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The Role of Mean-Variance Efficiency

We began the Chapter with an idealized picture of investors (including management) who are rational and risk-averse and formally analyze one course of action in relation to another. What concerns them is not only profitability but also the likelihood of it arising; a risk-return trade-off with which they feel comfortable and that may also be unique.

Thus, in a sophisticated mixed market economy where ownership is divorced from control, it follows that the objective of strategic financial management should be to implement optimum investment-financing decisions using risk-adjusted wealth maximising criteria, which satisfy a multiplicity of shareholders (who may already hold a diverse portfolio of investments) by placing them all in an equal, optimum financial position.

No easy task!

But remember, we have not only assumed that investors are rational but that capital markets are also reasonably efficient at processing information. And this greatly simplifies matters for management. Because today's price is independent of yesterday's price, efficient markets have no memory and individual security price movements are random. Moreover, investors who comprise the market are so large in number that no one individual has a comparative advantage. In the short run, "you win some, you lose some" but long term, investment is a fair game for all, what is termed a "martingale". As a consequence, management can now afford to take a linear view of investor behaviour (as new information replaces old information) and model its own plans accordingly.

What rational market participants require from companies is a diversified investment portfolio that delivers a maximum return at minimum risk.

What management need to satisfy this objective are investment-financing strategies that maximise corporate wealth, validated by simple linear models that statistically quantify the market's risk-return trade-off.

Like Fisher's Separation Theorem, the concept of linearity offers management a lifeline because in efficient capital markets, rational investors (including management) can now assess anticipated investment returns (ri) by reference to their probability of occurrence, (pi) using classical statistical theory.

If the returns from investments are assumed to be random, it follows that their expected return (R) is the expected monetary value (EMV) of a symmetrical, normal distribution (the familiar "bell shaped curve" sketched overleaf). Risk is defined as the variance (or dispersion) of individual returns: the greater the variability, the greater the risk.

Unlike the mean, the statistical measure of dispersion used by the market or management to assess risk is partly a matter of convenience. The variance (VAR) or its square root, the standard deviation (a = VVAR) is used.

When considering the proportion of risk due to some factor, the variance (VAR = a2) is sufficient. However, because the standard deviation (a) of a normal distribution is measured in the same units as (R) the expected value (whereas the variance (a2) only summates the squared deviations around the mean) it is more convenient as an absolute measure of risk.

Moreover, the standard deviation (a) possesses another attractive statistical property. Using confidence limits drawn from a Table of z statistics, it is possible to establish the percentage probabilities that a random variable lies within one, two or three standard deviations above, below or around its expected value, also illustrated below.

The Symmetrical Normal Distribution, Area under the Curveand Confidence Limits

Figure 1.1: The Symmetrical Normal Distribution, Area under the Curveand Confidence Limits

Armed with this statistical information, investors and management can then accept or reject investments according to the degree of confidence they wish to attach to the likelihood (risk) of their desired returns. Using decision rules based upon their optimum criteria for mean-variance efficiency, this implies management and investors should pursue:

- Maximum expected return (R) for a given level of risk, (s).

- Minimum risk (s) for a given expected return (R).

Thus, our conclusion is that if modern capital market theory is based on the following three assumptions:

(i) Rational investors,

(ii) Efficient markets,

(iii) Random walks.

The normative wealth maximisation objective of strategic financial management requires the optimum selection of a portfolio of investment projects, which maximises their expected return (R) commensurate with a degree of risk (s) acceptable to existing shareholders and potential investors.

Activity 3

If you are not familiar with the application of classical statistical formulae to financial theory, read Chapter Four of "Strategic Financial Management" (both the text and exercises) downloadable from

Each chapter focuses upon the two essential characteristics of investment, namely expected return and risk. The calculation of their corresponding statistical parameters, the mean of a distribution and its standard deviation (the square root of the variance) applied to investor utility should then be familiar.

We can then apply simple mathematical notation: (r, p? R, VAR, a and U) to develop a more complex series of ideas throughout the remainder of this text.

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