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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
|
|
|
(13) |
|
|
|
(14) |
where
and the integration is over the set of (m-1)- dimensional portfolios
The wealth
achieved from the using the universal portfolio is
From this we see that the portfolio
is uniform over the stocks, and the portfolio
at time k is the performance weighted average of all portfolios
.
As in the previous section we have
|
(15) |
where Fn denotes the empirical distribution of
,
i.e. it places mass
at each
xi.
We can easily prove that Sn* exceeds the maximum of the component stocks, the arithmetic mean, the geometric mean and that
is invariant under permutations of the sequence
.
- The target wealth exceeds the wealth from best stock:
|
(16) |
- The target wealth exceeds the Dow Jones (or the arithmetic mean):
|
(17) |
where
.
This assumes buy-and-hold strategies
,
where ej is the j-th basis vector.
Since the wealth ,
resulting from the universal portfolio, is the average of
Sn(b), or
|
(18) |
where
we can show that for the universal portfolio
|
(19) |
or, the wealth from the universal portfolio also exceeds value line index.
We can also prove that
is invariant under permutations of the sequence
.
This invariance entails that a stock market crash will have no worse consequences for wealth
than if the bad days of that time had been sprinkled out among the good.
The important question remains, how does
behave?
For a portfolio of two stocks, we consider the arbitrary stock vector sequence
|
(20) |
The portfolio choice can be transformed into a choice of one variable, namely
|
(21) |
and similarly
Sn(b) becomes
|
(22) |
Let's define
|
(23) |
Jn is generally known as the curvature or volatility index of a portfolio.
It can be shown that for any
and for any subsequence of times
such that the doubling rate Wn (b) satisfies the condition
|
(24) |
and where W(b) achieves its maximum at
the following holds true
|
(25) |
This means that the universal wealth is within a factor
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
.
Next: Real world example
Up: A study in portfolio
Previous: Stochastic Markets
Magnus Bjornsson
1998-05-12