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Portfolio Management and Information Theory

Let's denote a stock market for one investment period as $\mathbf{x} = (x_1, x_2, \ldots, x_m)^t \geq 0$, where xi is the price relative for the ith stock, i.e., the ratio of closing to opening price for stock i. A portfolio $\mathbf{b} = (b_1, b_2, \ldots, b_m)^t, b_i \geq 0, \sum b_i = 1$, is the proportion of the current wealth invested in each of the m stocks. Therefore, the wealth increase over one investment period using portfolio b is $S = \mathbf{b}^t \mathbf{x} = \sum b_i x_i$, where b and x are considered to be column vectors. If we consider an arbitrary sequence of stock vectors $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n$, we achieve wealth

\begin{displaymath}S_n(\mathbf{b}) = \prod_{i=1}^n \mathbf{b}^t \mathbf{x}_i,
\end{displaymath} (2)

with a constant rebalanced portfolio strategy b. The maximum wealth achievable on the given stocks is

\begin{displaymath}S_n^* = \max_{\mathbf{b}} S_n (\mathbf{b}).
\end{displaymath} (3)

Our goal is, of course, to achieve wealth as close to Sn* as possible. Since we generally reinvest every day in the stock market, the accumulated wealth at the end of a n days is the product of factors, one for each day of the market. As will be shown, defining a doubling rate for a portfolio is a good idea.

Definition 2   The doubling rate of a stock market portfolio b is defined as

\begin{displaymath}W(\mathbf{b}, F) = \int \log \mathbf{b}^t \mathbf{x}\ dF(\mathbf{x}) = E(\log \mathbf{b}^t \mathbf{x}),
\end{displaymath} (4)

where F(x) is the joint distribution of the vector of price relatives.

Definition 3   The optimal doubling rate W*(F) is defined as

\begin{displaymath}W^*(F) = \max_{\mathbf{b}} W(\mathbf{b}, F),
\end{displaymath} (5)

where the maximum is over all possible portfolios $b_i \geq 0$. Similarly, a portfolio b* that achieves the maximum of W(b, F) is called a log-optimal portfolio.

We can justify definition [*] by the following theorem

Theorem 2   Let $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n$ be i.i.d. according to F(x). Then

\begin{displaymath}\frac{1}{n} \log S_n^* \rightarrow W^* \text{ with probability 1.}
\end{displaymath} (6)

Therefore the investor's wealth grows as 2nW* using the log-optimal portfolio. An import property of W is that W(b, F) is concave in b and linear in F, and W* (F) is convex in F (for a proof, see [#!Cover:Elements!#].) That knowledge tells us that the set of log-optimal portfolios forms a convex set. The importance of these properties will come clear in the next section.

 
next up previous
Next: The Log-Optimal Portfolio Up: A study in portfolio Previous: The Market Portfolio
Magnus Bjornsson
1998-05-12