing the number of heads to at least 8 and earning him the full $100, is 0.7461. Mr. H should get $74.61, leaving $25.39 to Mr. T.

**DANIEL BERNOULLI AND THE ST. PETERSBURG PARADOX**

The wisdom of the early eighteenth century was that a gambler would play any game for which the expected net gain (expected winnings less the cost of playing) was positive. Stated differently, the gambler would calculate the expected winnings and, if pressed, would pay as much as that amount to play the game. But the St. Petersburg paradox, named for the city in which Bernoulli presented his answer, gives a game that real-world gamblers would not pay the actuarial value to play

A fair coin is flipped until a head comes up, at which point the game stops and the gambler receives $2 raised to the power of the number of that flip. The expected winning, which is the maximum amount a gambler would offer to play the game, is the sum of all the possible payoffs, each multiplied by the probability of its occurrence. The expected winning in the St. Petersburg game is the infinite series

(2)(1/2)+(2^{2})(1/2)^{2} +(2^{3})(1/2)^{3} + ....

The expected winning is thus the sum of 1, added an infinite number of times. The expected winning, in other words, is infinite. But no gambler would wager all his or her wealth to play.

Bernoulli's answer to the paradox was simple but profound in its effects. He argued that gamblers do not maximize the expected amount of their net winnings. Rather, they maximize their "moral utility," the expected satisfaction which the game provides. Furthermore, he argued, moral utility is subject to diminishing returns Each additional dollar of winnings adds a smaller amount to satisfaction. The result is that the game has a finite expected utility, even though the expected winnings are infinite.

Bernoulli's solution introduced risk -- and aversion to risk -- into the language of decision making; it was the first statement of a fundamental axiom of economic theory, that diminishing returns prevail; and it is the foundation of the most widely applied theory of "rational" decision making--that economic agents maximize their expected utility, not their expected wealth.

**SOME MILESTONES IN THE STORY OF RISK**

1654 The first use of probability analysis by Blaise Pascal, a brilliant mathematician, and Pierre de Fermat, lawyer and mathematics hobbyist, who jointly solved the problem of points.

1733 The development of the normal probability distribution by Abraham de Moivre, a French Protestant transplanted to England who never held a proper academic position.

1738 The analysis of risk as a factor shaping decisions emerged from the solution to the St. Petersburg paradox presented by Daniel Bernoulli, whose family of geniuses supports the thesis of Francis Galton's Hereditary Genius, that genius runs in the genes.

1764 Posthumous publication of the Rev. Thomas Bayes's analysis of the mixing of old and new information in the formation of probability estimates.

1877 Development of the concept of regression toward the mean, applied to human characteristics by Francis Galton, the snobbish cousin to Charles Darwin, who created the unfortunate pseudo-science of "eugenics."

1900Louis Bachelier's doctoral dissertation, "The Theory of Speculation," laid the foundation for modern finance and provided the underpinnings for modern option pricing models, although it was poorly received by his professors at the Sorbonne.

1952 The introduction of modern portfolio theory in a 14 page paper titled "Portfolio Selection," by Harry Markowitz, a graduate student in economics.

1953 Publication of The Theory of Games by John von Neumann, mathematician and early computer scientist, and economist Oskar Morgenstern. This book, written in the 1940s, analyzed human interactions as a source of risk.

1964 William Sharpe's extension of Markowitz's insights into an understanding of the role of risk as a factor in determining in security prices, in a paper titled "Capital Asset Prices."

1973 The development of modern option pricing theory by Fischer Black, an academic nomad, and economist Myron Scholes, published in a paper that was first rejected by many academic journals.

#### About the Authors

Peter Fortune