166 THE HANDBOOK OF PORTFOLIO MATHEMATICS
The sum of our probabilities equals 1. We have at least one scenario with a
negative result, and our mathematical expectation is positive:
(. (^1) −500, 000)+(. (^2) −200, 000)+...etc.=185, 000
We can, therefore, use the technique on this set of scenarios.
Notice first, however, that if we used the single most likely outcome
method, we would conclude that peace will be the future of this country,
and we would then act as though peace were to occur, as though it were a
certainty, only vaguely remaining aware of the other possibilities.
Returning to the technique, we must determine the optimalf.The op-
timalfis that value for f (between zero and one) which maximizes the
geometric mean, using Equations (4.16 to 4.18). Now, we obtain the terminal
wealth relative, or TWR using Equation (4.17). Finally, if we take Equation
(4.17) to thepiroot, we can find our average compound growth per play,
also called the geometric mean HPR, which will become more important
later on. We use Equation (4.18) for this.
Here is how to perform these equations. To begin with, we must decide
on an optimization scheme, a way of searching through thef values to
find that fwhich maximizes our equation. Again, we can do this with a
straight loop withffrom .01 to 1, through iteration, or through parabolic
interpolation.
Next, we must determine the worst possible result for a scenario among
all of the scenarios we are looking at, regardless of how small the probability
of that scenario’s occurrence are. In the example of XYZ Corporation, this
is−$500,000.
Now, for each possible scenario, we must first divide the worst possible
outcome by negativef.In our XYZ Corporation example, we will assume
that we are going to loop throughfvalues from .01 to 1. Therefore, we start
out with anfvalue of .01. Now, if we divide the worst possible outcome of
the scenarios under consideration by the negative value forf, we get the
following:
−$500, 000
−. 01
=50,000,000
Notice how negative values divided by negative values yield positive
results, and vice versa. Therefore, our result in this case is positive. Now,
as we go through each scenario, we will divide the outcome of the scenario
by the result just obtained. Since the outcome to the first scenario is also
the worst scenario—a loss of $500,000—we now have:
−$500, 000
50, 000, 000