The Geometry of Leverage Space Portfolios 345
Thus, if we start out with an initial active equity of 5%, then 13.63% is the
upside point where the dynamic would be expected to overtake the static,
and we would use the following for FRAC in Equations (10.10a) and (10.10b)
in determining the hedge ratio at the upside point,Y, dictated by Equation
(10.13):
FRAC=
(0. 5 + 1 .363)
(1+.1363)
=
. 1863
1. 1363
=. 1639531814
Thus, when we have an account which is up 13.63%, and we start with a 5%
initial active equity, we know that the active equity is then 16.39531814%.
Gradient Trading and Continuous Dominance
CONTINUOUS DOMINANCE
We have seen throughout this text, that trading at the optimal f for a
given market system or scenario spectrum (or the set of optimalfs for
multiple simultaneous scenario spectrums or market systems) will yield
the greatest growth asymptotically, that is, in the long run, as the number
of holding periods we trade for gets greater and greater. However, we have
seen in Chapter 5, with “Threshold to Geometric,” and in Chapter 6, that if
we have a finite number of holding periods and we know how many holding
periods we are going to trade for, what is truly optimal is somewhat more
aggressive even than the optimalfvalues; that is, it is those values forf
which maximize expected average compound growth (EACG).
Ultimately, each of us can only trade a finite number of holding
periods—none of us will live forever. Yet, in all but the rarest cases, we
do not know the exact length of that finite number of holding periods, so
we use the asymptotic limit as the next best approximation.
Now you will see, however, a technique that can be used in this case of
an unknown, but finite, number of holding periods over which you are going
to trade at the asymptotic limit (i.e., the optimalfvalues), which, if you are
trading any kind of a dilutedf(static or dynamic), allows for dominance
not only asymptotically, but forany given holding period in the future.
That is, we will now introduce a technique for a diluted f (which
nearly all money managers must use in order to satisfy the real-
world demands of clients pertaining to drawdowns) that not only en-
sures that an account will be at the highest equity in the very long