Bettor Math - 經濟

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作者: fizeau (.) 看板: EngTalk
標題: Bettor Math
時間: Sun May 4 00:56:21 2008

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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.

Reviewer Information
Elwyn Berlekamp, a professor of mathematics at the University of California
at Berkeley, is best known for his works on games and codes. In 1960 and
1962, he was John Kelly's research assistant. In 1967, he coauthored Claude
Shannon's last paper on information theory. In 1990, he managed a 55 percent
gain of Jim Simons's Medallion Fund.


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