4/12/00

Dawn
Levy, News Service (650) 725-1944; e-mail: dawnlevy@stanford.edu

**Universal portfolios take investors back
to the future**

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 Wash., 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 2^{3}
= 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."

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By Dawn Levy