168 THE HANDBOOK OF PORTFOLIO MATHEMATICS
calculating this, we can state that we must raise the TWR to the power of
1 to give us the geometric mean. Since anything raised to the power of 1
equals itself, we can say that, in this case, our geometric mean equals the
TWR. We therefore have a geometric mean of 1.003667853.
The answer we have just obtained in our example is our geometric
mean corresponding to anfvalue of .01. Now we move on to anfvalue of
.02, and repeat the whole process until we have found the geometric mean
corresponding to anfvalue of .02. We will proceed as such until we arrive
at that value forfwhich yields the highest geometric mean.
In the case of our example, we find that the highest geometric mean is
obtained at anfvalue of .57, which yields a geometric mean of 1.1106. Di-
viding our worst possible outcome to a scenario (−500,000) by the negative
optimalfyields a result of $877,192.35. In other words, if XYZ Corporation
wants to commit to marketing this new product in this remote country, they
will optimally commit this amount to this venture at this time. As time goes
by and things develop, the scenarios, their resultant outcomes, and proba-
bilities will likewise change. Thisfamount will then change as well. The
more XYZ Corporation keeps abreast of these changing scenarios, as well
as the more accurate the scenarios they develop as input are, the more ac-
curate their decisions will be. Note that if XYZ Corporation cannot commit
this $877,192.35 to this undertaking at this time, then they are too far beyond
the peak of thefcurve. It is the equivalent of the guy who has too many
commodity contracts with respect to what the optimalf says he should
have. If XYZ Corporation commits more than this amount to this project at
this time, the situation would be analogous to a commodity trader with too
few contracts.
There is an important point to note about scenarios and trading. What
you use for a scenario can be any of a number of things:
1.It can be, as in the previous example, the outcomes that a given trade
may take. This is useful if you are trading only one item. However, when
you trade a portfolio of items, you violate the rule that all holding period
lengths must be uniform.
2.If you know what the distribution of price outcomes will be, you can
use that for scenarios. For example, suppose you have reason to be-
lieve that prices for the next day for a given item are normally dis-
tributed. Therefore, you can discern your scenarios based on the nor-
mal distribution. For example, in the normal distribution, 97.72% of the
time, prices will not exceed 2 standard deviations to the upside, and
99.86% of the time they will not exceed 3 standard deviations to the up-
side. Therefore, as one scenario, you can have as the result something