The Geometry of Leverage Space Portfolios 355
FIGURE 10.7 Two-to-one coin toss, GRR atT= 30
from nearly 0,0 for both values off, to the optimal values forf(at .23,.23
in the two-to-one coin toss).
Discerning the GRR for more than one scenario spectrum traded si-
multaneously is simple, using Equation (10.18), regardless of how many
multiple simultaneous scenario spectrums we are looking at.
The next and final point to be covered to the left, which may be quite
advantageous for many money managers, is the point of inflection in the
TWR with respect tof.
Refer again to Figure 9.2 in Chapter 9. Notice that as we approach the
peak in the optimal ffrom the left, starting at 0, we gain TWR (vertical)
at an ever-increasing rate, up to a point. We are thus getting greater and
greater benefit for a linear increase in risk. However, at a certain point,
the TWR curve gains, but at a slower and slower rate for every increase in
f. This point of changeover, calledinflection, because it represents where
the function goes from concave up to concave down, is another important
point to the left for the money manager. The point of inflection represents
the point where the marginal increase in gain stops increasing and actually
starts to diminish for every marginal increase in risk. Thus, it may be an
extremely important point for a money manager, and may even, in some
cases, be optimal in the eyes of the money manager in the sense of what it
does in fact, maximize.
However, recall that Figure 9.2 represents the TWR after 40 plays. Let’s
look at the TWR after one play for the two-to-one coin loss, also simply
called the geometric mean HPR, as shown in Figure 10.8.
Interestingly, there isn’t any point here where the function goes from
concave up to concave down, or vice versa. There aren’t any points of
inflection. The whole thing is concave down.