Universal portfolio, part 2

Universal Portfolios has a classical structure: introduction of the problem, defining then proving a proposition, provide some illustration on real data, conclusion.

The introduction first defines a reference portfolio, the best constant rebalanced portfolio (BCRP).  The BCRP is found after all prices are known, so it is a rather tough reference to meet.  In particular, by definition, Cover shows that the BRCP:

• exceeds the best stock (proposition 2.1 in the article)
• exceeds the value line (proposition 2.2)
• exceeds the arithmetic mean (proposition 2.3) 

And still Cover then proves that a specific portofolio selection algorithm, called Universal Portfolio (UP) can asymptotically match the growth rate of the BCRP.  This comes by showing that the ratio between BCRP and UP goes asymptotically to zero but slower than as an exponential, and thus the ratio of the growth rates tends to 1.  The exact expression (6.1 in the article) for the ratio is

$$\frac{\widehat{S_n}}{S_n^{\text{*}}}\sim {\left(\sqrt{\frac{\text{2}\Pi }{n}}\right)}^{m-1}\frac{\left(m-1\right)}{{\left|J^{\text{*}}\right|}^{\text{1/2}}}$$
$\widehat{S_n}$ is the value of the Universal Portfolio after n periods
$S_n^{\text{*}}$ is the value of the BCRP after n periods
m is the number of stocks considered
$\left|J^{\text{*}}\right|$ is a measure of "curvature" of the time series of price relatives
The important aspect is that the ratio is not an exponential function of n and so the difference in growth rates between the two portfolios is asymptotically 0.

Unfortunately, the Universal Portfolio is a little bit disappointing, it effectively splits the initial investment across all possible CRP, so that it is sure to hit the best one.  This is only done on paper, there is only one real portfolio but its composition is matched to the combination of all possible CRP at any moment in time.  This happens to be difficult to do (exponential in the number of stocks for the obvious approach), with a number of simpler approximations available.

Universal Portfolios then uses a long sequence of historical prices to show real applications.  This will be the subject of the next post in this series.