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
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." SR