Forecast data that extends beyond point estimations to multi-valued outcomes may be converted to quasi-risk using the more sophisticated technique of mean-variance analysis. Based on classical probability theory, management assume cash flows are random variables, which conform to a normal distribution with a symmetrical bell-shaped curve as follows:

Figure 4.1: The Symmetrical Normal Distribution, Area under the Curve and Confidence Limits

The mean is derived by first multiplying a spectrum of annual cash flows C by respective probabilities Pi (subject to the proviso that 2Pi =1.0). Then the products C Pi for any number of cash flows (n) are summated to derive an expected monetary value (EMV) at time period t:

Next, the annual EMV series is discounted over the appropriate periods at a risk-free rate (avoiding double-counting) to determine its expected PV, (EPV). From this we subtract the investment cost, Io, to obtain a project's expected NPV, (ENPV) in the usual manner:

Obviously, EMV time-series and ENPV analyses improve upon point estimates. But project selection using ENPV maximisation alone cannot minimise risk because it doesn't calibrate the degree to which cash flows vary around their mean (business risk) or managerial reaction to this variability. To resolve the dilemma, the standard deviation is used to measure the average dispersion of cash flows from their EMV. Management then compare the standard deviation with the expected return to assess a project's risk-return profile; the interpretation being that for a given return, the lower the standard deviation the lower the risk and vice versa.

Assuming statistically normal returns, the standard deviation is determined as follows:

- Calculate the mean of the distribution (EMV) by multiplying each variable's value by its probability of occurrence and adding the products.

- Subtract the EMV from each possible value and square the result.

- Multiply each squared deviation around the mean by its probability to determine certainty equivalents and add them together. This sum is the variance.

- Calculate the square root of the variance. This is the standard deviation

In most texts the standard deviation (SD) is denoted by the Greek letter s or the term OVAR and the variance by a2 and VAR respectively. Using these conventions we can express the risk characteristics of a cash flow distribution algebraically given:

The project variance equals the weighted average of the sum of the squared deviations of each observable cash flow (Ci) from its mean cash flow (EMV) where each weight is represented by the cash flow's probability of occurrence (Pi). Because normal distributions are symmetrical (Figure 4.1) we square the deviations, otherwise their summation would be self-cancelling with a mean deviation of zero. However, squaring also introduces a scale change to the variables in relation to the EMV This is remedied by calculating the square root of the variance to produce the standard deviation, which is a measure of dispersion expressed in identical units to the mean of the distribution (£say).

4.4. Mean-Variance Analyses

To understand the role of mean-variance calculations in capital budgeting, consider the following calculations of EMV, a2 and s for a project's possible contribution per unit.

The first point to note is that best, worst and most likely states of the world all differ from the EMV of the cash flow distribution. Second, the most optimistic outcome is least likely to occur (£8 with a probability of 0.1).There is also a 70 percent chance of cash flows falling short of their EMV. So, what does the standard deviation of £0.943 tell us?

Refer back to Figure 4.1 which sketched the area under the standard normal curve and the probability that a variable's value lies within a number of standard deviations away from the mean. Because these probabilities are the same for any normal distribution they have long been quantified in tables based on the z statistic, which standardizes any variable's actual deviation from the mean by reference to the standard deviation. For a particular cash flow (G) drawn from a distribution with known mean and variance:

We then consult the table to establish the area under the normal curve between the right or left of z (plus or minus) to estimate the probability that the expected cash flow will be a given number of standard deviations away from the mean. Since a normal distribution is symmetrical, the probability of a variable deviating above and below the mean is given by 2z.

Activity 1

Seek out a z table, and with the previous sample data (EMV of £6.10 and s of £0.943) let us establish the probability of project contributions ranging from £6.50 to £5.50.

To determine the probability of contributions deviating above or below the mean as specified, we must first calculate the following z statistics using Equation (5).

£6.10 is (6.10 - 6.10) / 0.943 = zero s from the mean (obviously)

£6.50 is (6.50 - 6.10) / 0.943 = +0.42 s from the mean

£5.50 is (5.50 - 6.10 / 0.943 = -0.64 a from the mean

Next we consult the table for the area under the standard normal curve where z equals zero, 0.42 and 0.64 (i.e. 0.5000, 0.3372 and 0.2611).The mean-z areas are 0.1628 and 0.2389 respectively (i.e. 0.5000-0.3372 and 0.5000-0.2611). Thus, the total area under the curve, between +0.42 and -0.64, equals 0.4017 (i.e. 0.1628+0.2389). So, there is a roughly a 40 percent probability of the contribution ranging from £6.50 to £5.50.

Activity 2

Over many years, surveys reveal that probability analysis has gained ground (see Arnold and Hatzopoulus, 2000) one reason is that the risk-return profile for any normal distribution conveniently conforms to predetermined confidence limits.

Referring to Figure 4.1 and your z table, confirm that the percentage probability of any cash flow lying one, two, or three standard deviations above or below the EMV is given by the following confidence limits:

Applied to our previous Activity, now confirm there is a 99.74 percent probability that the distribution with an EMV of £6.10 and a standard deviation of £0.943 will have a contribution within the following range:

EMV ± 3d = £6.10 + [3 (£0.943)] to £6.10 - [3 (£0.943] = £8.93 to £3.27

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