Against the Gods: The Remarkable Story of Risk (72 page)

*Chapter 7 describes Jacob Bernoulli's achievements in detail. The Law of Large Numbers
says in essence that the difference between the observed value of a sample and its true value
will diminish as the number of observations in the sample increases.

*Peter Kinder has pointed out to me a great historical irony in all this. The Vikings and
other Norsemen who laid waste to Roman civilization and destroyed the repositories of
learning in the ninth century reappear in history as the Normans who brought back to the
West the achievements of Arabic learning in the twelfth century.

*One of those odd quirks that numbers can produce reveals that you can derive 0.618 if you
take the square root of 5, which is 2.24, subtract 1, and then divide by 2; this result is the
algebraic proof of Fibonacci's sequence of numbers.

In technical terms, the formula for the Fibonacci ratio is as follows: the ratio of the smaller
part to the larger part equals the ratio of the larger part to the whole.

*The Arabic term survives even in Russian, where it appears as tsifra, which is the word for
number.

*Readers who are not interested in the technicalities of this discussion can skip to page 53
without any loss of continuity.

*The full text of this correspondence, translated into English, appears in David, 1962,
Appendix 4.

*Mathematicians will note that what Pascal has really provided here is the binomial expansion, or the coefficients of each successive multiplication of (a + b) by itself. For example,
the first row is (a + b)° = 1, while the fourth row is (a + b)3 = 1a3 + 3a2b + 3abz + 1 b'.

*At this point, Pascal anticipates Daniel Bernoulli's epochal breakthrough in decision analysis in 1738, which we explore in detail in Chapter 6.

The Latin title for this book was Ars Cogitandi. See Hacking, 1975, pp. 12 and 24.

*The information on the quantity of bread a penny could buy provided a standard for
estimating the cost of living. In our own times, a package of goods and services is used as
the standard.

*The word "venery" descends from the Middle-French word vener, to hunt (from which
also comes the word "venison") and from Venus (from which comes the word "venereal)."
A venerable word indeed!

*Lloyd's, in short, is the ancestor of the huge Bloomberg business news network of our own
time.

*The trust business in Boston was founded by Nathaniel Bowditch in the 1810s to serve the
same market.

tThis principle applied to life insurance as well. The debts of a soldier who died in battle
were forgiven and did not have to be repaid.

In the United States it survived into the twentieth century. Here it was known as "industrial insurance" and usually covered only funeral expenses. My father-in-law had a little
book in which he kept a record of the weekly premiums he paid into such a policy.

`As usual, the essay was published in Latin. The Latin title of the publication in which it
appeared was Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tomus V.

*Newman is not easy to characterize, although his The World of Mathematics was a major
source for this book. He was a student of philosophy and mathematics who became a
highly successful lawyer and public servant. A one-time senior member of the editorial
board of Scientific American, he was an avid collector of scientific documents of great historical importance. He died in 1966.

*Daniel's uncle Jacob, who will play a major role in the next chapter, once wrote that "the
value of our expectation always signifies something in the middle between the best we can
hope for and the worst we can fear." (Hacking, 1975, p. 144.)

*With the assistance of Richard Sylla and Leora Mapper, the best information I have been
able to obtain about the value of ducats in the early 18th century is that one ducat could
have purchased the equivalent of about $40 in today's money. Baumol and Baumol,
Appendix, provides an approximate confirmation of this estimate. See also McKuster, 1978,
and Warren and Pearson, 1993.

*Bernoulli's solution to the paradox has been criticized because he fails to consider a game
in which the prize would rise at a faster rate than the rate Nicolaus had specified.
Nevertheless, unless there is a point where the player has zero interest in any additional
wealth, the paradox will ultimately come into play no matter what the rate is.

*A theoretical exploration into this question appears in Durand, 1959, which anticipated
the events described in the paragraphs immediately following.

*This is an oversimplification. The utility of any absolute loss depends on the wealth of the
loser. Here the implicit assumption is that the two players have equal wealth.

*He did have sufficient poetry in his soul to request that the beautiful Fibonacci spiral be
engraved on his tombstone, claiming that the way it could grow without changing its form
was "a symbol of fortitude and constancy in adversity: or even of the resurrection of our
flesh." He went on to ask that it be inscribed with the epitaph "Eadem mutata resurgo"
[However changed it is always the same]. See David, 1962, p. 139.

*At a later point in the correspondence with Jacob, Leibniz observed, "It is certain that
someone who tried to use modern observations from London and Paris to judge mortality
rates of the Fathers before the flood would enormously deviate from the truth." (Hacking,
1975, p. 164.)

*The franc/dollar exchange rate has been remarkably steady over the years at around 5:1.
Hence, 2,000 francs was the equivalent of $400 dollars of 1807 purchasing power. A dollar in 1807 bought about twelve times as much as today's dollar.

*The standard deviation was the device that de Moivre discovered for measuring the dispersion of observations around their mean. Approximately two-thirds of the observations
(68.26%) will fall within a range of plus or minus one standard deviation around the mean;
95.46% will fall within two standard deviations around the mean.

*Richard Price's experience reminds us that the data themselves must be of good quality.
Othe.-wise, GIGO: garbage in, garbage out.

Readers skilled in statistics will protest that I should have used the lognormal analysis in the
discussion that follows. For readers not so skilled, the presentation in this form is much
more intelligible and the loss of accuracy struck me as too modest to justify further complexity.

*Galton would surely have classified Cardano as eminent, but what would he have thought
of Cardano's disastrous progeny? Gauss, also an eminent man, scored better. He produced
five surviving children, of whom one was a distinguished engineer and two of whom emigrated to the United States to run successful businesses (but also to escape from their father's
domineering influence); one of them was also a brilliant linguist, a gambler, and a skillful
mathematician as well.

*1 am not related to Sanford Bernstein, by the way.

*Opposite tendencies are apparent in the historical record of interest rates, which reflect
"aversion" to the mean. A trend once in place has a higher probability of continuing than
reversing. Over two-year periods, the variance of the yield on 90-day Treasury bills is 2.2
times annual data; over eight-year periods, the variance is nearly 32 times the annual data;
for longer-term interest rates, the pattern is similar but muted.

*They called depressions "panics" in those days; "depression" was a euphemism coined for
the occasion. Later, "recession" became the accepted euphemism. One can only speculate
on how deep a recession would have to be before the experts would decide to call it a
"depression."

*Florence Nightingale was described by Edward Cook, one of her biographers, as a "passionate statistician." A compulsive collector of data in the tradition of Galton, she was also
an enthusiastic admirer of the work of Quetelet, which inspired her pioneering work in
medical and other social statistics. See Kendall and Plackett, 1977, pp. 310-327.

*It is conceivable, however, that the opposite might occur. Risk often serves as a stimulant.
Without risk, a society might turn passive in the face of the future.

*Knight rarely uses such arcane words. "Apodeictic" means incontestable, necessarily true
because logically certain.

*The feeling appears to have been mutual. Morgenstern took a dim view of Samuelson's
knowledge of mathematics. Complaining that "[von Neumann] says [Samuelson] has
murky ideas about stability," he predicted that "even in thirty years he won't absorb game
theory!" See Leonard, 1994, p. 494n.

tRopke, also a Christian, was far more emphatic than Morgenstern had been about his reasons for leaving Hitler's Germany.

*The word has a Latin root, from portare, to carry, and foglio, leaf or sheet. Portfolio has thus
come to mean a collection of paper assets.

*The standard deviation of the Mexican market alone leapt from 8% to 10% a month (four
times the monthly volatility of the S&P 500) to better than 15% a month during the first
half of 1995.

*For an extended and informative discussion of these issues, see TheJournal of Investing, Fall
1994.

*Jack Benny had a routine on a Sunday radio show in which he remained silent when confronted by a mugger demanding "Your money or your life." After a long pause, the mugger cried, "Come on!" "I'm thinking it over," Benny predictably responded.

*Kahneman has described his introduction to experimentation when one of his professors
told the story of a child being offered the choice between a small lollipop today or a larger
lollipop tomorrow. The child's response to this simple question correlated with critical
aspects of the child's life, such as family income, one or two parents present, and degree of
trust.

*An excellent review of this matter appears in "The Triumph of Indexing," a booklet published by the Vanguard Group of mutual funds in May 1995. This controversial subject
receives more detailed treatment later in this chapter.

*In a speech to the National Association of Realtors in May 1995, none other than the
Chairman of the Federal Reserve Board, Alan Greenspan, confirmed the piggy bank
metaphor: "It is hard to overestimate the importance of house price trends for consumer
psyches and behavior.... Consumers view their home equity as a cushion or security blanket against the possibility of future hard times." As a consequence of the growth in borrowing in the form of home equity loans, home equity has shrunk from 73% of home value
in 1983 to around 55% at this writing, provoking what the July 10, 1995, issue of Business
Week describes as "a major deterrent to buoyant spending."

tWe exclude financial corporations from these calculations to avoid double-counting.
Banks and other financial organizations re-lend to the nonfinancial sector most of the
money they borrow.

BAs usual, Shakespeare got there first. In Act 1, Scene 1, lines 168-171 of Timon of Athens,
the jeweler says to Timon, "My Lord, tis rated/As those which sell would give; but you
well know/Things of like value differing in their owners/Are prized by their masters."

*In Chapter 7 of Thaler, 1987, in fact, Thaler declared that von Neumann-Morgenstern
utility had failed in psychological testing. See p. 139.

This bald assertion should be interpreted broadly. Cross-cultural problems and concerns
for the health of the home country add to the value of domestic securities and detract from
the value of foreign securities.

*The bond even offered protection against the possibility that one pound sterling might
subsequently buy more or less than 25 francs. The French went off gold in 1870, at which
time one pound sterling could buy substantially more than 25 francs.

*This is an oversimplification to make the basic point. Most individual home mortgages are
packaged with other mortgages and sold off in the open market to a wide variety of
investors. In effect, the bankers have traded off the risks of prepayment to a market more
willing to bear that risk; these mortgage-backed securities are complex, volatile, and much
too risky for amateur investors to play around with.

*Black suspected that something more unpleasant was involved: that his lack of proper
warpaint in the form of a degree in economics excluded him from the tribal membership
that the editors considered essential for an appearance in the JPE.

*The literature on derivatives is massive, but I especially recommend the Fall 1994 issue of
the Journal of Applied Corporate Finance, which is entirely devoted to the subject, and
Smithson and Smith's book on managing risk (Smithson and Smith, 1995).

In July 1995, the Federal Reserve Board, the Treasury Department, and the FDIC
requested comments on a proposal to revise their requirements for commercial bank risk
controls on transactions involving foreign exchange, commodities, and trading in debt and
equity instruments. The document runs to 130 single-spaced pages. The so-called Basle
Committee, consisting of representatives of central bankers from major economies, has
issued the authoritative framework for the supervision of derivatives activities of banks and
securities firms; it was published as a Federal Reserve press release on May 16, 1995.

'al-Khowirizmi, the mathematician whose name furnished the root of the word "algorithm," would surely be astonished to see the offspring of what he launched nearly 1200
years ago.

`From 1871 to 1958, stock yields exceeded bond yields by an average of about 1.3 percentage points, with only three transitory reversals, the last in 1929. In an article in Fortune
magazine for March 1959 Gilbert Burke declared, "It has been practically an article of faith
in the U.S. that good stocks must yield more income than good bonds, and that when they
do not, their prices will promptly fall." (See Bank Credit Analyst, 1995.) There is reason to
believe that stocks yielded more than bonds even before 1871, which is the starting point
for reliable stock market data. Since 1958, bond yields have exceeded stock yields by an
average of 3.5 percentage points.