Ralph Vince - Portfolio Mathematics

The Geometry of Leverage Space Portfolios 325

you would now look to trade 19 units ($95,000/$5,000). However, under the
split equity technique you are now left with $45,000 of active equity and,
thus, you will look to trade 18 units ($45,000/$2,500).
Notice that with the split equity technique, the exact fraction of optimal
fthat we are using changes with the equity changes. We specify the fraction
we want to start with. In our example, we used an initial fraction of .5.
When the equity increases, this fraction of the optimalfincreases, too,
approaching 1 as a limit as the account equity approaches infinity. On the
downside, this fraction approaches 0 as a limit at the level where the total
equity in the account equals the inactive portion. This fact, that there is
built-in portfolio insurance with the split equity technique, is a tremendous
benefit and will be discussed at length later in this chapter.
Because the split equity technique has a fraction forfthat moves, we
will refer to it as a dynamic fractionalfstrategy, as opposed to the straight
fractionalf(which we will call astaticfractionalf) strategy.
Using the dynamic fractionalftechnique is analogous to trading an
account full out at the optimalflevels, where the initial size of the account
is the active equity portion.
So, we see that there are two ways to dilute an account down from
the full geometric optimal portfolio. We can trade a static fractional or a
dynamic fractionalf.Although the two techniques are related, they also
differ. Which is best?
To begin with, we need to be able to determine the arithmetic average
HPR for tradingngiven scenario spectrums simultaneously, as well as the
variance in those HPRs for thosensimultaneously traded scenario spec-
trums, for givenfvalues (f 1 ...fn) operating on those scenario spectrums.
These are given now as:

AHPR (f 1 ...fn)=

∑m

k= 1

[(

1 +

∑n

i= 1

(

fi∗

(

−PLk,i

BLi

)))

∗Probk

]

∑m

k= 1

Probk

(10.01)

where: n=The number of scenario spectrums (market systems or

portfolio components).

m=The possible number of combinations of outcomes

between the various scenario spectrums (market

systems) based on how many scenarios are in each

set.m=The number of scenarios in the first spectrum

* the number of scenarios in the second spectrum*...*

the number of scenarios in thenth spectrum.