Against the Gods: The Remarkable Story of Risk Page 22
In business, we seal a deal by signing a contract or by shaking hands. These formalities prescribe our future behavior even if conditions change in such a way that we wish we had made different arrangements. At the same time, they protect us from being harmed by the people on the other side of the deal. Firms that produce goods with volatile prices, such as wheat or gold, protect themselves from loss by entering into commodity futures contracts, which enable them to sell their output even before they have produced it. They pass up the possibility of selling later at a higher price in order to avoid uncertainty about the price they will receive.
In 1971, Kenneth Arrow, in association with fellow economist Frank Hahn, pointed up the relationships between money, contracts, and uncertainty. Contracts would not be written in money terms "if we consider an economy without a past or a future."18 But the past and the future are to the economy what woof and warp are to a fabric. We make no decision without reference to a past that we understand with some degree of certainty and to a future about which we have no certain knowledge. Contracts and liquidity protect us from unwelcome consequences even when we are coping with Arrow's clouds of vagueness.
Some people guard against uncertain outcomes in other ways. They call a limousine service to avoid the uncertainty of riding in a taxi or taking public transportation. They have burglar alarm systems installed in their homes. Reducing uncertainty is a costly business.
Arrow's idea of a "complete market" was based on his sense of the value of human life. "The basic element in my view of the good society," he wrote, "is the centrality of others.... These principles imply a general commitment to freedom.... Improving economic status and opportunity ... is a basic component of increasing freedom. `9 But the fear of loss sometimes constrains our choices. That is why Arrow applauds insurance and risk-sharing devices like commodity futures contracts and public markets for stocks and bonds. Such facilities encourage investors to hold diversified portfolios instead of putting all their eggs in one basket.
Arrow warns, however, that a society in which no one fears the consequences of risk-taking may provide fertile ground for antisocial behavior. For example, the availability of deposit insurance to the depositors of savings and loan associations in the 1980s gave the owners a chance to win big if things went right and to lose little if things went wrong. When things finally went wrong, the taxpayers had to pay. Wherever insurance can be had, moral hazard-the temptation to cheat-will be present.*
There is a huge gap between Laplace and Poincare on the one hand and Arrow and his contemporaries on the other. After the catastrophe of the First World War, the dream vanished that some day human beings would know everything they needed to know and that certainty would replace uncertainty. Instead, the explosion of knowledge over the years has served only to make life more uncertain and the world more difficult to understand.
Seen in this light, Arrow is the most modem of the characters in our story so far. Arrow's focus is not on how probability works or how observations regress to the mean. Rather, he focuses on how we make decisions under conditions of uncertainty and how we live with the decisions we have made. He has brought us to the point where we can take a more systematic look at how people tread the path between risks to be faced and risks to be taken. The authors of the Port-Royal Logic and Daniel Bernoulli both sensed what lines of analysis in the field of risk might lie ahead, but Arrow is the father of the concept of risk management as an explicit form of practical art.
The recognition of risk management as a practical art rests on a simple cliche with the most profound consequences: when our world was created, nobody remembered to include certainty. We are never certain; we are always ignorant to some degree. Much of the information we have is either incorrect or incomplete.
Suppose a stranger invites you to bet on coin-tossing. She assures you that the coin she hands you can be trusted. How do you know whether she is telling the truth? You decide to test the coin by tossing it ten times before you agree to play.
When it comes up eight heads and two tails, you say it must be loaded. The stranger hands you a statistics book, which says that this lop-sided result may occur about one out of every nine times in tests of ten tosses each.
Though chastened, you invoke the teachings of Jacob Bernoulli and request sufficient time to give the coin a hundred tosses. It comes up heads eighty times! The statistics book tells you that the probability of getting eighty heads in a hundred tosses is so slight that you will have to count the number of zeroes following the decimal point. The probability is about one in a billion.
Yet you are still not 100% certain that the coin is loaded. Nor will you ever be 100% certain, even if you were to go on tossing it for a hundred years. One chance in a billion ought to be enough to convince you that this is a dangerous partner to play games with, but the possibility remains that you are doing the woman an injustice. Socrates said that likeness to truth is not truth, and Jacob Bernoulli insisted that moral certainty is less than certainty.
Under conditions of uncertainty, the choice is not between rejecting a hypothesis and accepting it, but between reject and not-reject. You can decide that the probability that you are wrong is so small that you should not reject the hypothesis. You can decide that the probability that you are wrong is so large that you should reject the hypothesis. But with any probability short of zero that you are wrong-certainty rather than uncertainty-you cannot accept a hypothesis.
This powerful notion separates most valid scientific research from hokum. To be valid, hypotheses must be subject to falsification-that is, they must be testable in such fashion that the alternative between reject and not-reject is clear and specific and that the probability is measurable. The statement "He is a nice man" is too vague to be testable. The statement "That man does not eat chocolate after every meal" is falsifiable in the sense that we can gather evidence to show whether the man has or has not eaten chocolate after every meal in the past. If the evidence covers only a week, the probability that we could reject the hypothesis (we doubt that he does not eat chocolate after every meal) will be higher than if the evidence covers a year. The result of the test will be not-reject if no evidence of regular consumption of chocolate is available. But even if the lack of evidence extends over a long period of time, we cannot say with certainty that the man will never start eating chocolate after every meal in the future. Unless we have spent every single minute of his life with him, we could never be certain that he has not eaten chocolate regularly in the past.
Criminal trials provide a useful example of this principle. Under our system of law, criminal defendants do not have to prove their innocence; there is no such thing as a verdict of innocence. Instead, the hypothesis to be established is that the defendant is guilty, and the prosecution's job is to persuade the members of jury that they should not reject the hypothesis of guilt. The goal of the defense is simply to persuade the jury that sufficient doubt surrounds the prosecution's case to justify rejecting that hypothesis. That is why the verdict delivered by juries is either "guilty" or "not guilty."
The jury room is not the only place where the testing of a hypothesis leads to intense debate over the degree of uncertainty that would justify rejecting it. That degree of uncertainty is not prescribed. In the end, we must arrive at a subjective decision on how much uncertainty is acceptable before we make up our minds.
For example, managers of mutual funds face two kinds of risk. The first is the obvious risk of poor performance. The second is the risk of failing to measure up to some benchmark that is known to potential investors.
The accompanying chart20 shows the total annual pretax rate of return (dividends paid plus price change) from 1983 through 1995 to a stockholder in the American Mutual Fund, one of the oldest and largest equity mutual funds in the business. The American Mutual performance is plotted as a line with dots, and the performance of the Standard & Poor's Composite Index of 500 Stocks is represented by the bars.
Although American Mut
ual tracks the S&P 500 closely, it had higher returns in only three out of the thirteen years-in 1983 and 1993, when American Mutual rose by more, and in 1990, when it fell by less. In ten years, American Mutual did about the same as or earned less than the S&P.
Was this just a string of bad luck, or do the managers of American Mutual lack the skill to outperform an unmanaged conglomeration of 500 stocks? Note that, since American Mutual is less volatile than the S&P, its performance was likely to lag in the twelve out of thirteen years in which the market was rising. The Fund's performance might look a lot better in years when the market was declining or not moving up or down.
Nevertheless, when we put these data through a mathematical stress test to determine the significance of these results, we find that American Mutual's managers probably did lack skill.21 There is only a 20% probability that the results were due to chance. To put it differently, if we ran this test over five other thirteen-year periods, we would expect American Mutual to underperform the S&P 500 in four of the periods.
Many observers would disagree, insisting that twelve years is too small a sample to support so broad a generalization. Moreover, a 20% probability is not small, though less than 50%. The current convention in the world of finance is that we should be 95% certain that something is "statistically significant" (the modern equivalent of moral certainty) before we accept what the numbers indicate. Jacob Bernoulli said that 1,000 chances out of 1,001 were required for one to be morally certain; we require only one chance in twenty that what we observe is a matter of chance.
But if we cannot be 95% certain of anything like this on the basis of only twelve observations, how many observations would we need? Another stress test reveals that we would need to track American Mutual against the S&P 500 for about thirty years before we could be 95% certain that underperformance of this magnitude was not just a matter of luck. As that test is a practical impossibility, the best judgment is that the American Mutual managers deserve the benefit of the doubt; their performance was acceptable under the circumstances.
The next chart shows a different picture. Here we see the relative performance of a small, aggressive fund called AIM Constellation. This fund was a lot more volatile during these years than either the S&P Index or American Mutual. Note that the vertical scale in this chart is twice the height of the vertical scale in the preceding chart. AIM had a disastrous year in 1984, but in five other years it outperformed the S&P 500 by a wide margin. The average annual return for AIM over the thirteen years was 19.8% as compared with 16.7% for the S&P 500 and 15.0% for American Mutual.
Is this record the result of luck or skill? Despite the wide spread in returns between AIM and the S&P 500, the greater volatility of AIM makes this a tough question to answer. In addition, AIM did not track the S&P 500 as faithfully as American Mutual did: AIM went down one year when the S&P 500 was rising, and it earned as much in 1986, as in 1985, as the S&P was earning less. The pattern is so irregular that we would have a hard time predicting this fund's performance even if we were smart enough to predict the returns on the S&P 500.
Because of the high volatility and low correlation, our mathematical stress test reveals that luck played a significant role in the AIM case just as in the American Mutual case. Indeed, we would need a track record exceeding a century before we could be 95% certain that these AIM results were not the product of luck! In risk-management terms, there is a suggestion here that the AIM managers may have taken excessive risk in their efforts to beat the market.
Many anti-smokers worry about second-hand smoke and support efforts to making smoking in public places illegal. How great is the risk that you will develop lung cancer when someone lights up a cigarette at the next table in a restaurant or in the next seat on an airplane? Should you accept the risk, or should you insist that the cigarette be extinguished immediately?
In January 1993, the Environmental Protection Administration issued a 510-page report carrying the ominous title Respiratory Health Effects of Passive Smoking: Lung Cancer and Other Disorders.22 A year later, Carol Browner, the EPA Administrator, appeared before a congressional committee and urged it to approve the Smoke-Free Environment Act, which establishes a complex set of regulations designed to prohibit smoking in public buildings. Browner stated that she based her recommendation on the report's conclusion that environmental tobacco smoke, or ETS, is "a known human lung carcinogen."23
How much is "known" about ETS? What is the risk of developing lung cancer when someone else is doing the smoking?
There is only one way even to approach certainty in answering these questions: Check every single person who was ever exposed to ETS at any moment since people started smoking tobacco hundreds of years ago. Even then, a demonstrated association between ETS and lung cancer would not be proof that ETS was the cause of the cancer.
The practical impossibility of conducting tests on everybody or everything over the entire span of history in every location leaves all scientific research results uncertain. What looks like a strong association may be nothing more than the luck of the draw, in which case a different set of samples from a different time period or from a different locale, or even a different set of subjects from the same period and the same locale, might have produced contrary findings.
There is only one thing we know for certain: an association (not a cause-and-effect) between ETS and lung cancer has a probability that is some percentage short of 100%. The difference between 100% and the indicated probability reflects the likelihood that the ETS has nothing whatsoever to do with causing lung cancer and that similar evidence would not necessarily show up in another sample. The risk of coming down with lung cancer from ETS boils down to a set of odds, just as in a game of chance.
Most studies like the EPA analysis compare the result when one group of people is exposed to something, good or bad, with the result from a "control" group that is not exposed to the same influences. Most new drugs are tested by giving one group the drug in question and comparing their response with the response of a group that has been given a placebo.
In the passive smoking case, the analysis focused on the incidence of lung cancer among non-smoking women living with men who smoked. The data were then compared with the incidence of disease among the control group of non-smoking women living with nonsmoking companions. The ratio of the responses of the exposed group to the responses of the control group is called the test statistic. The absolute size of the test statistic and the degree of uncertainty surrounding it form the basis for deciding whether to take action of some kind. In other words, the test statistic helps the observer to distinguish between CONSTANTINOPLE and BZUXRQVICPRGAB and cases with more meaningful results. Because of all the uncertainties involved, the ultimate decision is often more a matter of gut than of measurement, just as it is in deciding whether a coin is fair or loaded.
Epidemiologists-the statisticians of health-observe the same convention as that used to measure the performance of investment managers. They usually define a result as statistically significant if there is no more than a 5% probability that an outcome was the result of chance.
The results of the EPA study of passive smoking were not nearly as strong as the results of the much larger number of earlier studies of active smoking. Even though the risk of contracting lung cancer seemed to correlate well with the amount of exposure-how heavily the male companion smoked-the disease rates among women exposed to ETS averaged only 1.19 times higher than among women who lived with non-smokers. Furthermore, this modest test statistic was based on just thirty studies, of which six showed no effect from ETS. Since many of those studies covered small samples, only nine of them were statistically significant.24 None of the eleven studies conducted in the United States met that criterion, but seven of those studies covered fewer than forty-five cases.25
In the end, admitting that "EPA has never claimed that minimal exposure to secondhand smoke poses a huge individual cancer risk,"26 the agency estimated that "approximately 3,000 Amer
ican nonsmokers die each year from lung cancer caused by secondhand smoke."27 That conclusion prompted Congress to pass the Smoke-Free Environment Act, with its numerous regulations on public facilities.
We have reached the point in the story where uncertainty, and its handmaiden luck, have moved to center stage. The setting has changed, in large part because in the 75 years or so since the end of the First World War the world has faced nearly all the risks of the old days and many new risks as well.
The demand for risk management has risen along with the growing number of risks. No one was more sensitive to this trend than Frank Knight and John Maynard Keynes, whose pioneering work we review in the next chapter. Although both are now dead-their most important writings predate Arrow's-almost all the figures we shall meet from now on are, like Arrow, still alive. They are testimony to how young the ideas of risk management are.
The concepts we shall encounter in the chapter ahead never occurred to the mathematicians and philosophers of the past, who were too busy establishing the laws of probability to tackle the mysteries of uncertainty.
rancis Galton died in 1911 and Henri Poincare died the following year. Their passing marked the end of the grand age of mea an era that reached back five centuries to Paccioli's game of balla. For it was his problem of the points (page 43) that had launched the long march to defining the future in terms of the laws of probability. None of the great mathematicians and philosophers of the past whom we have met so far doubted that they had the tools they needed to determine what the future held. It was only the facts that demanded attention. surement,
I do not mean to imply that Galton and Poincare finished the task: the principles of risk management are still evolving. But their deaths occurred-and their understanding of risk climaxed-on the eve of one of the great watersheds of history, the First World War.