Sunday, May 27, 2012

Universal portfolio, part 1

Thomas M. Cover was a well known Stanford professor working in the field of information theory.  He studied portfolio theory from an information theory standpoint, in a tradition started by John L. Kelly, Jr in A New Interpretation of Information Rate, Bell System Technical Journal 35: 917–926, 1956.  A large number of his portfolio theory articles are available on line.

Universal Portfolios. Mathematical Finance, 1(1): 1-29, January 1991 is the most interesting for me:

  • It is highly didactic and relatively easy to understand.
  • It includes a number of interesting graphs, contrary to many articles that restrict themselves to numeric tables.
  • It presents a portfolio approach that is guaranteed to work "reasonably well" under all conditions.
What not to like?  Reading this article led to some R code that eventually became the seed for logopt.

But what does "reasonably well"?  In Cover own words
We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market.
This seems sensational, and is also slightly misleading as I'll discuss later on.

Constant rebalanced portfolio (CRP)

In Cover's words again,
"Universal Portfolios," Cover [1991] introduces a portfolio that does as well to first order in the exponent as the best constant rebalanced portfolio
A constant rebalanced portfolio buys and sells shares so that the ratio of each stock in the portfolio is a constant.  Intrinsically this corresponds to sell winners to buy losers.  The evolution of the value of a constant rebalanced portfolio is easy to describe mathematically, and this is why Cover is able to derive a portfolio selection algorithm that is guaranteed to track (in a mathematical sense) the best possible constant rebalanced portfolio.

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