Before we can discuss rules for decision making under risk, we must first discuss how risk can be measured. The most direct method of measuring risk involves the characteristics of a probability distribution of outcomes associated with a particu- lar decision. This section will describe these characteristics.
Probability Distributions
A probability distribution is a table or graph showing all possible outcomes (payoffs) for a decision and the probability that each outcome will occur. The prob- abilities can take values between 0 and 1, or, alternatively, they can be expressed as percentages between 0 and 100 percent.1 If all possible outcomes are assigned probabilities, the probabilities must sum to 1 (or 100 percent); that is, the prob- ability that some other outcome will occur is 0 because there is no other possible outcome.
uncertainty A decision-making condition under which a manager cannot list all possible outcomes andyor cannot assign probabilities to the various outcomes.
probability distribution
A table or graph showing all possible outcomes or payoffs of a decision and the probabilities that each outcome will occur.
1If the probability of an outcome is 1 (or 100 percent), the outcome is certain to occur and no risk exists. If the probability of an outcome is 0, then that particular outcome will not occur and need not be considered in decision making.
To illustrate a probability distribution, we assume that the director of advertis- ing at a large corporation believes the firm’s current advertising campaign may result in any one of five possible outcomes for corporate sales. The probability distribution for this advertising campaign is as follows:
Outcome
(sales) Probability (percent)
47,500 units 10
50,000 units 20
52,500 units 30
55,000 units 25
57,500 units 15
Each outcome has a probability greater than 0 but less than 100 percent, and the sum of all probabilities is 100 percent (5 10 1 20 1 30 1 25 1 15). This probability distribution is represented graphically in Figure 15.1.
From a probability distribution (either in tabular or in graphical form), the riski- ness of a decision is reflected by the variability of outcomes indicated by the dif- ferent probabilities of occurrence. For decision-making purposes, managers often turn to mathematical properties of the probability distribution to facilitate a for- mal analysis of risk. The nature of risk can be summarized by examining the cen- tral tendency of the probability distribution, as measured by the expected value of the distribution, and by examining the dispersion of the distribution, as measured by the standard deviation and coefficient of variation. We discuss first the measure of central tendency of a probability distribution.
F I G U R E 15.1 The Probability Distribution for Sales Following an Advertising Campaign
0.30
50,000 52,500 55,000 57,500 0.20
Probability
0.10
0.20
0.30
0.25
0.15
Sales 47,500
0.10
0
Expected Value of a Probability Distribution
The expected value of a probability distribution of decision outcomes is the weighted average of the outcomes, with the probabilities of each outcome serving as the respective weights. The expected value of the various outcomes of a prob- ability distribution is
E(X) 5 Expected value of X 5 S
i51
n
piXi
where Xi is the ith outcome of a decision, pi is the probability of the ith outcome, and n is the total number of possible outcomes in the probability distribution.
Note that the computation of expected value requires the use of fractions or deci- mal values for the probabilities pi, rather than percentages. The expected value of a probability distribution is often referred to as the mean of the distribution.
The expected value of sales for the advertising campaign associated with the probability distribution shown in Figure 15.1 is
E(sales) 5 (0.10)(47,500) 1 (0.20)(50,000) 1 (0.30)(52,500) 1 (0.25)(55,000) 1 (0.15)(57,500)
5 4,750 1 10,000 1 15,750 1 13,750 1 8,625
5 52,875
While the amount of actual sales that occur as a result of the advertising cam- paign is a random variable possibly taking values of 47,500, 50,000, 52,500, 55,000, or 57,500 units, the expected level of sales is 52,875 units. If only one of the five levels of sales can occur, the level that actually occurs will not equal the expected value of 52,875, but expected value does indicate what the average value of the outcomes would be if the risky decision were to be repeated a large number of times.
Dispersion of a Probability Distribution
As you may recall from your statistics classes, probability distributions are gener- ally characterized not only by the expected value (mean) but also by the variance.
The variance of a probability distribution measures the dispersion of the distribu- tion about its mean. Figure 15.2 shows the probability distributions for the profit outcomes of two different decisions, A and B. Both decisions, as illustrated in Figure 15.2, have identical expected profit levels but different variances. The larger variance associated with making decision B is reflected by a larger dispersion (a wider spread of values around the mean). While distribution A is more compact (less spread out), A has a smaller variance.
The variance of a probability distribution of the outcomes of a given decision is frequently used to indicate the level or degree of risk associated with that deci- sion. If the expected values of two distributions are the same, the distribution with the higher variance is associated with the riskier decision. Thus in Figure 15.2, decision B has more risk than decision A. Furthermore, variance is often used to
expected value The weighted average of the outcomes, with the probabilities of each outcome serving as the respective weights.
mean of the distribution
The expected value of the distribution.
variance
The dispersion of a distribution about its mean.
compare the riskiness of two decisions even though the expected values of the distributions differ.
Mathematically, the variance of a probability distribution of outcomes Xi, de- noted by s2x, is the probability-weighted sum of the squared deviations about the expected value of X
Variance (X ) 5 s2x 5 S
i51
n
pi[Xi 2 E(X )]2
As an example, consider the two distributions illustrated in Figure 15.3. As is evident from the graphs and demonstrated in the following table, the two distri- butions have the same mean, 50. Their variances differ, however. Decision A has a smaller variance than decision B, and it is therefore less risky. The calculation of the expected values and variance for each distribution are shown here:
Decision A Decision B
Profit
(Xi) Probability
(pi) pi Xi [Xi 2 E(X)]2pi Probability
(pi) pi Xi [Xi 2 E(X)]2pi
30 0.05 1.5 20 0.10 3 40
40 0.20 8 20 0.25 10 25
50 0.50 25 0 0.30 15 0
60 0.20 12 20 0.25 15 25
70 0.05 3.5 20 0.10 7 40
E (X ) 5 50 sA2 5 80 E (X ) 5 50 sB2 5 130
Since variance is a squared term, it is usually much larger than the mean. To avoid this scaling problem, the standard deviation of the probability distribution is more commonly used to measure dispersion. The standard deviation of a prob- ability distribution, denoted by sx, is the square root of the variance:
sx 5 ẽ____________
Variance (X) standard deviation
The square root of the variance.
Probability
E (profit)
Distribution A
Distribution B F I G U R E 15.2
Two Probability Distributions with Identical Means but Different Variances
The standard deviations of the distributions illustrated in Figure 15.3 and in the preceding table are sA 5 8.94 and sB 5 11.40. As in the case of the variance of a probability distribution, the higher the standard deviation, the more risky the de- cision.
Managers can compare the riskiness of various decisions by comparing their standard deviations, as long as the expected values are of similar magnitudes.
For example, if decisions C and D both have standard deviations of 52.5, the two decisions can be viewed as equally risky if their expected values are close to one another. If, however, the expected values of the distributions differ substantially in magnitude, it can be misleading to examine only the standard deviations. Sup- pose decision C has a mean outcome of $400 and decision D has a mean outcome of $5,000 but the standard deviations remain 52.5. The dispersion of outcomes for decision D is much smaller relative to its mean value of $5,000 than is the dispersion of outcomes for decision C relative to its mean value of $400.
When the expected values of outcomes differ substantially, managers should measure the riskiness of a decision relative to its expected value. One such measure of relative risk is the coefficient of variation for the decision’s distri- bution. The coefficient of variation, denoted by y, is the standard deviation
coefficient of variation The standard deviation divided by the expected value of the probability distribution.
F I G U R E 15.3
Probability Distributions with Different Variances
30 Profit
40 50 60 70
5%
50%
Distribution A
Probability 20%
20%
5%
30 Profit
40 50 60 70
Distribution B 25%
Probability
10% 10%
25%
30%
divided by the expected value of the probability distribution of decision out- comes
y 5 Standard deviation __________________ Expected value 5 _____ s E(X)
The coefficient of variation measures the level of risk relative to the mean of the probability distribution. In the preceding example, the two coefficients of varia- tion are yC 5 52.5y400 5 0.131 and yD 5 52.5y5,000 5 0.0105.