Ralph Vince - Portfolio Mathematics

388 THE HANDBOOK OF PORTFOLIO MATHEMATICS

cover the same time period. That is, we may have our scenarios derived
such that they are the scenarios of outcomes for the next month—or the
next day, and so on.
We therefore consider each combination of scenarios for each scenario
spectrum. Thus, if we are looking at two scenario spectrums of our two-to-
one coin toss, we would then have the following four outcomes:

Game 1 + 2 + 2 − 1 − 1

Game 2 + 2 − 1 + 2 − 1

The reason we have four outcomes is that we have two scenario spec-
trums with two scenarios in each. Thus,nin this case will equal 4.
When you have more than one scenario spectrum:

n=

∏m

i= 1

#scenariosi (12.06)

where: m=The number of scenario spectrums you are including.

In other words,nis the product of all the scenario spectrums we are
considering.
And in our example here, since there are two scenario spectrums
(m=2), each with two scenarios, we haven=4.
The HPRs then for these four outcomes are the arithmetic average HPRs
across scenario spectrums. The arithmetic average is used simply because
an HPR represents the effect of trading one unit at a given value offon the
entire account.
So, if we assume we are going to trade at fvalues of .25,.25 in our
example, we then have the following:

Game 1 + 2 + 2 − 1 − 1

Game 2 + 2 − 1 + 2 − 1

Converted to HPRs at .25,.25:

Game 1 1.5 1.5 .75 .75

Game 2 1.5 .75 1.5 .75

Arithmetic mean:

1.5 1.125 1.125 .75

Thus, we haven=4, and the four values are (1.5, 1.125, 1.125, .75),
which we would then use in our analysis.