354 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 10.6 Two-to-one coin toss, GRR atT= 1
Notice that by doing this, if each scenario spectrum realizes its worst-case
scenario simultaneously, you will still have defined the maximum drawdown
percent for the entire portfolio.
Next, we move on to another important point to the left, which may be
of importance to certain money managers: thegrowth risk ratio,orGRR
(Figure 10.6). If we take the TWR as the growth, the numerator, and thef
used (or the sum of thefvalues used for portfolios) as representing risk,
since it represents the percentage of your stake you would lose if the worst
case scenarios(s) manifest, then we can write the growth risk ratio as:
GRRT=
TWRT
∑n
i= 1fi(10.18)
This ratio is exactly what its name implies, the ratio of growth (TWRT,
the expected multiple on our stake afterTplays) to risk (sum of thef
values, which represent the total percentage of our stake risked). If TWR is
a function ofT, then too is the GRR. That is, as T increases, the GRR moves
from that point where it is an infinitesimally small value forf, towards the
optimal f (see Figure 10.7). At infiniteT, the GRR equals the optimalf. Much
like the EACG, you can trade at thefvalue to maximize the GRR if you
know, a priori, what value forTyou are trying to maximize for.
The migration from an infinitesimally small value for f atT=1to
the optimalfatT=infinity happens with respect to all axes, although in
Figures 10.6 and 10.7 it is shown for trading one scenario spectrum. If you
were trading two scenario spectrums simultaneously, the peak of the GRR
would migrate through the three-dimensional landscape asTincreased,