Questions: Problem 9-15 Calculating Profitability Index [LO7] 1.43 points 8005824 Year Cash r 10 % 0 -815,100 1 10,400 3 9,300 3 5,800 a. What is the profitability index for the set of cash flows if the relevant discount rate is 11 percent? Note: Do not round intermediate calculations and round your answer to 3 decimal places, e.g., 32.161. b. What is the profitability index for the set of cash flows if the relevant discount rate is 16 percent? Note: Do not round intermediate calculations and round your answer to 3 decimal places, e.g., 32.161. What is the profitability index for the set of cash flows if the relevant discount rate is 23 percent? Note: Do not round intermediate calculations and round your answer to 3 decimal places, e.g., 32.161. a. Profitability index b. Profitability index c. Profitability index

Solution Steps

To calculate the profitability index (PI) for a set of cash flows, we need to compute the present value (PV) of future cash flows and divide it by the initial investment. The formula for the present value of cash flows is:

\[ \text{PV} = \sum \frac{\text{Cash Flow}_t}{(1 + r)^t} \]

where \( r \) is the discount rate and \( t \) is the year. The profitability index is then calculated as:

\[ \text{PI} = \frac{\text{PV of future cash flows}}{\text{Initial Investment}} \]

We will calculate the PI for each given discount rate: 11%, 16%, and 23%.

To find the profitability index (PI) at a discount rate of \( r = 0.11 \), we first calculate the present value (PV) of the future cash flows:

\[ \text{PV} = \frac{10,400}{(1 + 0.11)^1} + \frac{9,300}{(1 + 0.11)^2} + \frac{5,800}{(1 + 0.11)^3} \]

Calculating each term:

\[ \text{PV} = \frac{10,400}{1.11} + \frac{9,300}{1.2321} + \frac{5,800}{1.3676} \approx 9,369.369 + 7,548.546 + 4,243.243 \approx 21,161.158 \]

Now, we compute the profitability index:

\[ \text{PI}_{11} = \frac{\text{PV}}{|\text{Initial Investment}|} = \frac{21,161.158}{815,100} \approx 0.026 \]

Next, we calculate the profitability index at a discount rate of \( r = 0.16 \):

\[ \text{PV} = \frac{10,400}{(1 + 0.16)^1} + \frac{9,300}{(1 + 0.16)^2} + \frac{5,800}{(1 + 0.16)^3} \]

Calculating each term:

\[ \text{PV} = \frac{10,400}{1.16} + \frac{9,300}{1.3456} + \frac{5,800}{1.560896} \approx 8,965.517 + 6,903.225 + 3,709.677 \approx 19,578.419 \]

Now, we compute the profitability index:

\[ \text{PI}_{16} = \frac{\text{PV}}{|\text{Initial Investment}|} = \frac{19,578.419}{815,100} \approx 0.024 \]

Finally, we calculate the profitability index at a discount rate of \( r = 0.23 \):

\[ \text{PV} = \frac{10,400}{(1 + 0.23)^1} + \frac{9,300}{(1 + 0.23)^2} + \frac{5,800}{(1 + 0.23)^3} \]

Calculating each term:

\[ \text{PV} = \frac{10,400}{1.23} + \frac{9,300}{1.5129} + \frac{5,800}{1.860867} \approx 8,442.287 + 6,139.344 + 3,110.174 \approx 17,691.805 \]

Now, we compute the profitability index:

\[ \text{PI}_{23} = \frac{\text{PV}}{|\text{Initial Investment}|} = \frac{17,691.805}{815,100} \approx 0.022 \]

Final Answer