In this summary, you will learn
Greek mythology described the rolling of dice as part of creation, but Grecian scientists and philosophers did not form a theory of probability.
Europe’s adoption of Arabic numerals was an important step toward the development of probability theory during the Renaissance.
Blaise Pascal and Pierre de Fermat invented the mathematics of probability.
Daniel Bernoulli used mathematical concepts to analyze commercial transactions.
Francis Galton established the idea of regression to the mean.
Risk management, at its core, is about controlling business factors you can influence and avoiding those you cannot.
Human behavior often departs from theory’s rational prescriptions.
Although individuals make irrational decisions, markets usually perform as theory would suggest.
Behavioral finance explores the implications of emotional and psychological factors in economic decision making.
Despite its limitations, risk management theory probably has helped avoid more catastrophes than it has caused.
Gambling was part of the Greeks’ creation myth. Their gods played dice to decide who would rule the heavens, the seas and the underworld. However, the ancient Greek pioneers of science and philosophy did not undertake the study of probability. Nor, for that matter, did the philosophers of the Hebrew Talmud, although they used calculations of odds to determine what was permissible under the law. Not until the Renaissance did thinkers turn their attention to developing the mathematics of probability. Then, in order to move ahead, they needed numbers.
“This book tells the story of a group of thinkers whose remarkable vision revealed how to put the future at the service of the present.”
The Hindu numbering system originated in roughly A.D. 500. Arabs began using it in about A.D. 700, brought it out of India and spread it through the world. The concept of zero was one of the most profound innovations in mathematical history. The invention of zero allowed people to calculate using merely 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
“By showing the world how to understand risk, measure it and weigh its consequences, they converted risk taking into one of the prime catalysts that drives modern Western society.”
Leonardo Pisano, known as an adult – and today – as Fibonacci, explored the power and possibility of Arabic numbers in Liber Abaci, or Book of the Abacus. His work was published in 1202, a time when much of the Western world still relied on Roman numerals. The “Fibonacci ratio” mathematically expresses the balanced proportion that the Greeks called “the golden mean.” His mathematical explorations broke new ground in measurement, but the world was not yet ready to apply his findings to risk. Today, however, Fibonacci is a touchstone for investors and speculators.
“The recognition of risk management as a practical art rests on a simple cliché with the most profound consequences: When our world was created, nobody remembered to include certainty.”
Accomplished mathematician Luca Paccioli, a Franciscan monk, may not have invented double-entry accounting, but he did more to popularize it than anyone before him. A friend of Leonardo da Vinci, Paccioli proposed a puzzle about how to divide the stakes in an interrupted gambling game. The study of gambling was the nursery of the science of risk. Physician Girolamo Cardano, an avid, perhaps even addicted, gambler, wrote on many subjects – including mathematics. In a 16th-century treatise on games of chance, he made important progress toward building the science of probability. He did not solve Fibonacci’s ratio, but he calculated the odds of various dice throws. Even Galileo studied dice as part of his duty to his patron, the Grand Duke of Tuscany.
“Modern methods of dealing with the unknown start with measurement, with odds and probabilities. The numbers come first. But where did the numbers come from?”
However, the most important pioneers of probability were Frenchmen Blaise Pascal and Pierre de Fermat. Pascal was a mathematics prodigy. In his early teens, he joined a group that became the forerunner of the Académie des Sciences. Pascal worked with Fermat – a lawyer, extraordinary mathematician and superb linguist – to solve the puzzle of Fibonacci’s ratio. Pascal advanced the study of probability even further, developing an argument for belief in God that proved faith is the only rational response to the universe. He said that a person who obeys religious law and tries to pursue holiness is, in essence, betting on God’s existence. Conversely, a person who does not adhere to religious law is betting that there is no God. If there is a God, the reward for believing is infinite; if there is no God, the believer’s loss is finite, because human life is finite. Wagering a finite stake on a 50/50 chance of an infinite reward is eminently rational.
“Gambling is an ideal laboratory in which to perform experiments on the quantification of risk.”
Origins of Statistical Forecasting
Statistical sampling is fundamental to the use of statistics and to probability – and, therefore, it is also fundamental to risk management. John Graunt was the first to apply statistical sampling to a problem of financial risk. He published his seminal work, Natural and Political Observations Made Upon the Bills of Mortality, in 1660. A dealer in buttons, needles and other “notions,” Graunt tabulated the causes of death in London using city records, or “Bills of Mortality,” as his basis for reasoned conclusions and estimates. He also studied births, population distribution and other aspects of demography. He recognized that the most feared causes of death were not actually the most likely causes of death. More importantly, he pioneered the use of averages.
“Most people know...that flying in an airplane is far safer than driving...but some passengers will keep the flight attendants busy while others will snooze happily regardless of the weather.”
Edmund Halley followed up on Graunt’s work with data compiled in Breslau, Germany. He calculated the odds of a person of a given age dying in a given year. These calculations would eventually provide the basis for the life insurance industry. However, governments that were selling annuities to raise money generally ignored this rigorous approach to pricing risk, and sold annuities to all comers at a uniform price.
“The past, or whatever data we choose to analyze, is only a fragment of reality. That fragmentary quality is crucial in going from data to a generalization.”
In 1738, Daniel Bernoulli wrote, “The value of an item must not be based on its price, but rather on the utility that it yields.” Bernoulli was 38 years old at the time, a member of a family of extraordinary mathematicians. Utility was certainly a new concept. Bernoulli meant something beyond expected value, which is the calculation of the relative probabilities of various results or returns. He knew that people have different risk preferences. Though the odds of being struck by lightning are small, a person who is terrified of lightning would place a high value on protection against it – perhaps higher than the probabilities would justify. Bernoulli understood the concept of diminishing marginal utility, the idea that, if you already own a thing, you do not value getting more of it as highly as you’d value getting something you don’t possess.
“The normal distribution forms the core of most systems of risk management.”
Bernoulli also grasped the importance of the intangible asset now called “human capital.” He recognized that even beggars have skill, albeit skill at begging. He saw that he could put a price on that ability: the amount of money a beggar would accept to forswear begging. Bernoulli also realized that the satisfaction people derive from increasing wealth diminishes as they get richer. This perception has pivotal implications for risk management since, for instance, people generally fear losses more than they yearn for gain. Centuries later, it became clear that people were not as rational as Bernoulli, a man of the Enlightenment, supposed.
“Galton moves us into the world of everyday life, where people breathe, sweat, copulate and ponder their future.”
Normality and Randomness
Carl Friedrich Gauss, an intellectually snobbish math prodigy, made an important discovery about geometry at age 18 and published his first book in 1801, at age 21. Gauss accidentally discovered “normal distribution” or “the bell curve,” a crucial concept in risk management. Although he originated many mathematical and astronomical findings, Gauss wasn’t thinking about risk management at the time. Rather, he was trying to perform a geodesic survey, using “the curvature of the earth to improve the accuracy of geographic measurements.” He discovered that he could check his measurements by comparing them to the average of all observations and calculating their distance from the mean.
“The asymmetry between the way we make decisions involving gains and decisions involving losses is one of the most striking findings of Prospect Theory.”
Francis Galton and Regression to the Mean
Francis Galton, a Victorian, an inventor and an explorer of Africa, was a pioneer of eugenics. Indeed, he coined the word. He used the normal distribution to explore differences among various groups of people, trying to prove that heredity was destiny. However, his work was somewhat frustrating. He discovered that the principle of regression to the mean prevailed. That is, particularly tall or exceptionally gifted parents tended to produce slightly shorter or less gifted children who were closer to average than to outstanding.
“Theories of how people make decisions and choices seem to have become detached from everyday life in the real world.”
Galton’s principle of regression to the mean matters greatly when forecasting market prices. Wall Street has numerous sayings that caution traders against expecting extraordinary performance to continue, such as “Buy low and sell high.” In fact, regression to the mean actually makes it advisable to fire a money manager who has been doing extraordinarily well, and to hire one who has been doing quite poorly. Odds are that the high performer will have a period of low performance and the poor performer will experience high performance. That’s what regression to the mean indicates. However, that’s not how people make such decisions. Research by behavioral economists shows that people do not take a dispassionate, rational view of performance. Rarely thinking for the long run, they assign too much importance to recent occurrences.
“Throughout most of the history of stock markets...it never occurred to anyone to define risk with a number.”
As powerful as the notion of regression to the mean is, a word of caution is in order. Times do change. The world of today is not the world of the Industrial Revolution. Regression to the mean occurs, but sometimes only over the very long term. When President Herbert Hoover reassured the U.S. that prosperity was almost at hand, he wasn’t lying. He was making a reasonable forecast based on his experience with prior recessions and depressions. Nothing like the Great Depression had ever occurred before, so it wasn’t part of his database for calculating the mean.
“Even though millions of investors would readily plead guilty to acting in defiance of rationality, the market – where it really counts – acts as though rationality prevailed.”
“Value depends entirely upon utility,” wrote William Stanley Jevons, who was convinced scholars could develop an applicable theory of value by using the mathematics of probability to measure utility. However, experience demonstrates that even experts simply cannot get enough data to confirm the statistics needed for such a precise measurement. In fact, mathematicians often must work without comprehensive information. In its absence, they have to make estimates.
Nobel Laureate Kenneth Arrow observed, however, that – as a rule – people have more confidence in their estimates than the facts justify. Even though they are not likely to collect on the policies, they pay premiums to insurance companies because they want to protect themselves against extreme outcomes. They try to reduce uncertainty and to protect themselves against risks, for instance, by purchasing burglar alarms. Arrow’s issue is not measuring probability. He focuses on discerning how people make decisions when they’re aware that they don’t know – but they don’t know the extent of their ignorance.
John von Neumann was a physicist and an author. He invented game theory, which offers rational analysis of decision making. He co-authored the Theory of Games and Economic Behavior with Oskar Morgenstern. The Federal Communications Commission used game theory’s principles to auction off a portion of the communications spectrum in 1993, and it has had other practical applications.
Measuring Investment Risk
Nobelist Harry Markowitz applied advanced mathematics to selecting stocks in a portfolio. His most important discovery was the role of diversification in reducing risk. He proved that an investor could hold a portfolio of securities that were, treated separately and distinctly, quite risky – and yet have a reasonably low-risk portfolio. How could this be? The individual stocks’ risks offset, balanced and countered one another. Markowitz upended traditional ways of thinking about investing, but critics level many reasonable objections to his theory, particularly that investors are not really as rational as his model supposes. Other objections focus on his use of variance as a reasonable expression of risk.
Israeli psychologists Daniel Kahneman and Amos Tversky recognized that people follow decision-making patterns that don’t seem appropriate according to rational analysis. Prospect theory proposes that people tend to use heuristics, rules of thumb and shortcuts when they have a limited understanding of all the factors that are really involved. For example, investors are likely to arrive at different decisions about the right course of action if the problem before them is framed differently. That is, most folks don’t think like mathematicians. As Kahneman and Tversky wrote, “Theories of choice are at best approximate and incomplete.”
In the 1990s, derivatives became widely popular. These highly mathematically sophisticated financial instruments essentially represented side bets on the performance of various markets. Used responsibly, and for risk protection, derivatives played an important role in enabling investors to take advantage of opportunities in an uncertain environment. However, derivatives proved disastrous for companies that became speculators instead of hedgers. These companies “ignored one of the most fundamental principles of investment theory: You cannot expect to make large profits without taking the risk of large losses.”