Against the Gods: The Remarkable Story of Risk (22 page)

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.

Bernoulli introduced another novel idea that economists today consider a driving force in economic growth-human capital. This idea
emerged from his definition of wealth as "anything that can contribute
to the adequate satisfaction of any sort of want .... There is then
nobody who can be said to possess nothing at all in this sense unless he
starves to death."

What form does most people's wealth take? Bernoulli says that tangible assets and financial claims are less valuable than productive capacity, including even the beggar's talent. He suggests that a man who can
earn 10 ducats a year by begging will probably reject an offer of 50
ducats to refrain from begging: after spending the 50 ducats, he would
have no way of supporting himself. There must, however, be some
amount that he would accept in return for a promise never to beg again. If that amount were, for instance, 100 ducats, "we might say that
[the beggar] is possessed of wealth worth one hundred."

Today, we view the idea of human capital-the sum of education,
natural talent, training, and experience that comprise the wellspring of
future earnings flows-as fundamental to the understanding of major
shifts in the global economy. Human capital plays the same role for an
employee as plant and equipment play for the employer. Despite the
enormous accretions of tangible wealth since 1738, human capital is still
by far the largest income-producing asset for the great majority of people. Why else would so many breadwinners spend their hard-earned
money on life-insurance premiums?

For Bernoulli, games of chance and abstract problems were merely
tools with which to fashion his primary case around the desire for wealth
and opportunity. His emphasis was on decision-making rather than on
the mathematical intricacies of probability theory. He announces at the
outset that his aim is to establish "rules [that] would be set up whereby
anyone could estimate his prospects from any risky undertaking in light
of one's specific financial circumstances." These words are the grist for
the mill of every contemporary financial economist, business manager,
and investor. Risk is no longer something to be faced; risk has become
a set of opportunities open to choice.

Bernoulli's notion of utility-and his suggestion that the satisfaction
derived from a specified increase in wealth would be inversely related to
the quantity of goods previously possessed-were sufficiently robust to
have a lasting influence on the work of the major thinkers who followed. Utility provided the underpinnings for the Law of Supply and
Demand, a striking innovation of Victorian economists that marked the
jumping-off point for understanding how markets behave and how buyers and sellers reach agreement on price. Utility was such a powerful
concept that over the next two hundred years it formed the foundation
for the dominant paradigm that explained human decision-making and
theories of choice in areas far beyond financial matters. The theory of
games-the innovative twentieth century approach to decision-making
in war, politics, and business management-makes utility an integral
part of its entire system.

Utility has had an equally profound influence on psychology and
philosophy, for Bernoulli set the standard for defining human rationality. For example, people for whom the utility of wealth rises as they grow richer are considered by most psychologists-and moralists-as
neurotic; greed was not part of Bernoulli's vision, nor is it included in
most modern definitions of rationality.

Utility theory requires that a rational person be able to measure
utility under all circumstances and to make choices and decisions
accordingly-a tall order given the uncertainties we face in the course
of a lifetime. The chore is difficult enough even when, as Bernoulli
assumed, the facts are the same for everyone. On many occasions the
facts are not the same for everyone. Different people have different
information; each of us tends to color the information we have in our
own fashion. Even the most rational among us will often disagree about
what the facts mean.

Modern as Bernoulli may appear, he was very much a man of his
times. His concept of human rationality fitted neatly into the intellectual
environment of the Enlightenment. This was a time when writers, artists, composers, and political philosophers embraced the classical ideas of
order and form and insisted that through the accumulation of knowledge mankind could penetrate the mysteries of life. In 1738, when
Bernoulli's paper appeared, Alexander Pope was at the height of his
career, studding his poems with classical allusions, warning that "A little
learning is a dangerous thing," and proclaiming that "The proper
study of mankind is man." Denis Diderot was soon to start work on a
28-volume encyclopedia, and Samuel Johnson was about to fashion the
first dictionary of the English language. Voltaire's unromantic viewpoints on society occupied center stage in intellectual circles. By 1750,
Haydn had defined the classical form of the symphony and sonata.

The Enlightenment's optimistic philosophy of human capabilities
would show up in the Declaration of Independence and would help
shape the Constitution of the newly formed United States of America.
Carried to its violent extreme, the Enlightenment inspired the citizens
of France to lop off the head of Louis XVI and to enthrone Reason on
the altar of Notre Dame.

Bernoulli's boldest innovation was the notion that each of useven the most rational-has a unique set of values and will respond
accordingly, but his genius was in recognizing that he had to go further than that. When he formalizes his thesis by asserting that utility is inversely proportionate to the quantity of goods possessed, he opens up
a fascinating insight into human behavior and the way we arrive at
decisions and choices in the face of risk.

According to Bernoulli, our decisions have a predictable and systematic structure. In a rational world, we would all rather be rich than
poor, but the intensity of the desire to become richer is tempered by
how rich we already are. Many years ago, one of my investment counsel clients shook his finger at me during our first meeting and warned
me: "Remember this, young man, you don't have to make me rich. I
am rich already!"

The logical consequence of Bernoulli's insight leads to a new and
powerful intuition about taking risk. If the satisfaction to be derived
from each successive increase in wealth is smaller than the satisfaction
derived from the previous increase in wealth, then the disutility caused
by a loss will always exceed the positive utility provided by a gain of
equal size. That was my client's message to me.

Think of your wealth as a pile of bricks, with large bricks at the foundation and with the bricks growing smaller and smaller as the height
increases. Any brick you remove from the top of the pile will be larger
than the next brick you might add to it. The hurt that results from losing a brick is greater than the pleasure that results from gaining a brick.

Bernoulli provides this example: two men, each worth 100 ducats,
decide to play a fair game, like tossing coins, in which there is a 50-50
chance of winning or losing, with no house take or any other deduction from the stakes. Each man bets 50 ducats on the throw, which
means that each has an equal chance of ending up worth 150 ducats or
of ending up worth only 50 ducats.

Would a rational person play such a game? The mathematical
expectation of each man's wealth after the game has been played with
this 50-50 set of alternatives is precisely 100 ducats (150 + 50 divided
by 2), which is just what each player started with. The expected value
for each is the same as if they had not decided to play the game in the
first place.

Bernoulli's theory of utility reveals an asymmetry that explains why
an even-Steven game like this is an unattractive proposition. The 50
ducats that the losing player would drop have greater utility than the 50
ducats that the winner would pocket. Just as with the pile of bricks, los
ing 50 ducats hurts the loser more than gaining 50 ducats pleases the winner.*
In a mathematical sense a zero-sum game is a loser's game when it is valued in terms of utility. The best decision for both is to refuse to play this game.

Bernoulli uses his example to warn gamblers that they will suffer a loss of utility even in a fair game. This depressing result, he points out, is:

Nature's admonition to avoid the dice altogether .... [E]veryone who bets any part of his fortune, however small, on a mathematically fair game of chance acts irrationally .... [T]he imprudence of a gambler will be the greater the larger part of his fortune which he exposes to a game of chance.

Most of us would agree with Bernoulli that a fair game is a loser's game in utility terms. We are what psychologists and economists call "risk-averse" or "risk averters." The expression has a precise meaning with profound implications.