BUSF_A01.qxd

investment decisions 7 Portfolio theory and its relevance to real

7.2 The expected value/variance (or mean/variance) criterion

We saw in Chapter 6 that just knowing the expected value of the possible outcomes of a

project is not an adequate basis on which to make decisions. We need some measure

of how the various possible outcomes are arrayed around the expected or mean value.

A popular statistical measure of dispersion is the variance(or its square root, the

standard deviation). The variance (σ^2 ) is defined as follows:

σ^2 = pi[xi−E(x)]^2

where piis the probability of outcome xi, E(x) its expected value or mean, and there are

npossible outcomes.

n

∑

i= 1

A project will yield one of three NPVs whose amount and probabilities of occurrence are as

follows:

NPV Probability

£

−1,000 0.2

−1,000 0.5

+2,000 0.3

What is the expected NPV of the project and its variance?

Example 7.1

The variance is a particularly useful device for our present purposes. It can be

shown (Levy and Sarnat 1988) that if we assume that the range of possible outcomes

from a decision is distributed symmetrically around the expected value (as with both

of the distributions in Figure 6.7 on page 173), then knowledge of the expected value

and the variance (or standard deviation) is all that risk-averse investors need to enable

them to select between two risky investments. In other words, provided that the

investor is risk-averse and that the dispersion of possible outcomes is symmetrical,

then selecting investments on the basis of their expected (or mean) value and variance

will maximise the investor’s utility of wealth. You may recall from Chapter 6 that, as

far as we can judge, nearly all of us are risk-averse, with some of us more risk-averse

than others.

ENPV =(0.2 ×−1,000) +(0.5 ×1,000) +(0.3 ×2,000)

=£900

s^2 =0.2(−1,000 −900)^2 +0.5(1,000 −900)^2 +0.3(2,000 −900)^2

=(0.2 ×3,610,000) +(0.5 ×10,000) +(0.3 ×1,210,000)

s^2 =1,090,000 (in £^2 )

Solution

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