STOCHASTIC

1. MODELS THAT HAVE A SINGLE
DECISION POINT
MANAGEMENT SCIENCE
Vol. It, No. la, AuguM, i9tt
PriviU in U.S.A.
INVESTMENT ANALYSIS UNDER UNCERTAINTY^
ROBERT WILSON
Graduate School of Business, Stanford University
Investment projects are described in this paper as a pattern of uncertain cash flows
over time; i.e., as a cash flow pattern over an event tree. Whereas the portfolio problem
of selecting projects generally requires simultaneous consideration of all available
projects, here we seek sufficient conditions for accepting and rejecting individual
projects. These conditions are formulated as mathematical programming problems
which are amenable to routine application at subordinate levels of an organization.
"Investment is, in essence, present sacrifice for Suture benefit. But the present is relatively
well known, whereas the future is always an enigma. Investment is also, therefore,
certain sacrifice for uncertain benefit." 1
1. The State Description of Uncertain Events
When one says that the return in the next year from an investment project is uncertain,
it usually means that the return will depend upon prevailing conditions or
circumstances in the interim, but precisely which conditions will obtain is not known
at present. If the prevailing conditions, or "state of the world," were known, however,
then also the return would be known. That is, one knows well enough the effect of the
prevailing conditions on the return to be able to predict the return that will be attained
in each state of the world. Of course, how well the effects must be known depends
upon the precision demanded of the prediction, but in practice it often suffices
to take account only of major sources of uncertainty. A state description is a specification
of possible states of the world which is, for purposes of analysis, sufficiently fine
to enable one to regard the predicted return in each state as certain if that state should
obtain. We will not formalize the construction of a state description, but rather take
it to be a datum.
The time dimension of a state description can be made explicit by constructing an
event tree specifying the sequence of states that can obtain. Figure 1 depicts an example
of an event tree over four time periods. Each node of the tree represents a point in
time, and each arc represents a state of the world prevailing up to the point in time of
its successor node at the right-hand end point. The label &,*... on an arc records that
the state is compounded of an event indexed by t in the first period, followed by an
event indexed by j in the second period, then an event indexed by k in the third period,
and so on. The essential feature of an event tree is the delineation of the possible events
that can follow after the state of the world in one period to determine the state of the
world in the succeeding period. It is important to recognize that, as defined here, a
state embodies the entire sequence of events up to and including the period it describes.
For this reason, the set of events that can possibly occur in the next period
may be different for different nodes representing that same point in time; cf. Figure 1,
in which the sets of events that might obtain in the last period depend upon the node
attained at time 3.
* Received January 1967; revised March 1968.
t This study was supported, in part, by funds made available by the Ford Foundation to the
Graduate School of Business, Stanford University. However, the conclusions, opinions and other
statements in this publication are those of the author and are not necessarily those of the Ford
Foundation.
1 Quotation from J. Hirshleifer [1].
MODELS THAT HAVE A SINGLE DECISION POINT