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196 Applying simulation to decision problems
to minimize the risk or uncertainty which her company faces. If we
compare products A and B we see that, while they offer the same
expected return, product B is much more risky. Product A is therefore
said to dominate B. B is also dominated by C, which for the same level
of risk offers higher expected profits. For the same reason, D dominates
E. The non-dominated products, A, C and D, are therefore said to lie
on the efficient frontier, and only these products would survive the
screening process and be considered further. The choice between A, C
and D will depend on how risk averse the decision maker is. Product
A offers a low expected return but also a low level of risk, while at the
other extreme, C offers high expected returns but a high level of risk.
The utility approach, which we discussed above, could now be used to
compare these three products.
Note that for the mean–standard deviation screening process to be
valid it can be shown that a number of assumptions need to be made.
First, the probability distributions for profit should be fairly close to
the normal distribution shape shown in Figure 7.9(a) (in many practical
situations this is likely to be the case: see Hertz and Thomas 1 ). Second, the
decision maker should have a utility function which not only indicates
risk aversion but which also has (at least approximately) a quadratic
form. This means that the function can be represented by an equation of
the form:
U(x) = c + bx + ax 2
where x = a given sum of money, U(x) = the utility of this sum of money
and a, b and c are other numbers which determine the exact nature of
the utility function.
For example, Figure 7.9(b) shows the utility functions
and
U(x) = 0 + 0.4x − 0.04x 2
U(x) = 0 + 0.25x − 0.01x 2
(where x = monetary sums in millions of dollars) for monetary values
between $0 and $5 million. Of course, not all utility functions have the
quadratic form, but as Markowitz 4 argues, ‘the quadratic nevertheless
shows a surprising flexibility in approximating smooth, concave curves’.
Having considered the different ways in which the results of simulations
can be compared, how was a decision made in the case of
the Elite Pottery Company? First, it was established that the Managing
Director’s utility function for profit was concave (i.e. he was risk