The Geometry of Mean Variance Portfolios 281
we get very high positive correlation. This mistakenly happens because
their regression lines are rising a little each day. Yet on most days the equity
change is zero. Therefore, the difference is negative. The preponderance of
slightly negative days with both market systems, then, mistakenly results
in high positive correlation.
Completing the Loop
One thing you will readily notice about unconstrained portfolios (portfolios
for which the sum of the weights is greater than 1 and NIC shows up as a
market system in the portfolio) is that the portfolio is exactly the same for
any given level of E—the only difference being the degree of leverage. (This
isnottrue for portfolios lying along the efficient frontier(s) when the sum
of the weights is constrained). In other words, the ratios of the weightings
of the different market systems to each other are always the same for any
point along the unconstrained efficient frontiers (AHPR or GHPR).
For example, the ratios of the different weightings between the differ-
ent market systems in the geometric optimal portfolio can be calculated.
The ratio of Toxico to Incubeast is 102.5982% divided by 49.00558%, which
equals 2.0936. We can thus determine the ratios of all the components in
this portfolio to one another:
Toxico/Incubeast= 2.0936
Toxico/LA Garb= 2.5490
Incubeast/LA Garb= 1.2175Now, we can go back to the unconstrained portfolio and solve for differ-
ent values for E. What follows are the weightings for the components of the
unconstrained portfolios that have the lowest variances for the given values
of E. You will notice that the ratios of the weightings of the components are
exactly the same:
E=.1 E=.3Toxico .4175733 1.252726
Incubeast .1994545 .5983566
LA Garb .1638171 .49145Thus, we can state thatthe unconstrained efficient frontiers are the
same portfolio at different levels of leverage.This portfolio, the one that
gets levered up and down with E when the sum of the weights constraint