Year Cash Flow0 −$8,300          1 3,900          2 3,300          3 3,100             a. What is the profitability index for the cash flows if the relevant discount rate is 8 percent?   b. What is the profitability index for the cash flows if the relevant discount rate is 13 percent?   c. What is the profitability index for the cash flows if the relevant discount rate is 24 percent?

The profitability index (PI) is calculated as the present value of future cash flows divided by the initial investment.

a. If the discount rate is 8 percent, the PI is calculated as follows:

Year 1: 3900 / (1+0.08)^1 = 3601.85 Year 2: 3300 / (1+0.08)^2 = 2824.31 Year 3: 3100 / (1+0.08)^3 = 2419.26

Sum of these values = 3601.85 + 2824.31 + 2419.26 = 8845.42

Initial investment = 8300

So, PI = 8845.42 / 8300 = 1.0657

b. If the discount rate is 13 percent, the PI is calculated as follows:

Year 1: 3900 / (1+0.13)^1 = 3451.33 Year 2: 3300 / (1+0.13)^2 = 2584.07 Year 3: 3100 / (1+0.13)^3 = 2057.02

Sum of these values = 3451.33 + 2584.07 + 2057.02 = 8092.42

Initial investment = 8300

So, PI = 8092.42 / 8300 = 0.975

c. If the discount rate is 24 percent, the PI is calculated as follows:

Year 1: 3900 / (1+0.24)^1 = 3145.16 Year 2: 3300 / (1+0.24)^2 = 2145.16 Year 3: 3100 / (1+0.24)^3 = 1696.76

Sum of these values = 3145.16 + 2145.16 + 1696.76 = 6987.08

Initial investment = 8300

So, PI = 6987.08 / 8300 = 0.8416

So, the profitability index for the cash flows at 8 percent, 13 percent, and 24 percent discount rates are 1.0657, 0.975, and 0.8416 respectively.