'Universal portfolios'
take investors back to the future
Photo:
L.A. Cicero
BY DAWN LEVY
 If you've played the stock market in recent
years, odds are you've felt the nail-biting exuberance of
a kid at an amusement park. The ride can be wild, and
many an investor has lost his lunch. Or shirt.
Wouldn't it be great to
have the security of hindsight? A time machine, perhaps?
You could travel to the future to find the next Cisco
Systems, jump back to the past to buy stock, and laugh
your way to the bank. Or you could avoid market crashes
like those of 1929 and 1987.
Tom Cover has the
next-best thing to a time machine: He has an algorithm --
a computational procedure -- that uses the past to
predict the future. It works as well or better than
hindsight, outperforming a pretty good investment
strategy: diversifying your stock portfolio and hoping
that performance of superstars will more than make up for
money wasted on losers.
Cover, professor of
statistics and the Kwoh-Ting Li Professor of Electrical
Engineering at Stanford, described his investment
strategy during an invited talk in Washington, D.C., at
the annual meeting of the American Association for the
Advancement of Science (AAAS) in February. The strategy
uses an algorithm that mirrors universal data-compression
algorithms to create the so-called "universal
portfolio." Each day the stock proportions in the
universal portfolio are readjusted to track a constantly
shifting "center of gravity" where performance
is optimal and investment desirable. The result? The
universal portfolio performs as well as the best
strategies that keep a constant proportion of wealth in
each stock would have performed in hindsight, "no
matter how the market wiggles and squirms," Cover
says.
To create a universal
portfolio, the investor buys very small amounts of every
stock in a market -- no small task in itself. The New
York Stock Exchange, for example, lists 3,025 companies.
In essence, the universal investor mimics the buy order
of a sea of investors using all possible "constant
rebalanced" strategies, in which the amount of money
invested in each stock is adjusted each day to achieve a
fixed proportion.
The bad news? Universal
portfolios need to be rebalanced daily to keep the
highest-return investments near the center of gravity of
the constant rebalanced portfolios, which makes investing
a high-maintenance activity. The good news? The algorithm
does not model the market as an independent, static
entity unresponsive to declared wars, oil gluts and the
introduction of rival technologies. In fact, it does not
attempt to model the market at all.
"Imagine we have, for
a simple example, two stocks," Cover explains.
"A good constant rebalanced portfolio might invest,
say, one-fourth in one stock and three-fourths in the
other. At the end of the day, the wealth you have in each
stock would not be exactly one-fourth, three-fourths
because the prices of the stocks change, so you would do
the necessary buying and selling to restore it to
one-fourth, three-fourths."
Cover's universal
portfolio algorithm invests uniformly in all
constant rebalanced portfolio strategies. The result is a
strategy that is nearly optimal. Cover has shown, for any
sequence of stock market outcomes, that this mixture of
investments has as high a compound growth rate in the
long run as the best constant rebalanced portfolio. Over
time, the best strategy (that is, the best constant
rebalanced portfolio) fights its way to the top of the
fiscal food chain.
Economic Darwinism?
"Yes, but nobody dies in this Darwinism," Cover
explains. "The unfit investments still survive, but
at exponentially reduced levels of wealth. The surviving
investments dominate your holdings."
Cover earned a bachelor's
degree in physics from the Massachusetts Institute of
Technology, and both master's and doctoral degrees in
electrical engineering from Stanford. As a graduate
student, he was intrigued by the work of statisticians
David Blackwell at the University of California,
Berkeley, and Herbert Robbins at Columbia, who developed
a robust theory for playing repeated games, such as
predicting the outcome of coin flips.
He was a contract
statistician for the California State Lottery from 1986
to 1994 while at Stanford, designing tests of the lottery
balls and wheels, analyzing the payoff structure of
games, and finding ways to beat the lottery so the state
could devise ways to protect itself from fraud.
His interest in the
mathematics of gaming lends itself well to another form
of gambling -- stock market investment. But whereas
gamblers and investors rely on intuition and advice,
Cover utilizes equations.
"A good theorem is
like a joke," Cover says. "You're led to
believe something and then a surprise causes you to
laugh. A good theorem makes something very clear that you
didn't think was, or it flies in the face of your
intuition. The joke with universal portfolios is that you
seem to get something for nothing."
If you think it odd that
an electrical engineer and statistician would ponder the
stock market, it all adds up. Cover is a pioneer in
information theory, a field that treats all information
as quantifiable but ignores the semantic content of
messages. Information theory has been applied in fields
as diverse as wireless communication, data compression
and deep space communications to transmit information
without errors. The field was born in 1948 when research
mathematician Claude Shannon provided a theory that laid
the foundation for phone and Internet communications.
With Joy A. Thomas, formerly of IBM, Yorktown Heights,
N.Y., Cover wrote what many consider the benchmark
textbook on modern information theory. He has written
more than 115 papers. In 1990, the Information Theory
Society of the Institute of Electrical and Electronics
Engineers (IEEE), the world's largest technical
professional organization, gave him the Claude E. Shannon
Award, the highest honor in information theory. In 1997,
the IEEE gave him the Richard W. Hamming medal (a gold
medal and $10,000) for "fundamental work in
information theory, statistics and pattern
recognition."
Joining the Stanford
faculty in 1964, Cover was named professor in 1972. He
directed Stanford's Information Systems Laboratory from
1988 to 1996 and currently leads a research group in
information theory. His work has influenced areas as
diverse as broadcasting of high-definition television,
bandwidth compression, mobile telephones and theory of
stock market investment. In 1972 he introduced the
concept of superposition in broadcast channels, which
made it possible to send information simultaneously from
one transmitter to multiple receivers. His paper on the
topic is credited as one of the pioneering works in
network information theory.
Cover is a member of the
National Academy of Engineering and a Fellow of the IEEE,
the Institute for Mathematical Statistics and the
American Association for the Advancement of Science. He
is a past president of the IEEE Information Theory
Society.
One aspect of information
theory is data compression. "The beauty of it is,
the mathematics of growth-rate-optimal investment turns
out to be parallel to the mathematics for optimal data
compression," Cover says. Thus universal investment
algorithms are a counterpart to the universal data
compression algorithms used to compress voice, fax and
computer files.
Theory meets the real
world
How well do universal
investment algorithms do on real data? Consider the cases
of Iroquois Brands Ltd. and Kin Ark Corp., two stocks
chosen for their volatility on the New York Stock
Exchange. Cover looked at 20 years of data -- that's
about 6,000 trading days -- ending in 1985. With the
buy-and-hold strategy, every dollar invested in Iroquois
is worth eight dollars after 20 years. With Kin Ark,
every dollar invested earned four.
The best constant
rebalanced portfolio would have achieved 74 dollars for
each dollar invested. But because the universal algorithm
always lags behind the center of gravity by a day, it
falls short of this theoretical maximum and achieves only
39 dollars. Still, not too shabby!
A key feature of the
algorithm is that the return on investment is
exponential, like compound interest. A good way to
visualize the tremendous growth potential of an exponent
(a number "raised" to some power, like 23
= 8) is to know the legend of the king who unknowingly
gave away his kingdom to a peasant who had done him a
favor. "I'll give you anything," the grateful
monarch is said to have promised. The peasant looked at
the king's chess board and asked for one grain of wheat
on the first square, two grains on the second square,
four on the third square and so on. The innumerate king
agreed and unwittingly gave away all his wealth.
With the universal
portfolio algorithm, profit grows exponentially, Cover
says, and the average of exponential growth rates has the
same growth rate as the maximum.
"This is an automatic
investment algorithm in the stock market," Cover
says. "The portfolio rides the stocks and lives off
the fluctuations. It essentially puts a little bit of
money on every possible rebalanced investment algorithm,
and the surviving algorithms -- the ones that made most
of the money -- make enough so that your money grows at
the same rate as if you had used the best algorithm to
start with."
So who wants to be a
millionaire? The math-apt can read the paper that first
detailed Cover's algorithm (T. Cover. Universal
Portfolios. Mathematical Finance, 1(1): 1-29, January
1991). The subject of his AAAS talk was more recent work
with one of his 50 former Ph.D. students, Erik
Ordentlich, and one of his current Ph.D. students, David
Julian. (His other current students are Assaf Zeevi,
Joshua Singer, Michael Baer, Arak Sutivong and Jon Yard.)
That work incorporates side information, such as the
state of the economy and the price of oil, into the
algorithm.
The algorithm is
"somewhat ponderous," Cover says. "The
performance of the algorithm, although good relative to
the best portfolio in hindsight, is still slow in
responding in an absolute sense. It sometimes requires
hundreds of days before the initial conditions wash out,
leaving the 'fittest' rebalanced portfolio dominating the
performance. It's guiding thinking, but no one's making
money off it yet."
And Cover's algorithm has
a catch. It ignores the brokerage fees affixed to each
stock trade. "The transaction costs will eat you
up," he says. This is true even considering lower
transaction fees available over the Internet that improve
performance. "But there's a nice theoretical patch
that will allow you to include transaction costs. You
trade only when you get far enough away from the optimal
investment proportions. This results in less frequent
trades, but a lower growth rate as well." SR
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