Against the Gods: The Remarkable Story of Risk Read online

  So far, the accomplishments of the theory appear modest compared to its claims. Its practitioners have managed to cup the butterfly in their hands, but they have not yet traced all the airflows impelled by the flutterings of its wings. But they are trying.

  In recent years, other sophisticated innovations to foretell the future have surfaced, with strange names like genetic algorithms and neural networks.7 These methods focus largely on the nature of volatility; their implementation stretches the capability of the most high-powered computers.

  The objective of genetic algorithms is to replicate the manner in which genes are passed from one generation to the next. The genes that survive create the models that form the most durable and effective offspring.* Neural networks are designed to simulate the behavior of the human brain by sifting out from the experiences programed into them those inferences that will be most useful in dealing with the next experience. Practitioners of this procedure have uncovered behavior patterns in one system that they can use to predict outcomes in entirely different systems, the theory being that all complex systems like democracy, the path of technological development, and the stock market share common patterns and responses."

  These models provide important insights into the complexity of reality, but there is no proof of cause and effect in the recognition of patterns that precede the arrival of other patterns in financial markets or in the spin of a roulette wheel. Socrates and Aristotle would be as skeptical about chaos theory and neural networks as the theorists of those approaches are about conventional approaches.

  Likeness to truth is not the same as truth. Without any theoretical structure to explain why patterns seem to repeat themselves across time or across systems, these innovations provide little assurance that today's signals will trigger tomorrow's events. We are left with only the subtle sequences of data that the enormous power of the computer can reveal. Thus, forecasting tools based on nonlinear models or on computer gymnastics are subject to many of the same hurdles that stand in the way of conventional probability theory: the raw material of the model is the data of the past.

  The past seldom obliges by revealing to us when wildness will break out in the future. Wars, depressions, stock-market booms and crashes, and ethnic massacres come and go, but they always seem to arrive as surprises. After the fact, however, when we study the history of what happened, the source of the wildness appears to be so obvious to us that we have a hard time understanding how people on the scene were oblivious to what lay in wait for them.

  Surprise is endemic above all in the world of finance. In the late 1950s, for example, a relationship sanctified by over eighty years of experience suddenly came apart when investors discovered that a thousand dollars invested in low-risk, high-grade bonds would, for the first time in history, produce more income than a thousand dollars invested in risky common stocks.* In the early 1970s, long-term interest rates rose above 5% for the first time since the Civil War and have dared to remain above 5% ever since.

  Given the remarkable stability of the key relationships between bond yields and stocks yields, and the trendless history of long-term interest rates over so many years, no one ever dreamed of anything different. Nor did people have any reason for doing so before the development of contracyclical monetary and fiscal policy and before they had experienced a price level that only went up instead of rising on some occasions and falling on others. In other words, these paradigm shifts may not have been unpredictable, but they were unthinkable.

  If these events were unpredictable, how can we expect the elaborate quantitative devices of risk management to predict them? How can we program into the computer concepts that we cannot program into ourselves, that are even beyond our imagination?

  We cannot enter data about the future into the computer because such data are inaccessible to us. So we pour in data from the past to fuel the decision-making mechanisms created by our models, be they linear or nonlinear. But therein lies the logician's trap: past data from real life constitute a sequence of events rather than a set of independent observations, which is what the laws of probability demand. History provides us with only one sample of the economy and the capital markets, not with thousands of separate and randomly distributed numbers. Even though many economic and financial variables fall into distributions that approximate a bell curve, the picture is never perfect. Once again, resemblance to truth is not the same as truth. It is in those outliers and imperfections that the wildness lurks.

  Finally, the science of risk management sometimes creates new risks even as it brings old risks under control. Our faith in risk management encourages us to take risks we would not otherwise take. On most counts, that is beneficial, but we must be wary of adding to the amount of risk in the system. Research reveals that seatbelts encourage drivers to drive more aggressively. Consequently, the number of accidents rises even though the seriousness of injury in any one accident declines.* Derivative financial instruments designed as hedges have tempted investors to transform them into speculative vehicles with sleigh-rides for payoffs and involving risks that no corporate risk manager should contemplate. The introduction of portfolio insurance in the late 1970s encouraged a higher level of equity exposure than had prevailed before. In the same fashion, conservative institutional investors tend to use broad diversification to justify higher exposure to risk in untested areas-but diversification is not a guarantee against loss, only against losing everything at once.

  Nothing is more soothing or more persuasive than the computer screen, with its imposing arrays of numbers, glowing colors, and elegantly structured graphs. As we stare at the passing show, we become so absorbed that we tend to forget that the computer only answers questions; it does not ask them. Whenever we ignore that truth, the computer supports us in our conceptual errors. Those who live only by the numbers may find that the computer has simply replaced the oracles to whom people resorted in ancient times for guidance in risk management and decision-making.

  At the same time, we must avoid rejecting numbers when they show more promise of accuracy than intuition and hunch, where, as Kahneman and Tversky have demonstrated, inconsistency and myopia so often prevail. G.B. Airy, one of many brilliant mathematicians who have served as director of Britain's Royal Observatory, wrote in 1849, "I am a devoted admirer of theory, hypothesis, formula, and every other emanation of pure intellect which keeps erring man straight among the stumbling-blocks and quagmires of matter-of-fact observations."9

  The central theme of this whole story is that the quantitative achievements of the heroes we have met shaped the trajectory of progress over the past 450 years. In engineering, medicine, science, finance, business, and even in government, decisions that touch everyone's life are now made in accordance with disciplined procedures that far outperform the seat-of-the-pants methods of the past. Many catastrophic errors of judgment are thus either avoided, or else their consequences are muted.

  Cardano the Renaissance gambler, followed by Pascal the geometer and Fermat the lawyer, the monks of Port-Royal and the ministers of Newington, the notions man and the man with the sprained brain, Daniel Bernoulli and his uncle Jacob, secretive Gauss and voluble Quetelet, von Neumann the playful and Morgenstern the ponderous, the religious de Moivre and the agnostic Knight, pithy Black and loquacious Scholes, Kenneth Arrow and Harry Markowitz-all of them have transformed the perception of risk from chance of loss into opportunity for gain, from FATE and ORIGINAL DESIGN to sophisticated, probability-based forecasts of the future, and from helplessness to choice.

  Opposed though he was to mechanical applications of the laws of probability and the quantification of uncertainty, Keynes recognized that this body of thought had profound implications for humanity:

  The importance of probability can only be derived from the judgment that it is rational to be guided by it in action; and a practical dependence on it can only be justified by a judgment that in action we ought to act to take some account of it.

  It is for this reas
on that probability is to us the "guide of life," since to us, as Locke says, "in the greatest part of our concernment, God has afforded only the Twilight, as I may so say, of Probability, suitable, I presume, to that state of Mediocrity and Probationership He has been pleased to place us in here."10

  INTRODUCTION

  1. Quoted in Keynes, 1921, frontispiece to Chapter XXVIII.

  2. Personal conversation.

  3. Arrow, 1992, p. 46.

  CHAPTER 1

  1. Quoted in Ignatin and Smith, 1976, p. 80. The quotation is from Book I, Chapter X, of The Wealth of Nations.

  2. Keynes, 1936, p. 159.

  3. Ibid., p. 150.

  4. This entire paragraph is from Bolen, 1976.

  5. Excellent background on all this may be found in David, 1962, pp. 2-21.

  6. See David, 1962, p. 34.

  7. Hayano, 1982.

  8. Johnson, 1995.

  9. See David, p. 2.

  10. Sambursky, 1956, p. 36.

  11. Ibid., p. 37.

  12. Ibid., pp. 36-40.

  13. Rabinovitch, 1969.

  14. Frankfort, 1956; quoted in Heilbroner, 1995, p. 23. See also David, 1962, pp. 21-26.

  15. See Eves, 1983, p. 136.

  CHAPTER 2

  1. Most of the background and biographical material on Fibonacci comes from the Encyclopedia Brittanica; Eves, 1983, p. 161; Hogben, 1968, p. 250; and Garland, 1987.

  2. Two stimulating commentaries on the Fibonacci numbers are Garland, 1987, and Hoffer, 1975. The examples here are drawn from those two sources.

  3. The background material presented here comes primarily from Hogben, 1968, Chapter I.

  4. See Hogben, 1968, p. 35; also Eves, 1983, Chapter I.

  5. See Hogben, 1968, p. 36 and pp. 246-250.

  6. The background material on Diophantus is from Turnbull, 1951, p. 113.

  7. Ibid., p. 110.

  8. Ibid., p. 111.

  9. See Hogben, 1968, pp. 244-246.

  10. From Newman, 1988a, p. 433.

  11. The background material on al-Khowarizmi is primarily from Muir, 1961, and Hogben, 1968.

  12. Hogben, 1968, p. 243.

  13. See Hogben, 1968, Chapter VI, for an extended and stimulating discussion of the development of algebra and the uses of zero.

  14. The background material on Omar Khayyam is from Fitzgerald.

  15. Hogben, 1968, p. 245.

  CHAPTER 3

  1. The background material on Paccioli comes primarily from David, 1962, pp. 36-39, and Kemp, 1981, pp. 146-148.

  2. The material on Paccioli and Leonardo is from Kemp, 1981, pp. 248-250.

  3. David, 1962, p. 37.

  4. Sambursky, 1956.

  5. Ibid.

  6. Ibid.

  7. The background material on Cardano and the quotations are primarily from Ore, 1953, and Morley, 1854, with some quotes from Cardan, 1930.

  8. David, 1962, p. 61.

  9. See Sarton, 1957, pp. 29-32; also Muir, 1961, pp. 35-38.

  10. Hacking, 1975, p. 18. The complete discussion, which runs through Chapter 3, "Opinion," is well worth careful study.

  11. Hacking, 1975, p. 22.

  12. Ibid., p. 26.

  13. Ibid., p. 44.

  14. David, 1962, p. 58.

  15. Kogelman and Heller, 1986, pp. 164-165.

  16. The background material on Galileo is primarily from David, 1962, Chapter 7, pp. 61-69.

  17. David, 1962, p. 65.

  18. Ibid., p. 13.

  19. Stigler, 1988.

  CHAPTER 4

  1. The background material on Pascal is from Muir, 1961, pp. 77-100; David, 1962, pp. 34-79; and Hacking, 1975, pp. 55-70.

  2. See David, 1962, p. 74.

  3. Muir, 1961, p. 90.

  4. Ibid., p. 93.

  5. Ibid., p. 94.

  6. Ibid., p. 95.

  7. David, 1961, p. 69; see also Appendix 4.

  8. See Huff, 1959, pp. 63-69.

  9. See Hogben, 1968, p. 551; see also Hacking, 1975, pp. 58-59.

  10. See David, 1962, pp. 71-75.

  11. Turnbull, 1951, p. 130.

  12. Ibid., p. 131.

  13. See Hogben, 1968, pp. 277-279; see also David, 1962, p. 34.

  14. Turnbull, 1951, p. 131; also Eves, 1984, p. 6.

  15. I am grateful to Stanley Kogelman for helping me work out these examples.

  16. This point, and the quotation from Pascal that follows, are from Guilbaud, 1968; the translation is mine.

  17. David, 1962, p. 252.

  18. All the material that follows is from Hacking, 1975, Chapter 8, "The Great Decision," pp. 63-70.

  19. Hacking, p. 62.

  20. The material about the Port-Royal monastery is from Hacking, 1975, pp. 73-77.

  21. Ibid., p. 25.

  22. Ibid., p. 74.

  23. Ibid., p. 77.

  24. Ibid., p. 77.

  25. Ibid., p. 77.

  26. Ibid., p. 77.

  CHAPTER 5

  1. I am grateful to Stigler (1977) for this description and to Stephen Stigler personally for drawing the Trial of the Pyx to my attention.

  2. The background material on Graunt is from Muir, 1961; David, 1962; and Newman, 1988g. (Direct quotations from Natural and Political Obligations are primarily from Newman.)

  3. Newman, 1988g, p. 1394.

  4. The background material on Petty is from Hacking, 1975, pp. 102-105. 5. The material about Wilkins and the Royal Society is from Hacking, 1975, pp. 169-171.

  6. Graunt, p. 1401.

  7. Ibid., p. 1401.

  8. Hacking, 1975, p. 103.

  9. I am grateful to Stephen Stigler for making this point clear to me. 10. See Hacking, 1975, pp. 103-105.

  11. The illustration is from Stigler, 1996.

  12. David, 1962, p. 107. An extended explanation of Graunt's calculations and estimating procedure appears on pp. 107-109.

  13. Hacking, 1975, p. 107.

  14. Ibid., p. 110.

  15. See discussion in Hacking, 1975, pp. 105-110.

  16. The background material on Naumann and Halley and the quotations from Halley are primarily from Newman, 1988g, pp. 1393-1396 and 1414-1432.

  17. See discussion in Hacking, 1975, pp. 111-121.

  18. The material that follows on the history of insurance in general and Lloyd's in particular is from Flower and Jones, 1974; also Hodgson, 1984.

  19. Macaulay, 1848, p. 494. For Macaulay's full and fascinating story of the English national debt, see the entire chapter that runs from p. 487 to p. 498.

  20. Flower and Jones, 1974.

  21. American Academy of Actuaries, 1994, and Moorehead, 1989.

  22. Interesting background material on the role of the Monte dei Paschi may be found in Chichilnisky and Heal, 1993.

  23. See, in particular, Townsend, 1995, and Besley, 1995.

  24. Flower and Jones, 1974, p. 13.

  CHAPTER 6

  1. Bernoulli, Daniel, 1738.

  2. The background material on the Bernoulli family is from Newman, 1988f.

  3. Bell, 1965, p. 131.

  4. Newman, 1988f, p. 759.

  5. Ibid.

  6. Ibid.

  7. Ibid.

  8. This story and the quotes are from David, 1962, pp. 133-135. 9. Stigler, 1993.

  10. All Bernoulli quotations are from Bernoulli, 1738.

  11. An extended and lucid example of expected utilities and risk may be found in Bodie, Kane, and Marcus, 1992, Chapter 7, pp. 183-209. See also Kritzman, 1995, Chapter 3, pp. 21-32.

  12. Todhunter, 1949. See also Bassett, 1987, and the list of references therein.

  13. Siegel, 1994, Chapter 8, pp. 95-104.

  CHAPTER 7

  1. Background material on Jacob Bernoulli is from Newman, 1988f.

  2. Hacking, 1975, p. 166; see also Kendall, 1974.

  3. Gesammelte Werke (ed. Pertz and Gerhardt), Halle 1855, vol. 3, pp. 71-97. I am grateful to Marta Steele and Doris Bullard for the translation into English. Chapter
XXX of Keynes, 1921, has an illuminating discussion of this exchange between Leibniz and Bernoulli.

  4. An excellent analysis of Ars Conjectandi may be found in David, 1962, pp. 133-139 and in Stigler, 1986, pp. 63-78.

  5. Bernoulli, Jacob, 1713, p. 1430.

  6. Ibid., p. 1431.

  7. Hacking, 1975, p. 145.

  8. Ibid., p. 146.

  9. Ibid., p. 163.

  10. David, 1962, p. 137.

  11. Stigler, 1986, p. 71. This book was an invaluable source of information for this chapter.

  12. The background material on de Moivre is from Stigler, 1986, Chapter 2, and David, 1962, Chapter XV.

  13. Stigler, 1986, p. 85.

  14. This example is freely adapted from Groebner and Shannon, 1993, Chapter 20.

  15. The background material on Bayes is from Stigler, 1986, and Cone, 1952.

  16. Groebner and Shannon, 1993, p. 1014.

  17. Stigler, 1986, p. 123.

  18. Cone, 1952, p. 50.

  19. Ibid., p. 41.

  20. Ibid., pp. 42-44.

  21. Bayes, 1763.

  22. Price's letter of transmittal and Bayes's essay are reprinted in Kendall and Plackett, 1977, pp. 134-141.

  23. An excellent description of this experiment may be found in Stigler, 1986, pp. 124-130.

  24. Smith, 1984. This paper contains an excellent analysis of the Bayesian approach.

  25. David, 1962, p. 177.

  CHAPTER 8

  1. The biographical material on Gauss is primarily from Shaaf, 1964, and from Bell, 1965.

  2. Schaaf, 1964, p. 40.

  3. Bell, 1965, p. 310.

  4. The biographical background on Laplace is from Newman, 1988d, pp. 1291-1299.

  5. Newman, 1988d, p. 1297.

  6. Ibid., p. 1297.

  7. Ibid., p. 1297.

  8. Bell, 1965, p. 324.

  9. Ibid., p. 307.