STOCHASTIC

B-654 ROBERT WI1-SON
4. The Investment Problem Under Uncertainty
In principle, the object of the investment analysis is to select from the set of available
projects a feasible package of projects to be undertaken which is preferred above
all others. This package then constitutes an optimal sequential investment strategy.
The strategy is sequential, of course, since it ordinarily includes deferred projects
scheduled to be launched conditional upon, and in a way determined by, the interim
states of the world. In practice, however, dependencies among projects usually preclude
a complete analysis due to the combinatorial complexities involved in considering
all possible packages. Instead, the more practical goal is established of successively
seeking projects to adjoin to the existing policy. In this way, step by step, successively
more preferred policies are attained. In part, of course, this approach is dictated by
the administrative necessity of considering major projects one or two at a time. This
goal, finding a desirable project to adjoin to an existing policy, is the one that will be
considered henceforth. We shall, however, allow a project to have multiple aspects,
such as both financing and expenditures, both present and deferred.
The object of choice in project selection is the dividend stream, as displayed in (3)
and (4), for the resulting policy, or at least that is what we will assume. This means
that each project is to be evaluated in terms of its effect on the dividend stream as
measured by its cash flow description (1). It does not mean that retained earnings are
prohibited, since the sequential investment strategy may well provide for reinvestment
of returns in deferred projects; but, returns which are not reinvested are paid
out as dividends, which surely must be mainly what matters to present (and potential)
shareholders. The personal ambitions of professional managers, however, are prohibited
as criteria for choices among projects.
For the methodology by which to evaluate alternative dividend streams we appeal
to the axiomatic theory of consistent decision making under uncertainty (e.g., [5]).
We suppose that each present and potential shareholder assesses a utility measure for
dividend streams and a probability measure over the states of the world, and seeks to
maximize the expected utility of his dividend stream (of course, if present and potential
shareholders disagree appropriately than a mutually favorable exchange can be constructed).
Although the dividends are shared jointly, nevertheless, the investment
decision is made in common by the shareholders or their representative, the professional
manager. There remains, therefore, the necessity of a transition from the utility
and probability measures of the shareholders to comparable measures for the firm as
an entity, as well as an accounting of the effects of security markets. The Appendix
briefly discusses two approaches to this problem. Although the theory of this transition
is incomplete, we shall in any case here presume that the firm possesses a utility
measure for dividend streams and a probability measure over states of the world.
Let U(tif, d 1 ,