BOOK REVIEW
Bettor Math
Elwyn Berlekamp
Fortune's Formula: The Untold Story of the Scientific Betting
System that Beat the Casinos and Wall Street. William
Poundstone. x + 367 pp. Hill and Wang, 2005. $27.
Every investor must decide how to partition her portfolio among many
possible investments. Plausible strategies range from
"diversify" to "focus."
In a paper published in 1956, John L. Kelly of Bell Labs formulated
the asset-allocation problem in terms of an idealized model for
which he derived some quantitative results. He used colorful
racetrack terminology reminiscent of the classic Damon Runyon movie
Guys and Dolls: Suppose that one goes to the racetrack
with an available bankroll, B. Suppose further that one
knows for each horse the correct probability that it will win the
next race. Suppose further that the betting odds are at least
slightly inconsistent with this information. And finally, suppose
that each race is merely one of a very long sequence of betting
opportunities. Kelly found criteria for deciding how much one should
then bet on each horse in each race.
Kelly observed that, under similar idealized assumptions, the same
formulation could also be applied to investments. In the idealized
model, the portfolio manager has an accurate probability
distribution on the future performance of each asset in the universe
of potential investments. Kelly's methodology then provides a
quantitative specification of how big a position to take in each of
the candidate assets. Not surprisingly, the fraction of one's
portfolio to be invested in any asset that has a negative expected
rate of return will be zero. Most assets with positive expected
rates of return will merit the investment of some positive fraction
of the portfolio. Among assets with similar expected rates of
return, those whose returns are relatively stable will be weighted
more heavily than those whose future returns have significant risks
of substantial losses, even when these risky investments also have
some chance of large gains. All of these qualitative features of
Kelly's performance criteria concur with conventional wisdom. What
distinguishes Kelly's work from that of his predecessors is his
quantitative specificity and the fact that he succeeded in proving
that, under his assumptions, in the very long run the bankroll of an
investor who followed his criteria would eventually surpass the
bankroll of anyone following any other strategy.
Kelly also derived a formula for the rate at which this bankroll
would grow. This formula is related to a fundamental
information-theoretic notion that Claude Shannon (now widely
considered to be the father of the information age) had introduced
in 1948. Shannon had shown that noise on a communication channel
need not impose any bound on the reliability with which information
can be communicated across it, because the probability of
transmitting a very long file inaccurately can be made arbitrarily
small by using sufficiently sophisticated coding techniques, subject
to a constraint that the ratio of the length of the source file to
the length of the encoded file must be less than a number called the
channel capacity. Kelly showed that the asymptotically optimum asset
allocation could be determined by solving a system of equations that
maximized the log of one's capital. In his horse-track
jargon, Kelly also showed that the resulting optimal compound growth
rate could be viewed as the capacity of a hypothetical noisy channel
over which the bettor was getting the information that distinguished
his odds from those of the track. Kelly's betting system, expressed
mathematically, is known as the Kelly criterion.
The title of Kelly's paper, "A New Interpretation of the
Information Rate," highlighted his discovery of a situation in
which Shannon's celebrated capacity theorem applied even though no
coding was contemplated. The paper, which appeared in the Bell
System Technical Journal, initially attracted a modest
audience among information theorists but went unnoticed by
economists and professors of finance courses in business schools.
Perhaps it would have received more attention if it had had another
title. "Information Theory and Gambling" was the title
that Kelly himself used for an earlier draft of his paper, but that
title was rejected by AT&T executives.
The phrase "Fortune's Formula," which could have served as
the title of Kelly's paper, was coined by the mathematician Ed Thorp
as the title for a paper he wrote in 1961 about a strategy for
winning at blackjack. It is now also the title of William
Poundstone's new book, which tells stories of gamblers and investors
over the past 150 years and how some of them have been influenced by
the Kelly criterion. The style is somewhat like that of the business
pages of a good newspaper, with no formulas or equations but
occasional graphs. There are many sources, most of which are
reliable. Even though there are many footnotes, the tone sometimes
changes from that of a science journalist to that of a gossip
columnist. There are biographical sketches not only of Kelly (who
died in 1965 of a heart attack at age 41) and such relevant
intellectual titans as Claude Shannon and Paul Samuelson (the father
of modern economics), but also of many other characters. The career
of the legendary Thorp, who became a successful, innovative
financial entrepreneur, is treated at considerable length.
Ed Thorp analyzed the game of blackjack far more deeply than anyone
had ever done before, and he devised card-counting schemes to gain
an edge, especially toward the end of a deck that is not reshuffled
after every deal. He wrote a bestseller, Beat the Dealer,
on how to win at blackjack. Earlier in his career, when he was a
mathematics instructor at MIT, he met Claude Shannon, and he brought
Claude and Betty Shannon with him as partners on one of his early
weekend forays to Las Vegas. Later, he discovered and exploited a
number of pricing anomalies in the securities markets and made a
significant fortune. Thorp's first hedge fund, Princeton-Newport,
achieved an annualized net return of 15.1 percent over 19 years, and
in May 1998, Thorp reported that his own investments had an
annualized 20 percent return over 28.5 years.
Poundstone pursues a sequence of increasingly tenuous connections
among moneymaking schemes and scams, some blatantly illegal and some
with reputed mob connections, ranging all the way back in time to
wire services that predated Alexander Graham Bell, and into the
current political world of Rudy Giuliani. The reader can only wonder
how much is fact, how much is literary license and how much is
sensationalism. Marketing copy included on the book's dust jacket,
characterizing Kelly as "gun-toting" and Shannon as
"neurotic," falls squarely into the category of sensationalism.
In later sections of the book, the patient reader will find some
interesting graphs and an overview of a now long-standing academic
and philosophical debate about the relevance and appropriateness of
the Kelly criterion. Most people with academic training in physics,
mathematics, operations research, computer science or engineering
view the Kelly criterion as a useful quantitative guideline for
investing, to be used along with others. They also view most large
institutional money managers and economists as too risk-averse; the
latter folks view the former as too risk-prone. Some extremely
risk-averse business-school professors espouse a doctrine called the
efficient-market hypothesis. Whenever some money manager achieves
significantly above-average returns, adherents of that hypothesis
strive to explain away the accomplishment: Perhaps the manager is a
lucky survivor of an unrepeatable strategy that took very big risks
on a few very large bets; perhaps he or she depended heavily on
inside knowledge or engaged in illegal activity.
No one who has made a legitimate fortune in the markets believes the
efficient-market hypothesis. And conversely, no one who believes the
efficient-market hypothesis has ever made a large fortune investing
in the financial markets, unless she began with a moderately large
fortune. Of the stories presented in Fortune's Formula, the
case of Ed Thorp presents the greatest challenge to the
efficient-market hypothesis. Poundstone devotes only a single
paragraph to the even stronger cases of Ken Griffin, D. E. Shaw and
Jim Simons, presumably because financial wizards as successful as
these have always been unwilling to discuss their formulas in public.
General readers seeking a broad overview of certain aspects of the
field of financial mathematics and its practitioners will find the
latter portions of Poundstone's book the most informative. Readers
who enjoy a gossipy approach to business history will find the
earlier portions more to their liking. Any experienced,
quantitatively oriented investor will, without reading Poundstone's
book, already know that she needs to estimate the likely
distributions of returns of the various investments she is
considering. This is quite difficult, because for some promising
investments, historical data are very limited, and for others, there
are good reasons to question whether the historical patterns are
likely to persist into the future. So in practice, the allocation
problem that Kelly's formula addresses is only one of the two main
parts of the investor's puzzle. Poundstone recognizes this
implicitly, but some readers would benefit from a more explicit
statement of the dichotomy.
In my experience, abstract financial mathematics is the only truly
significant commonality between the world of finance and the world
of racetracks and casinos. Poundstone has been lured by Kelly's
colorful terminology into seriously overemphasizing the relevance
and importance of whatever other relationships might exist.
Portrayal of the seamy side of business is a genre that runs at
least as far back as the novels of Charles Dickens. Readers who are
looking for something in that vein as well as a light introduction
to financial mathematics will find things to relish in Poundstone's book.