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Universal Portfolios

In this section a portfolio selection procedure will be considered with the goeal of performing as well as if we knew the empirical distribution of future market performance. The universal adaptive portfolio strategy is the performance weighted strategy specified by
$\displaystyle \mathbf{\hat{b}}_1 = \left( \frac{1}{m}, \frac{1}{m}, \ldots, \frac{1}{m} \right),$     (13)
$\displaystyle \mathbf{\hat{b}}_{k+1} = \frac{\int \mathbf{b} S_k (\mathbf{b} ) d\mathbf{b}}{\int S_k (\mathbf{b}) d\mathbf{b}},$     (14)

where

\begin{displaymath}S_k (\mathbf{b}) = \prod_{i=1}^k \mathbf{b}^t \mathbf{x}_i\end{displaymath}

and the integration is over the set of (m-1)- dimensional portfolios

\begin{displaymath}B = \{ \mathbf{b} \in R^m : b_i \geq 0, \sum_{i=1}^m b_i = 1\}.\end{displaymath}

The wealth $\hat{S}_n$ achieved from the using the universal portfolio is

\begin{displaymath}\hat{S}_n = \prod_{k=1}^n \mathbf{b}_k^t \mathbf{x}_k.\end{displaymath}

From this we see that the portfolio $\mathbf{\hat{b}}_1$ is uniform over the stocks, and the portfolio $\mathbf{\hat{b}}_k$ at time k is the performance weighted average of all portfolios $\mathbf{b} \in B$. As in the previous section we have

\begin{displaymath}S_n^* = \max_b s_n (\mathbf{b}) = \max_b \prod_{i=1}^n \mathbf{b}^t \mathbf{x}_i = e^{nW^*(F_n)},
\end{displaymath} (15)

where Fn denotes the empirical distribution of $\mathbf{x}_1, \ldots, \mathbf{x}_n$, i.e. it places mass $\frac{1}{n}$ at each xi. We can easily prove that Sn* exceeds the maximum of the component stocks, the arithmetic mean, the geometric mean and that $S_n^* (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ is invariant under permutations of the sequence $\mathbf{x}_1, \ldots, \mathbf{x}_n$. Since the wealth $\hat{S}_n$, resulting from the universal portfolio, is the average of Sn(b), or

\begin{displaymath}\hat{S}_n = \prod_{k=1}^n \mathbf{\hat{b}}_k^t \mathbf{x}_k = \frac{\int S_n(\mathbf{b}) d\mathbf{b}}{\int d \mathbf{b}},
\end{displaymath} (18)

where

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

we can show that for the universal portfolio

\begin{displaymath}\hat{S}_n \geq \left( \prod_{j_1}^m S_n (\mathbf{e}_j) \right)^{1/m},
\end{displaymath} (19)

or, the wealth from the universal portfolio also exceeds value line index. We can also prove that $\hat{S}_n$ is invariant under permutations of the sequence $\mathbf{x}_1, \ldots, \mathbf{x}_n$. This invariance entails that a stock market crash will have no worse consequences for wealth $\hat{S}_n$ than if the bad days of that time had been sprinkled out among the good.
The important question remains, how does $\hat{S}_n / S_n^*$ behave? For a portfolio of two stocks, we consider the arbitrary stock vector sequence

\begin{displaymath}\mathbf{x}_i = (x_{i1}, x_{i2}) \in \mathbf{R}_+^2, i=1,2, \ldots.
\end{displaymath} (20)

The portfolio choice can be transformed into a choice of one variable, namely

\begin{displaymath}\mathbf{b} = (b, 1-b), \text{ $0 \leq b \leq 1$ },
\end{displaymath} (21)

and similarly Sn(b) becomes

\begin{displaymath}S_n(b) = \prod_{i=1}^n(bx_{i1} + (1-b)x_{i2}), \text{ $0 \leq b \leq 1$ .}
\end{displaymath} (22)

Let's define

\begin{displaymath}J_n^* = - W^{''}(b_n^*) = \int \frac{(x_{i1} - x_{i2})^2}{(\mathbf{b}_n^{*^t} \mathbf{x})^2}dF_n(\mathbf{x}).
\end{displaymath} (23)

Jn is generally known as the curvature or volatility index of a portfolio. It can be shown that for any $\mathbf{x}_1, \mathbf{x}_2, \ldots \in \mathbf{R}_+^2$ and for any subsequence of times $n_1, n_2, \ldots$ such that the doubling rate Wn (b) satisfies the condition

\begin{displaymath}W_n^{''}(b_n^*) \rightarrow W^{''} (b^*),
\end{displaymath} (24)

and where W(b) achieves its maximum at $b^* \in (0,1)$ the following holds true

\begin{displaymath}\frac{\hat{S}_n}{S_n^*} \sim \sqrt{\frac{2\pi}{n J_n^*}}.
\end{displaymath} (25)

This means that the universal wealth is within a factor $C/\sqrt{n}$ of the (presumably) exponentially large Sn*. In fact, it can be further shown that every additional stock in the universal portfolio costs an additional factor of $1/\sqrt{n}$.

 
next up previous
Next: Real world example Up: A study in portfolio Previous: Stochastic Markets
Magnus Bjornsson
1998-05-12