# Chap7 - Chapter 7 Some Alternative Investment Rules 7.1 a...

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Chapter 7: Some Alternative Investment Rules 7.1 a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. Project A : Cumulative Undiscounted Cash Flows Year 1 = \$4,000 Cumulative Undiscounted Cash Flows Year 2 = \$4,000 +\$3,500 = \$7,500 Payback period = 2 Project A has a payback period of two years. Project B : Cumulative Undiscounted Cash Flows Year 1 = \$2,500 Cumulative Undiscounted Cash Flows Year 2 = \$2,500+\$1,200 = \$3,700 Cumulative Undiscounted Cash Flows Year 3 = \$2,500+\$1,200+\$3,000 = \$6,700 Project B ’s cumulative undiscounted cash flows exceed the initial investment of \$5,000 by the end of year 3. The payback period for project B is 3 years. Companies can calculate a more precise value using fractional years. To calculate the fractional payback period for Project B, find the fraction of year 3’s cash flows that is needed for the company to have cumulative undiscounted cash flows of \$5,000. Divide the difference between the initial investment and the cumulative undiscounted cash flows as of year 2 by the undiscounted cash flow of year 3. Payback period = 2 + (\$5,000 - \$3,700) / \$3,000 = 2.43 Since project A has a shorter payback period than project B has, the company should choose project A . b. Discount each project’s cash flows at 15 percent. Choose the project with the highest NPV. Project A = -\$7,500 + \$4,000 / (1.15) + \$3,500 / (1.15) 2 + \$1,500 / (1.15) 3 = -\$388.96 Project B = -\$5,000 + \$2,500 / (1.15) + \$1,200 / (1.15) 2 + \$3,000 / (1.15) 3 = \$53.83 The firm should choose Project B since it has a higher NPV than Project A has. 7.2 a. After 6 years, the cumulative cash flows will be \$900,000; after 7 years they will be \$1,050,000. Therefore the payback period is 7 years. Answers to End-of-Chapter Problems B-57

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b. Since the discounted payback period will always be greater than the undiscounted payback period when there are positive cash inflows, start the approximation at year 7. Cumulative Discounted Cash Flows Year 7 = \$150,000 A 7 0.1 = \$730,262.82 Cumulative Discounted Cash Flows Year 8 = \$150,000 A 8 0.1 = \$800,238.93 Cumulative Discounted Cash Flows Year 9 = \$150,000 A 9 0.1 = \$863,853.57 Cumulative Discounted Cash Flows Year 10 = \$150,000 A 10 0.1 = \$921,685.07 Cumulative Discounted Cash Flows Year 11 = \$150,000 A 11 0.1 = \$974,259.15 Cumulative Discounted Cash Flows Year 12 = \$150,000 A 12 0.1 = \$1,022,053.77 The cumulative discounted cash flows exceed the initial investment of \$1,000,000 by the end of year 12, so the payback period for the project is 12 years. The discounted payback period is 12 years. c. Apply the perpetuity formula, discounted at 10 percent, to calculate the PV of the annual cash inflows. NPV = -\$1,000,000 + \$150,000 / 0.1 = \$500,000 The NPV of the project is \$500,000. 7.6 The internal rate of return is the discount rate at which the NPV of the project’s cash flows equals zero. Set the project’s cash flows, discounted at the internal rate of return (IRR), equal to zero. Solve for the IRR. IRR(Project A) = C 0 + C 1 / (1+IRR) + C 2 / (1+IRR) 2 0 = -\$3,000 + \$2,500 / (1+IRR) + \$1,000 / (1+IRR) 2 IRR = 0.1287 IRR(Project B) = C 0 + C 1 / (1+IRR) + C 2 / (1+IRR) 2 0 = -\$6,000 + \$5,000 / (1+IRR) + \$2,000 / (1+IRR) 2 IRR = 0.1287 Note that since Project B’s cash flows are two times those of Project A, the IRR’s of both projects are the same.
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