STOCHASTIC

P if and only if
INVESTMENT ANALYSIS UNDER UNCERTAINTY B-655
(6) Z.-.y.*.- Pijk-[U(do + c,, d, + c., •••) - U(A,,d,, •••)) a 0.
The remaining discussion is devoted to procedures for implementing this criterion.
Since much of the remaining discussion is devoted to circumventing the calculations
required by (6), it may be well to illustrate using the data from Figure 2. Suppose
t/(d°, d\ d 2 ) = XXofrtW), where U,(d) = d - aV. The calculations are given
below for an example in which do, di, da, etc are all zero under the present policy.
Since the expected utility (1881.4) is positive
(
0
1
2
it
10" •
.5 X 10-'
io-'
01
1.00
.95
.90
i
1
1
1
2
2
3
1
2
3
1
2
f'i
.30
.24
.06
.32
.08
V
33,217
6,982
-19,595
-16,151
-42,692
PiiV
9,965.1
1,675.7
-1,175.7
-5,168.3
-3,415.4
1,881.4
this project satisfies (6). The expected utility is the same as the utility of a stream consisting
of $1,884 at t = 0 and zero thereafter. By way of contrast, the expected present
monetary value of the project, using the P'B as discount factors, is $3,020.
One restriction is imposed on the utility function U: in mathematical terms, we
require that the utility function be concave. Essentially, this means (1) that the utility
function must evidence risk aversion, or at least not risk preference; and (2) that the
intertemporal indifference surfaces for dividends must be convex, so that there is
always nonincreasing marginal utility from exchanging dividends at one time for
another time. As a consequence of concavity, the utility function possesses gradients
(the vector of partial derivatives) everywhere, although they need not be unique. In
addition, of course, a utility function must always be nondecreasing in each of its
arguments.
We shall not here develop the construction of utility and probability measuies, but
see [3] and [4].
5. The Methodology of Investment Analysis Under Uncertainty
In principle, the criterion (6) suffices entirely for the selection of investment projects
: one simply computes the cash flow description for a project, and if (6) is satisfied,
then the project is amended to the current policy. In practice, nevertheless, the
situation is quite different. The main reason, in brief, is that a package of projects is
considerably more than the sum of its components. To repeat, an investment is, in
essence, a present certain sacrifice for future uncertain benefits. Conceptually, we can
distinguish two characteristic rationale for investments. First, there is the smoothing
effect. For example, a fairly even distribution of dividends over time is usually preferred
to a lump-sum liquidation at any one time. Via investment, therefore, the firm
can defer present dividends for future dividends so as to smooth the pattern of the
dividend stream. Second, there is the diversification effect. For example, two projects
each of which yields a large return in one state and a small return in another state,
but the role of the states is opposite for the two projects, may be acceptable together
when neither is separately; because, separately the large variation in returns is in-
PART IV. DYNAMIC MODELS REDUCIBLE TO STATIC MODELS