396 THE HANDBOOK OF PORTFOLIO MATHEMATICS
simultaneous outcome for both scenario spectrums, wherein each combi-
nation of scenarios from both scenario spectrums occur.
Coin Toss 1,Coin Toss 2
.25,.25
2,2
2,− 1
−1,2
−1,− 1So in this file, the first outcome sees both scenario spectrums gaining
two units. The next outcome sees Coin Toss 1 gaining two units, while Coin
Toss 2 loses one unit (–1). Then Coin Toss 1 loses one unit (–1) and Coin
Toss 2 gains two units. For the last outcome, they both lose one unit (–1).
(Thus,n=4 in this file. In all data files, therefore, since the first two lines are
scenario spectrum name(s) and respectivefvalue(s),nequals the number
of lines in the file minus 2.)
To this point, we have not alluded to the probabilities of the scenario
outcomes. Rather, as if the scenario outcomes were like a stream of trades,
or a stream of coin toss results, we have quietly assumed for simplicity’s sake
that there has been an equal probability of occurrence with each scenario
outcome. In other words, we have been inexplicitly saying to this point that
for each scenario (or individual combinations of scenarios from multiple
spectrums occurring simultaneously), the probability of thekth outcome
among thenqoutcomes is:
pk=^1/
nq (12.07)Usually, however, we do not have the luxury of the convenience of all
scenarios having the same probability of occurrence.^6
To address this, we return now to Equation (12.05). We will discuss
first the case of a single scenario spectrum. In this case, we not only have
outcomes for each scenario [which comprise the HPRs used in Equation
(^6) Note, however, that if weweretalking about scenarios made up of individual coin
tosses, or of results of trading a given market system over a given day, or if we did
use purely empirical data in discerning our scenario spectrums and probabilities,
we could use Equation (12.07) for the said probabilities. In other words, if, say, we
used the last 24 trading months and examined the prices of stock ABC, we could
conceivably create a scenario spectrum of 24 bins, each with an outcome of those
months, each with a probability given in (12.07).