Ross5eChap07sm - Chapter 7 Net Present Value and Other...

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Chapter 7: Net Present Value and Other 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 cash flows Year 1 = $4,000 = $4,000 Cumulative cash flows Year 2 = $4,000 +3,500 = $7,500 Payback period = 2 years Project B: Cumulative cash flows Year 1 = $2,500 = $2,500 Cumulative cash flows Year 2 = $2,500 + 1,200 = $3,700 Cumulative cash flows Year 3 = $2,500 + 1,200 + 3,000 = $6,700 Companies can calculate a more precise value using fractional years. To calculate the fractional payback period, 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 Payback period = 2.43 years 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: NPV = –$7,500 + $4,000 / 1.15 + $3,500 / 1.15 2 + $1,500 / 1.15 3 NPV = –$388.96 Project B: NPV = –$5,000 + $2,500 / 1.15 + $1,200 / 1.15 2 + $3,000 / 1.15 3 NPV = $53.83 The firm should choose Project B since it has a higher NPV than Project A has. 7.2 To calculate the payback period, we need to find the time that the project has recovered its initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial cost is $3,000, the payback period is: Payback = 3 + ($480 / $840) = 3.57 years Answers to End–of–Chapter Problems B–57
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There is a shortcut to calculate the payback period if the future cash flows are an annuity. Just divide the initial cost by the annual cash flow. For the $3,000 cost, the payback period is: Payback = $3,000 / $840 = 3.57 years For an initial cost of $5,000, the payback period is: Payback = 5 + ($800 / $840) = 5.95 years The payback period for an initial cost of $7,000 is a little trickier. Notice that the total cash inflows after eight years will be: Total cash inflows = 8($840) = $6,720 If the initial cost is $7,000, the project never pays back. Notice that if you use the shortcut for annuity cash flows, you get: Payback = $7,000 / $840 = 8.33 years. This answer does not make sense since the cash flows stop after eight years, so there is no payback period. 7.3 When we use discounted payback, we need to find the value of all cash flows today. The value today of the project cash flows for the first four years is: Value today of Year 1 cash flow = $7,000/1.14 = $6,140.35 Value today of Year 2 cash flow = $7,500/1.14 2 = $5,771.01 Value today of Year 3 cash flow = $8,000/1.14 3 = $5,399.77 Value today of Year 4 cash flow = $8,500/1.14 4 = $5,032.68 To find the discounted payback, we use these values to find the payback period. The discounted first year cash flow is $6,140.35, so the discounted payback for an $8,000 initial cost is: Discounted payback = 1 + ($8,000 – 6,140.35)/$5,771.01 = 1.32 years For an initial cost of $13,000, the discounted payback is: Discounted payback = 2 + ($13,000 – 6,140.35 – 5,771.01)/$5,399.77 = 2.20 years Notice the calculation of discounted payback. We know the payback period is between two and three years, so we subtract the discounted values of the Year 1 and Year 2 cash flows from the initial cost.
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