358 THE HANDBOOK OF PORTFOLIO MATHEMATICS
increases, the distance between these advantageous points and optimalf
diminishes.
Suppose a money manager uses daily HPRs and wants to be optimal
(with respect to inflection or GRR) over the course of the current quarter
(63 days). He would use a value of 63 forTand set himself at those coordi-
nates to be optimal for each quarter.
When we begin working in more than two dimensions, that is, when we
are dealing with more than one scenario spectrum, we enter an altogether
more complicated problem.
The solution can be expressed mathematically as that point where the
second partial derivatives of the TWR [Equation (9.04), raised to the power
ofT, the number of holding periods at which we are seeking the points of
inflection] with respect to a each particularfequals zero, and each point is
to the left (on its axis) of the peak. This becomes ever more complicated in
that such a point, where the second partials of the TWR with respect to each
fequaling zero may not, depending upon the parameters of the scenario
spectrums themselves and how high or lowTis, exist. IfTequals one, the
TWR equals the geometric mean HPR, which is upside down parabolic—
it doesn’t have any points of inflection! Yet asTapproaches infinity, the
point(s) of inflection approach the optimalf(s)! Shy of infiniteT, there may
not be in most cases, such a conveniently common point of inflection with
respect to all axes.
∗
All of this brings us right back to the notion of then+1 dimensional
terrain in leverage space, if you will, the axes of which correspond to thef
values of the different scenario sets, is to act as aframeworkfor analyzing
portfolio construction and quantity determination through time. There is
so much more to be done in working with this new framework. This chap-
ter is not the end-all on the subject. Rather, it is a mere introduction to
an altogether new and, I believe, better way of determining asset alloca-
tion. Almost certainly, portfolio strategists, applied mathematicians, asset
allocators, and programmers have much new fertile ground to work. Truly,
there is a great deal to be done in analyzing, working with, and adding to
this new framework, the rewards of which cannot yet even be determined.
More importantly, whether one attempts to actively employ the Leverage
Space Model, the tenets of The New Framework, as expressed here, are at
work and apply to him regardless.
∗Remember that the primary thing gained by diversification, that is, trading more
than one scenario spectrum, or working in more than two dimensions, is that you
increaseT, the number of holding periods in a given period of time—you do not
reduce risk. In light of this, someone looking to maximize the marginal increase in
gain to a marginal increase inrisk, may well opt to trade only one scenario spectrum.