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The Log-Optimal Portfolio

The results of the Karush-Kuhn-Tucker conditions (or KKT conditions) are in the following theorem

Theorem 3   Assume that $f(\mathbf{x}), g_1(\mathbf{x}), g_2(\mathbf{x}), \ldots, g_m(\mathbf{x})$ are differentiable functions (satisfying certain regularity conditions.) Then

\begin{displaymath}\mathbf{x}^* = (x_1^*, x_2^*, \ldots, x_n^*)\end{displaymath}

can be an optimal solution for the non-linear programming problem only if there exist m numbers $u_1, u_2, \dots, u_m$ such that all the following KKT conditions are satisfied:

\begin{eqnarray*}& & \begin{array}{l}
...{ for $i=1, 2, \ldots, m$ .}

Which gives as a corollary, that if f(x) is a concave function and $g_1(\mathbf{x}), g_2(\mathbf{x}), \ldots, g_m(\mathbf{x})$ are convex functions, then $\mathbf{x}^* = (x_1^*, x_2^*, \ldots, x_n^*)$ is an optimal solution iff all the conditions of the theorem are satisfied. From this and the fact that we are trying to find an optimal solution for b, maximizing the concave function W(b, F) over a convex set $\mathbf{b} \in B$, we can derive the following theorem

Theorem 4   The log-optimal portfolio b* for a stock market X (i.e., the portfolio that maximizes the doubling rate), satisfies the following necessary and sufficient conditions:

E\left( \frac{X_i}{\mathbf{b}^{*^t} \mathbf{X}} \right) \b...
... > 0$ ,} \\
\leq 1 & \text{if $b^*_i = 0$ .}
\end{displaymath} (7)

This tells us that the expected value of the ratio between a price relative i and the corresponding wealth relative is equal to 1 if the i component of the portfolio is non-zero, and $\leq 1$ if the component is zero. Two interesting things can be derived from this theorem. The first is that

\begin{displaymath}E \log \frac{S}{S^*} \leq 0 \text{, for all $S$ } \Leftrightarrow E \frac{S}{S^*} \leq 1 \text{, for all $S$ }
\end{displaymath} (8)

where S* = b*t x is the random wealth resulting from the log-optimal portfolio b* and S is the wealth resulting from any other portfolio b. We have now shown that the log-optimal portfolio, in addition to maximizing the asymptotic growth rate, also maximizes the wealth relative for one day. Another consequence is that the expected proportion of wealth in stock i at the end of the day is the same as the proportion invested in stock i at the beginning of the day. Stated more precisely as

\begin{displaymath}E \left( \frac{b_i^* X_i}{\mathbf{b^{*^t}X}} \right) = b^*_i E \left( \frac{X_i}{\mathbf{b^{*^t}}X} \right) = b^*_i 1 = b_i^*.\end{displaymath}

But what if an investor were to use causal investment strategy? We can prove that with probability 1, the conditionally log-optimal investor will not do any worse than any other investor who uses a causal investment strategy. Let

\begin{displaymath}S_n = \prod_i^n \mathbf{b}_i^t \mathbf{X}_i
\end{displaymath} (9)

be the wealth after n days for an investor who uses portfolio bi on day i. Let

\begin{displaymath}W^* = \max_\mathbf{b} W(\mathbf{b},F) = \max_\mathbf{b} E \log \mathbf{b}^t X
\end{displaymath} (10)

be the maximal doubling rate and let b* be a portfolio that achieves that rate. From this we get

\begin{displaymath}E \log S_n^* = nW^* \geq E \log S_n,
\end{displaymath} (11)

that is, b* (satisfying equation [*]) maximizes the expected log wealth and that the wealth Sn* is equal to 2nW* to first order in the exponent, with high probability. In fact, we can prove a much stronger result, which shows that the log-optimal portfolio will do as well or better than any other portfolio to first order in the exponent.
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Next: Side Information Up: Portfolio Management and Information Previous: Portfolio Management and Information
Magnus Bjornsson