that returns are distributed normally. To calculate the
certainty equivalents, Freunds’ more stringent as-
sumptions that returns are distributed multivariate
normal and that the agent’s utility function is a
negative exponential are also required.
Formally, a negative exponential utility function
can be specified as
(1) U [W(X) ] =-exp [-ΘW(X) ] ; Xεlo,
where wealth (W) is a function of an investment
bundle (X), θ is the Pratt-Arrow absolute risk aver-
sion coefficient, and Io is the set of feasible invest-
ment bundles. Wealth is generated by investing in
the feasible bundle X, and if the returns on X are
multivariate normal, then W(X) ~ N[μ(X), σ2(X)]
Bussey has shown that under this specification, the
expected utility of the negative exponential is
equivalent to
(2) E{U[W(X)]} = -exp.{- θ[μ(X) -1 σ2 (X)]} .
Ch∞sing the vector of activities, X, to maximize
expected utility in (2) yields the same solution as
choosing X to maximize
(3) Z = μ(X)-∣σ2(X)
because (2) ia a monotonic transformation of (3).
In addition to yielding the same maximum, thereby
simplifying the process of finding the utility maxi-
mizing portfolio, (2) also allows calculation of the
certainty equivalent for a risky investment. The cer-
tainty equivalent is simply the certain level of wealth
for which the decision-maker is indifferent with
respect to a risky alternative. To compute the certain-
ty equivalent for a risky opportunity, an expenditure
function or inverse utility function is set equal to
expected utility. Specifically, we are interested in
determining the certainty equivalent, W*(X), that
yields the same level of utility as E(U[W(X)]}.
Substituting W*(X) for W(X) in (1) and solving for
W*(X) yields the certainty equivalent
(4) W*(X) = ∣ln(E{U[W(X)]} ).
Substituting (2) into (4) and simplifying, the cer-
tainty equivalent is
(5) W(X) = μ(X) -1 σ2(X).
The certainty equivalent of a risky investment is
equal to the objective function, Z. The above deriva-
tion also has a heuristic explanation. By definition,
a certainty equivalent has no variance, otherwise it
would not be certain. To find the certainty
equivalent, a utility function is set equal to the level
of expected utility of a risky alternative. Because a
certain outcome has no variance, μ(X) is equal to
expected utility (Z). For any set of assumptions in
which expected utility is maximized by maximizing
(3), Z defines the certainty equivalent.
THE MARGINAL BENEFIT AND COST OF
ADDITIONAL DIVERSIFICATION
A change in the feasible set can be used to derive
the marginal benefit and cost from additional diver-
sification opportunities. Once the risky investment
opportunity is expressed in terms of a certainty
equivalent, standard concepts of deterministic con-
sumer behavior become applicable. For example,
given that preferences are monotonically increasing
in wealth, a consumer will always prefer more
wealth. Therefore, the consumer will prefer an alter-
native with a higher certainty equivalent. The cer-
tainty equivalent includes an adjustment for risk
preferences. Hence, the agent, in choosing an invest-
ment with the greater certainty equivalent, is con-
sidering his or her risk preference. If a consumer is
faced with two risky alternatives and the certainty
equivalent of the first is greater than the certainty
equivalent of the second, the agent will prefer the
first. Further, the maximum price that agents will
pay for the first, given that they already have the
second, is the difference in the certainty equivalents.
Because the marginal benefit can be defined as the
most an agent is willing to pay for an item, the
marginal benefit of the additional diversification
opportunity is the change in certainty equivalents.
Mean-variance studies typically have examined
diversification based on gross margins (returns
minus variable costs, Adams et al.). Incremental
fixed costs play an important role in determining the
desirability of diversification. These incremental
fixed costs constitute the marginal costs of diver-
sification, which are often not considered. The mar-
ginal costs of diversification can be determined by
calculating the net present value of the incremental
fixed costs and amortizing those costs over the life
of the investment The amortized fixed costs can be
either subtracted from the mean return ( μ) in (2) or
compared directly with the marginal benefit defined
above. If the marginal incremental fixed costs are
subtracted from (2), then the investment would be
desirable when the marginal benefit is positive.
It may not be appropriate to subtract the fixed
costs of diversification from the returns above vari-
able costs given the Iumpiness of an investment A
solution for a risk programming problem often in-
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