What are the IRRs for the project What discount rate results in the maximum NPV

What are the irrs for the project what discount rate

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What are the IRRs for the project? What discount rate results in the maximum NPV for this project? How can you determine that this is the maximum NPV? (the highlighted part is NOT required) Ans : The equation for the IRR of the project is: 0 = –$75,000 + $155,000 / (1 + IRR) – $65,000 / (1 + IRR) 2 From Descartes’ Rule of Signs, we know there are either zero IRRs or two IRRs since the cash flows change signs twice. We can rewrite this equation as: 0 = –$75,000 + $155,000X – $65,000X 2 where X = 1 / (1 + IRR) (the highlighted part of the solution is NOT required) This is a quadratic equation. We can solve for the roots of this equation with the quadratic formula: X = b ± b 2 4 ac 2 a Remember that the quadratic formula is written as: 0 = a X 2 + b X + c In this case, the equation is: 0 = –$65,000X 2 + $155,000X – $75,000 X = 155 , 000 ± ( 155,000 ) 2 4 (− 75 , 000 )(− 65 , 000 ) 2 (− 65 , 000 ) X = 155 , 000 ± 4,525,000,000 2 (− 65 , 000 ) X = 155 , 000 ± 67 , 268.12 130 , 000 Solving the quadratic equation, we find two Xs: X = .6749, 1.7098 Since: X = 1 / (1 + IRR) 1.7098 = 1 / (1 + IRR) IRR = –.4151, or – 41.51% 5
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Capital budgeting And: X = 1 / (1 + IRR) .6749 = 1 / (1 + IRR) IRR = .4818, or 48.18% To find the maximum (or minimum) of a function, we find the derivative and set it equal to zero. The derivative of this IRR function is: 0 = –$155,000(1 + IRR) –2 + $130,000(1 + IRR) –3 –$155,000(1 + IRR) –2 = $130,000(1 + IRR) –3 –$155,000(1 + IRR) 3 = $130,000(1 + IRR) 2 –$155,000(1 + IRR) = $130,000 IRR = $130,000 / $155,000 – 1 IRR = – .1613, or –16.13% To determine if this is a maximum or minimum, we can find the second derivative of the IRR function. If the second derivative is positive, we have found a minimum and if the second derivative is negative we have found a maximum. Using the reduced equation above, that is: –$155,000(1 + IRR) = $130,000 The second derivative is –$262,722.18, therefore we have a maximum. 6
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Capital budgeting Chapter 6: 2. Incremental Cash Flows Which of the following should be treated as an incremental cash flow when computing the NPV of an investment? a. A reduction in the sales of a company’s other products caused by the investment. Ans: Yes, the reduction in the sales of the company’s other products, referred to as erosion, should be treated as an incremental cash flow. These lost sales are included because they are a cost (a revenue reduction) that the firm must bear if it chooses to produce the new product. b. An expenditure on plant and equipment that has not yet been made and will be made only if the project is accepted. Ans : Yes, expenditures on plant and equipment should be treated as incremental cash flows. These are costs of the new product line. However, if these expenditures have already occurred (and cannot be recaptured through a sale of the plant and equipment), they are sunk costs and are not included as incremental cash flows. C.Costs of research and development undertaken in connection with the product during the past three years No, the research and development costs should not be treated as incremental cash flows. The costs of research and development undertaken on the product during the past three years are sunk costs and should not be included in the evaluation of the project. Decisions made and costs incurred in the past cannot be changed. They should not affect the decision to accept or reject the project.
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