STOCHASTIC

INVESTMENT ANALYSIS UNDER UNCERTAINTY B-661
Of more fundamental importance, however, is that this exclusion criterion derives
via (18) from a tangential linear approximation to the utility function U, which is
the borderline case between risk aversion and risk preference. In concrete terms, therefore,
the exclusion criterion says to reject any project of interest only to a gambler.
Clearly, such a project will be of little value fo» diversification purposes (although
under a different policy it might be, and if the investment analysis proceeds by successive
amendments to the policy this must be kept in mind). On the other hand, any
project which fails both the inclusion and exclusion criteria is a candidate for diversification
purposes, since the inclusion criterion says that it cannot be justified
singly or with complementary financing, and the exclusion criterion says that it is
potentially valuable if, perhaps with other projects, it conforms to the firm's risk
aversion. It is risk aversion, of course, that justifies diversification.
8* A Mathematical Programming Formulation
Up to now we have treated investment projects as discrete entities to be either
accepted or rejected. More generally a project might be of variable intensity, although
in the special case of a discrete project there would be only one feasible positive intensity.
The formulation in this more general situation delineates the structure of the
investment problem more clearly.
We pose the problem in terms of mathematical programming. Let the set of all
available projects, including financing projects, be indexed by n = 1, • • • , N. For an
initial policy P let D(P) as in (2) be the dividend stream, and for each project n let
the cash flow description be
(24) C.(* I P) = [*"(*); |e,"(*)| ; UUz)}; • • •],
where the vector of project intensities is x = (ii, • • • , x„ , • • • xx). Often, of course,
the cash flow description for project n will depend only on x, . For the set of available
projects define
(25) 7o(s) = E'-i