164 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Therefore, oftentimes, scenario planning puts us in a position where we
must make a decision regarding how much of a resource to allocate today,
given the possible scenarios of tomorrow. This is the true heart of scenario
planning: quantifying it.
First, we must define each unique scenario. Second, we must assign a
probability of that scenario’s occurrence. Being a probability means that this
number is between 0 and 1. We need not consider any further scenarios with
a probability of 0. Note that these probabilities are not cumulative. In other
words, the probability assigned to a given scenario is unique to that scenario.
Suppose we are decision makers for XYZ Manufacturing Corporation. Two
of the many scenarios we have are as follows. In one scenario, we have the
probability of XYZ Manufacturing filing for bankruptcy with a probability
of .15, and, in another scenario, we have XYZ being put out of business by
intense foreign competition with a probability of .07. Now, we must ask
if the first scenario, filing for bankruptcy, includes filing for bankruptcy
due to the second scenario, intense foreign competition. If it does, then
the probabilities in the first scenario must not take the probabilities of the
second scenario into account, and we must amend the probabilities of the
first scenario to be.08 (.15−.07).
Just as important as the uniqueness of each probability to each scenario
is that the sum of the probabilities of all of the scenarios we are considering
must equal 1 exactly. They must equal not 1.01 nor .99, but 1.
For each scenario, we now have a probability of just that scenario
assigned. We must now also assign an outcome result. This is a numerical
value. It can be dollars made or lost as a result of a scenario’s manifesting
itself; it can be units of utility or medication or anything. However, our
output is going to be in the same units that we put in.
You must have at least one scenario with a negative outcome in order
to use this technique. This is mandatory.
A last prerequisite to using this technique is that the arithmetic math-
ematical expectation, the sum of all of the outcome results times their re-
spective probabilities [Equation (1.01a)], must be greater than zero. If the
arithmetic mathematical expectation equals zero or is negative, the follow-
ing technique cannot be used.^5 That is not to say that scenario planning
itself cannot be used. It can and should. However, optimalfcan be incor-
porated with scenario planning only when there is a positive, mathematical
expectation.
Lastly, you must try to cover as much of the spectrum of outcomes as
possible. In other words, you really want to account for 99% of the possible
outcomes. This may sound nearly impossible, but many scenarios can be
(^5) However, later in the text we will be using scenario planning for portfolios, and,
therein, a negative arithmetic mathematical expectation will be allowed and can
possibly benefit the portfolio as a whole.