The Mean-Variance Paradox

Returning to the normative objective of financial management, if capital is rationed or investments are mutually exclusive and choices must be made, project selection using either the highest expected return or the lowest standard deviation does not necessarily maximise wealth or minimise risk. Consider the risk-return profiles of two projects (A and B):

ENPV selects B but sNPV selects A. But which maximises wealth and which is less risky?

To resolve the paradox, what we need is a relative rather than absolute statistical measure of project variability around its mean value that builds on confidence limits. One lifeline is the coefficient of variation:

This is interpreted as the smaller the coefficient, the lower the risk. Applied to the data:

Thus, Project A seems more risky than Project B because it involves £0.60 of risk ((a NPV) for every £1.00 of ENPV, rather than £0.45 (a NPV) for every £1.00 of ENPV.

The coefficient of variation is important because it tries to encapsulate the fundamental, twin objectives of corporate wealth maximisation, which we can summarise as follows:


MAX: ENPV, given □ NPV (i.e. maximise the ENPV for a given degree of variability)

MIN: □ NPV, given ENPV (i.e. minimise the variability of returns for a given ENPV).

Unfortunately, the coefficient of variation is not a selection criterion because it ignores investment size, thereby assuming that managerial risk attitudes Me constant, even though intuitively we know that rational investors become more risk averse as their stake increases. Add zeros to the previous project data and note that the coefficients remain the same.

To remove this anomaly, management must predetermine a desired minimum ENPV for any investment (Io) expressed as a profitability index to satisfy all their corporate investors.

Then comparison must be made to expected indices for proposed investments, incorporating confidence limits based upon an appropriate number of standard deviations. These define subjective managerial risk attitudes. So, the objective function for project selection becomes:

Activity 3

Assume management require a benchmark [MIN:NPV / I0] = £0.15 to satisfy stakeholders. Use the previous data to derive the left-hand side of Equation (7) three standard deviations from the mean for each project (A and B) that cost £100k and £120k, respectively.

Which project, if either, is acceptable?

Recall that Project B was preferable using the coefficient of variation. Note now, however, that if management require almost complete certainty (99.74%) neither project is acceptable using the expected profitability index, although Project A minimizes losses.

Certainty Equivalence and Investor Utility

The ultimate test of mean-variance analysis depends upon investor risk attitudes. In our previous example, risk aversion signals rejection. Yet risk-seekers (speculators) might actually accept Project B because investing £120k rather than £100k might yield £470k as opposed to £140k three standard deviations above the mean, whilst the equivalent downside loss is only £70k compared with £50k. So, how do we circumvent this risk-return paradox?

One solution is to dispense with the standard deviation altogether and calibrate an individual's subjective attitude towards risk expressed in terms of units of utility, rather than monetary gains and losses, associated with investments. Given this utility function, we then calculate the certain cash equivalent of the distribution of uncertain cash flows discounted at the risk-free rate for any project and assess its viability for the investor using wealth maximisation criteria.

Consider the data set overleaf that signals a project acceptance using Equation 2 as follows:

Consider the data set overleaf that signals a project acceptance using Equation 2 as follows:

Now look at the utility data, which rejects the project. To understand why, assume you ask the investor to enter a game with a 50/50 chance of receiving nothing or £100k to which we attach arbitrary utility values of zero and one respectively. Next you ask what the game is worth. The investor's response is £40k. This represents their indifference between certain cash and the game. Thus, three points on the individual's utility curve associated with certain cash equivalents can be obtained (shown in bold) based on the following equation of indifference:

If the game's entry price was £50k he would walk away. However, other scale points, such as £50k (with a value of 0.65 say) can be established by gaming cash amounts for known utilities. If the procedure is repeated exhaustively, the investor's utility function consistent with his risk attitude will emerge, like the profile plotted in Figure 4.2 overleaf.

The curve's geometry (if not its specific values) applies to any rational investor. Except for small gambles relative to current wealth, it reveals risk aversion, denoted by the convex shape of the function (looking from above). Near the origin, the concave sector denotes risk preference. Note that the utility of one for £100k is only twice that of 0.5 for £40k (which we originally calculated) but more than half the utility of £200k, as risk aversion sets in.

Returning to our example, the application of Equation (8) using the investor's utility curve reveals that despite a positive ENPV the project should be rejected. The utility of its cost exceeds the cash equivalent of the expected utility of the discounted cash flow distribution.

Review Activity

Summarise the problems that confront practicing financial managers who use certainty cash equivalents, rather than mean-variance as a basis for investment appraisal.

The Investor Utility Curve

Figure 4.2: The Investor Utility Curve

Summary and Conclusions

ENPV maximisation using the certainty cash equivalents of expected utilities is more sophisticated than mean-variance analysis because it not only incorporates probabilistic estimates of a project's outcomes but also the investor's risk psychology. But remember:

- Utility functions, like project probability distributions, are subjective, differ from individual to individual, susceptible to change and must be combined (somehow) for group decisions.

- Certainty cash equivalents, like mean-variance analyses, not only depend upon the borrowing and reinvestedsment assumptions of the basic NPV model but must also utilize gains and losses discounted at a risk-free rate to avoid the duplication of risk


Arnold, G.C. and Hatzopoulos, P.D., "The Theory-Practice Gap in Capital Budgeting: Evidence from the United Kingdom", Journal of Business Finance and Accounting, Vol. 25 (5) and (6), June/July, 2000.

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