Against the Gods: The Remarkable Story of Risk (39 page)

Another French mathematician, born about a century after Laplace,
gave further emphasis to the concept of cause and effect and to the
importance of information in decision-making. Jules-Henri Poincare,
(1854-1912) was, according to James Newman,

... a French savant who looked alarmingly like a French savant. He
was short and plump, carried an enormous head set off by a thick
spade beard and splendid mustache, was myopic, stooped, distraught
in speech, absent-minded and wore pince-nez glasses attached to a
black silk ribbon.5

Poincare was another mathematician in the long line of child prodigies
that we have met along the way. He grew up to be the leading French
mathematician of his time.

Nevertheless, Poincare made the great mistake of underestimating
the accomplishments of a student named Louis Bachelier, who earned
a degree in 1900 at the Sorbonne with a dissertation titled "The Theory
of Speculation."6 Poincare, in his review of the thesis, observed that
"M. Bachelier has evidenced an original and precise mind [but] the
subject is somewhat remote from those our other candidates are in the
habit of treating." The thesis was awarded "mention honorable," rather
than the highest award of "mention tres honorable," which was essential
for anyone hoping to find a decent job in the academic community.
Bachelier never found such a job.

Bachelier's thesis came to light only by accident more than fifty
years after he wrote it. Young as he was at the time, the mathematics
he developed to explain the pricing of options on French government
bonds anticipated by five years Einstein's discovery of the motion of
electrons-which, in turn, provided the basis for the theory of the random walk in finance. Moreover, his description of the process of speculation anticipated many of the theories observed in financial markets
today. "Mention honorable"!

The central idea of Bachelier's thesis was this: "The mathematical
expectation of the speculator is zero." The ideas that flowed from that
startling statement are now evident in everything from trading strategies
and the use of derivative instruments to the most sophisticated techniques of portfolio management. Bachelier knew that he was onto
something big, despite the indifference he was accorded. "It is evident," he wrote, "that the present theory solves the majority of problems in the study of speculation by the calculus of probability."

But we must return to Poincare, Bachelier's nemesis. Like Laplace,
Poincare believed that everything has a cause, though mere mortals are
incapable of divining all the causes of all the events that occur. "A
mind infinitely powerful, infinitely well-informed about the laws of
nature, could have foreseen [all events] from the beginning of the centuries. If such a mind existed, we could not play with it at any game of
chance, for we would lose."7

To dramatize the power of cause-and-effect, Poincare suggests what
the world would be like without it. He cites a fantasy imagined by
Camile Flammarion, a contemporary French astronomer, in which an
observer travels into space at a velocity greater than the speed of light:

[F] or him time would have changed sign [from positive to negative].
History would be turned about, and Waterloo would precede
Austerlitz.... [A]ll would seem to him to come out of a sort of chaos
in unstable equilibrium. All nature would appear to him delivered
over to chance.8

But in a cause-and-effect world, if we know the causes we can predict the effects. So "what is chance for the ignorant is not chance for
the scientist. Chance is only the measure of our ignorance."9

But then Poincare asks whether that definition of chance is totally
satisfactory. After all, we can invoke the laws of probability to make predictions. We never know which team is going to win the World
Series, but Pascal's Triangle demonstrates that a team that loses the first
game has a probability of 22/64 of winning four games before their
opponents have won three more. There is one chance in six that the
roll of a single die will come up 3. The weatherman predicts today that
the probability of rain tomorrow is 30%. Bachelier demonstrates that
the odds that the price of a stock will move up on the next trade are
precisely 50%. Poincare points out that the director of a life insurance
company is ignorant of the time when each of his policyholders will
die, but "he relies upon the calculus of probabilities and on the law of
great numbers, and he is not deceived, since he distributes dividends to
his stockholders."10

Poincare also points out that some events that appear to be fortuitous are not; instead, their causes stem from minute disturbances. A
cone perfectly balanced on its apex will topple over if there is the least
defect in symmetry; and even if there is no defect, the cone will topple
in response to "a very slight tremor, a breath of air." That is why,
Poincare explained, meteorologists have such limited success in predicting the weather:

Many persons find it quite natural to pray for rain or shine when they
would think it ridiculous to pray for an eclipse.... [O)ne-tenth of a
degree at any point, and the cyclone bursts here and not there, and
spreads its ravages over countries it would have spared. This we could
have foreseen if we had known that tenth of a degree, but ... all
seems due to the agency of chance.11

Even spins of a roulette wheel and throws of dice will vary in response to slight differences in the energy that puts them in motion.
Unable to observe such tiny differences, we assume that the outcomes
they produce are random, unpredictable. As Poincare observes about
roulette, "This is why my heart throbs and I hope everything from
luck."12

Chaos theory, a more recent development, is based on a similar
premise. According to this theory, much of what looks like chaos is in
truth the product of an underlying order, in which insignificant perturbations are often the cause of predestined crashes and long-lived bull
markets. The New York Times of July 10, 1994, reported a fanciful application of chaos theory by a Berkeley computer scientist named James Crutchfield, who "estimated that the gravitational pull of an electron,
randomly shifting position at the edge of the Milky Way, can change
the outcome of a billiard game on Earth."

Laplace and Poincare recognized that we sometimes have too little
information to apply the laws of probability. Once, at a professional
investment conference, a friend passed me a note that read as follows:

We can assemble big pieces of information and little pieces, but we
can never get all the pieces together. We never know for sure how good
our sample is. That uncertainty is what makes arriving at judgments so
difficult and acting on them so risky. We cannot even be 100% certain
that the sun will rise tomorrow morning: the ancients who predicted
that event were themselves working with a limited sample of the history
of the universe.

When information is lacking, we have to fall back on inductive reasoning and try to guess the odds. John Maynard Keynes, in a treatise on
probability, concluded that in the end statistical concepts are often useless: "There is a relation between the evidence and the event considered, but it is not necessarily measurable."13

Inductive reasoning leads us to some curious conclusions as we try
to cope with the uncertainties we face and the risks we take. Some of
the most impressive research on this phenomenon has been done by
Nobel Laureate Kenneth Arrow. Arrow was born at the end of the First
World War and grew up in New York City at a time when the city was
the scene of spirited intellectual activity and controversy. He attended
public school and City College and went on to teach at Harvard and
Stanford. He now occupies two emeritus professorships at Stanford,
one in operations research and one in economics.

Early on, Arrow became convinced that most people overestimate
the amount of information that is available to them. The failure of economists to comprehend the causes of the Great Depression at the
time demonstrated to him that their knowledge of the economy was
"very limited." His experience as an Air Force weather forecaster during the Second World War "added the news that the natural world was
also unpredictable." 14 Here is a more extended version of the passage
from which I quoted in the Introduction:

To me our knowledge of the way things work, in society or in
nature, comes trailing clouds of vagueness. Vast ills have followed a
belief in certainty, whether historical inevitability, grand diplomatic
designs, or extreme views on economic policy. When developing
policy with wide effects for an individual or society, caution is needed
because we cannot predict the consequences."15

One incident that occurred while Arrow was forecasting the weather
illustrates both uncertainty and the human unwillingness to accept it.
Some officers had been assigned the task of forecasting the weather a
month ahead, but Arrow and his statisticians found that their long-range
forecasts were no better than numbers pulled out of a hat. The forecasters
agreed and asked their superiors to be relieved of this duty. The reply
was: "The Commanding General is well aware that the forecasts are no
good. However, he needs them for planning purposes."16

In an essay on risk, Arrow asks why most of us gamble now and
then and why we regularly pay premiums to an insurance company.
The mathematical probabilities indicate that we will lose money in
both instances. In the case of gambling, it is statistically impossible to
expect-though possible to achieve-more than a break-even, because
the house edge tilts the odds against us. In the case of insurance, the
premiums we pay exceed the statistical odds that our house will burn
down or that our jewelry will be stolen.

Why do we enter into these losing propositions? We gamble because we are willing to accept the large probability of a small loss in the
hope that the small probability of scoring a large gain will work in our
favor; for most people, in any case, gambling is more entertainment
than risk. We buy insurance because we cannot afford to take the risk
of losing our home to fire-or our life before our time. That is, we prefer a gamble that has 100% odds on a small loss (the premium we must
pay) but a small chance of a large gain (if catastrophe strikes) to a gam ble with a certain small gain (saving the cost of the insurance premium)
but with uncertain but potentially ruinous consequences for us or our
family.

Arrow won his Nobel Prize in part as a result of his speculations
about an imaginary insurance company or other risk-sharing institution
that would insure against any loss of any kind and of any magnitude, in
what he describes as a "complete market." The world, he concluded,
would be a better place if we could insure against every future possibility. Then people would be more willing to engage in risk-taking,
without which economic progress is impossible.

Often we are unable to conduct enough trials or take enough samples to employ the laws of probability in making decisions. We decide
on the basis of ten tosses of the coin instead of a hundred. Consequently,
in the absence of insurance, just about any outcome seems to be a matter of luck. Insurance, by combining the risks of many people, enables
each individual to enjoy the advantages provided by the Law of Large
Numbers.