## These steps can best be illustrated by Example Example Developing an NPW Probability Distribution

Consider the BMC's transmission-housing project. If the unit sales (X) and unit price (Y) were to vary with the following probabilities, determine the NPW probability distribution. Here we assume the situation where both random variables are independent. In other words, observing a typical outcome for random variable X does not have any influence on predicting the outcome for random variable Y.

Demand (X) Probability Unit Price (Y) Probability

Discussion. If the product demand X and the unit price Y are independent random variables, the PW (15%) will also be a random variable. To determine the NPW distribution, we need to consider all the combinations of possible outcomes. The first possibility is the event where x = 1600 and y = \$48. With these values as input in Table 17.5.1, we compute the resulting NPW outcome to be \$5574. Since X and Y are considered to be independent random variables, the probability of this joint event is

P(x = 1600, y= \$48) = px = 1600)p/ = \$48) = (0.20)(0.30) = 0.06

 Event No. x y P(x,y) Cumulative Joint Probability NPW 1 1600 \$4S.00 0.06 0.06 \$5,574 2 1600 \$50.00 0.10 0.16 \$12,010 3 1600 \$53.00 0.04 0.20 \$21,664 4 2000 \$4S.00 0.1S 0.38 \$32,123 5 2000 \$50.00 0.30 0.68 \$40,168 6 2000 \$53.00 0.12 0.80 \$52,236 7 2400 \$4S.00 0.06 0.86 \$58,672 S 2400 \$50.00 0.10 0.96 \$68,326 9 2400 \$53.00 0.04 1.00 \$82,808
FIGURE 17.5.2 NPW probability distributions: when X and Y are independent (Example 17.5.3).

There are eight other possible joint outcomes. Substituting these pairs of values in Table 17.5.1, we obtain the NPWs and their joint probabilities in Table 17.5.4 and its NPW distribution as depicted in Figure 17.5.2.

Solution. The NPW probability distribution in Table 17.5.4 indicates that the project's NPW varies between \$5574 and \$82,808, but that there is no loss under any of the circumstances examined. From the cumulative distribution, we further observe that there is a 0.38 probability that the project would realize an NPW less than that forecast for the base-case situation (\$40,168). On the other hand, there is a 0.32 probability that the NPW will be greater than this value. Certainly, the probability distribution provides much more information on the likelihood of each possible event, as compared with the scenario analysis presented in Table 17.5.3.