Probability Distribution for PW for ElectroToys’ New Product Development (Example 18.6 continued) ElectroToys’ new product idea has an uncertain first cost for development and uncertain annual net revenue, as shown in the accompanying table. Assume that the first cost occurs at time 0 and that the

Question

Probability Distribution for PW for ElectroToys’ New Product Development (Example 18.6 continued)
ElectroToys’ new product idea has an uncertain first cost for development and uncertain annual net revenue, as shown in the accompanying table. Assume that the first cost occurs at time 0 and that the net annual revenues begin at the end of year 1. ElectroToys uses an interest rate of 12%, and new toys have a life of 5 years. What is the probability distribution for the present worth if the product is developed and maintained for the full 5 years?

P	Net Revenue	P	First Cost
.3	−$5,000	.2	$20,000
.5	10,000	.5	30,000
.2	30,000	.3	50,000

Accepted Answer

The first step is to identify the nine combinations of first cost and net revenue. Each first cost is paired with each net revenue. Exhibit 18.9 summarizes this, along with the probability of each

EXHIBIT 18.9 Probability distribution of PW with two independent variables (ElectroToys’ product development)

P . PW	PW	Joint P	Annual Net Return (P)	First Cost (P)
−$2281	−$38,024	.06	−$5000 (.3)	20,000 (.2)
1605	16,048	.10	10,000 (.5)	20,000 (.2)
3526	88,143	.04	30,000 (.2)	20,000 (.2)
−7204	−48,024	.15	−5000 (.3)	30,000 (.5)
1512	6048	.25	10,000 (.5)	30,000 (.5)
7814	78,143	.10	30,000 (.2)	30,000 (.5)
−6122	−68,024	.09	−5000 (.3)	50,000 (.3)
−2093	−13,952	.15	10,000 (.5)	50,000 (.3)
3489	58,143	.06	30,000 (.2)	50,000 (.3)
Expected value $245

combination. Based on the assumption of statistical independence, the probability of the first cost is multiplied by the probability of the net revenue to find the probability of the combination. For example, the probability of the high first cost and the low revenue is .3 · .3 = .09.
The last column of the table in Exhibit 18.9 is included to show that the original expected value of $245 that was calculated in Example 18.6 is, indeed, exact. These calculations are only valid if the probability distributions for the first cost and the net revenue are independent. The graph in Exhibit 18.9 lists the PWs in increasing order to make the probability distribution clearer.
Exhibit 18.9 shows the advantage of the greater detail in the probability distribution. The decision tree of Exhibit 18.4 correctly increased this example’s PW by canceling the project after 1 year if net annual returns were negative, which had a probability of .3. Exhibit 18.9 shows that the P(loss) or P(PW< 0) is much higher at .45—or nearly one chance in two. The .15 probability that a high first cost and an intermediate annual return will lead to a PW of −$13,952 cannot be avoided. Once the high first cost has occurred, it is a sunk cost. On the other hand, once the negative annual returns have been observed for year 1, they can be avoided for years 2 through 5.

Question 18.15: Probability Distribution for PW for ElectroToys’ New Product...

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