Let's denote a stock market for one investment period as
,
where xi is the price relative for the ith stock, i.e., the ratio of closing to opening price for stock i. A portfolio
,
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
,
where
b and
x are considered to be column vectors.
If we consider an arbitrary sequence of stock vectors
,
we achieve wealth
(2)
with a constant rebalanced portfolio strategy
b.
The maximum wealth achievable on the given stocks is
(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
(4)
where
F(x) is the joint distribution of the vector of price relatives.
Definition 3
The optimal doubling rateW*(F) is defined as
(5)
where the maximum is over all possible portfolios
.
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
be i.i.d. according to
F(x). Then
(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.