total project cost= $200mil.

Annual cash flows= $24mil. a year for 20yrs(+or -$4mil, per yr.)
Calculate payback period, NP, profitability index, and estimate the IRR.

To calculate the payback period, NPV, profitability index, and estimate the IRR, we need to step-by-step perform the following calculations:

1. Payback Period:
The payback period is the time it takes for the initial investment to be recovered. In this case, we have annual cash flows of $24 million for 20 years.

To calculate the payback period:
a. Subtract the annual cash flows from the initial investment until the cumulative cash flows become positive.
b. Divide the remaining negative value by the cash flow of the next year.
c. Add the number of years to the whole number portion of the division result to get the payback period.

Example:
Given annual cash flow = $24 million.
Initial investment = $200 million.

Year 1: -$200 million + $24 million = -$176 million
Year 2: -$176 million + $24 million = -$152 million
Year 3: -$152 million + $24 million = -$128 million
Year 4: -$128 million + $24 million = -$104 million
Year 5: -$104 million + $24 million = -$80 million
Year 6: -$80 million + $24 million = -$56 million
Year 7: -$56 million + $24 million = -$32 million
Year 8: -$32 million + $24 million = -$8 million
Year 9: -$8 million + $24 million = $16 million (positive)

The payback period is 8 years + [(-$8 million)/$24 million] = 8.33 years.

2. Net Present Value (NPV):
NPV is the sum of the present value of all cash flows in relation to the discount rate. The discount rate represents the opportunity cost of investing in the project.

To calculate NPV:
a. Calculate the present value of each cash flow by dividing it by (1 + discount rate)^year.
b. Sum up all the present values to obtain the NPV.

Example:
Given annual cash flow = $24 million.
Discount rate = x% (to be determined)

Year 1: $24 million / (1 + x%)^1
Year 2: $24 million / (1 + x%)^2
Year 3: $24 million / (1 + x%)^3
...
Year 20: $24 million / (1 + x%)^20

Add all the calculated present values to get the NPV.

3. Profitability Index:
Profitability index measures the value created per unit of investment. It is calculated by dividing the present value of future cash flows by the initial investment.

To calculate the profitability index:
a. Calculate the present value of each cash flow (as explained in step 2).
b. Divide the sum of present values by the initial investment.

Example:
Profitability index = (Present value of all cash flows) / Initial investment.

4. Estimating the Internal Rate of Return (IRR):
The IRR is the discount rate at which the NPV becomes zero.

To estimate the IRR, you can use a trial-and-error method or financial software. With a trial-and-error approach:
a. Guess an initial discount rate.
b. Calculate the NPV using the guessed discount rate.
c. Adjust the discount rate until the NPV is closest to zero.
d. The final discount rate is the estimated IRR.

Alternatively, you can use financial software or Excel functions to estimate the IRR more efficiently.

Remember, the discount rate is the key factor in calculating NPV, profitability index, and estimating the IRR. It is typically based on the opportunity cost of capital and the risk associated with the project.