# Unit 10: Math Problems

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question
Calculating NPV and IRR: A project that provides annual cash flows of \$2,145 for eight years costs \$8,450 today. 1. Is this a good project if the required return is 8 percent? 2. What if it's 24 percent? 3. At what discount rate would you be indifferent between accepting the project and rejecting it?
1. Steps: -a = 2,145 -b = 8 -irr(-8450,a,b) -IRR = 19.13% -The investment should be taken because the rate of return is higher than the required return 2. If it is 24% then the investment should not be taken because the IRR is only 19.13% 3. If the required return was 19.13% then you should be indifferent since the IRR is when the NPV is equal to zero
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Calculating PI: -Year 0: -27,500 -Year 1: 15,800 -Year 2: 13,600 -Year 3: 8,300 1. What is the profitability index for the set of cash flows if the relevant discount rate is 10 percent? 2. What if the discount rate is 15 percent? 3. If it is 22 percent?
1. Steps: -npv(10,0,{15800,13600,8300}) -NPV = 31,839.20 -PI = 31,839.20 / 27,500 = 1.158 2. Steps: -npv(15,0,{15800,13600,8300}) -NPV = 29,480.10 -PI = 29,480.10 / 27,500 = 1.072 3. Steps: -npv(22,0,{15800,13600,8300}) -NPV = 26,659 -PI = 26,659 / 27,500 = 0.969
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Calculating Payback: -Year 0: -60,000 -Year 1: 23,000 -Year 2: 28,000 -Year 3: 19,000 -Year 4: 9,000 Should the company accept this payback if the cutoff is 3 years?
Steps: 1. Year 1 + Year 2 = 51,000 2. 60,000 - 51,000 = 9,000 3. 9,000/19,000 = .47 4. Payback would be after 2.47 years 5. Project should be accepted because the payback would be in 2.47 years which is before the cutoff
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Calculating Payback: -Year 0: -105,000 -Year 1: 21,000 -Year 2: 26,000 -Year 3: 29,000 -Year 4: 260,000 Should the company accept this payback if the cutoff is 3 years?
Steps: 1. Year 1 + Year 2 + Year 3 = 76,000 2. 105,000 - 76,000 = 29,000 3. 29,000/260,000 = .11 4. Payback would be after 3.11 years 5. Project should be rejected since you wouldnt be payed back until 3.11 years but the cutoff is 3 years
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Calculating AAR: The plant has an installation cost of \$12.5 million, which will be depreciated straight-line to zero over its four-year life. If the plant has projected net income of \$1,368,000, \$1,935,000, \$1,738,000, and \$1,310,000 over these four years, what is the project's average accounting return (AAR)?
Steps: 1. Average Net Income = (1,368,000 + 1,935,000 + 1,738,000 + 1,310,000) / 4 = 1,587,750 2. Average Book Value = 12,500,000 / 2 = 6,250,000 3. AAR = Average Net Income / Average Book Value 4. AAR = 1,587,750 / 6,250,000 = 25.40%
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Calculating IRR: Project A Project B -Year 0: -78,500 -78,500 -Year 1: 43,000 21,000 -Year 2: 29,000 28,000 -Year 3: 23,000 34,000 -Year 4: 21,000 41,000 1. What is the IRR for each of these projects? 2. If you apply the IRR decision rule, which project should the company accept? 3. Is this decision necessarily correct?
1. Steps: Project A -a = 43,000, 29,000, 23,000, 21,000 -b = 1, 1, 1, 1 -irr(-78,500,a,b)-> IRR = 20.70% Project B -a = 21,000, 28,000, 34,000, 41,000 -b = 1, 1, 1, 1 -irr(-78500,a,b)-> IRR = 18.73% 2. Project A should be accepted because the IRR is greater 3. This may not be a correct decision, however, because the IRR criterion has a ranking problem for mutually exclusive projects.
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Calculating NPV: Project A Project B -Year 0: -78,500 -78,500 -Year 1: 43,000 21,000 -Year 2: 29,000 28,000 -Year 3: 23,000 34,000 -Year 4: 21,000 41,000 1. If the required return is 11 percent, what is the NPV for each of these projects? 2. Which project will you choose if you apply the NPV decision rule?
1. Steps: Project A -npv(11,-78500,{43000,29000,23000,21000}) -NPV = \$14,426.50 Project B -npv(11,-78500,{21000,28000,34000,41000}) -NPV = \$15,012.80 2. Project B because it has a greater net present value
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NPV versus IRR: Project A Project B -Year 0: -78,500 -78,500 -Year 1: 43,000 21,000 -Year 2: 29,000 28,000 -Year 3: 23,000 34,000 -Year 4: 21,000 41,000 1. Over what range of discount rates would you choose Project A? 2. Project B? 3. At what discount rate would you be indifferent between these two projects?
1. Steps: -You need to subtract the inflows from A by the inflows from B -Year 0: 0, Year 1: 22,000, Year 2: 1,000,Year 3: -11,000, Year 4: -20,000 -irr(0,{22000,1000,-11000,-20000}) irr = 12.21% -At discount rates above 12.21% choose A 2. At discount rates below 12.21% choose B 3. You would be indifferent at a discount rate of 12.21%
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Problems with Profitability Index: Cash Flow 1 Cash Flow 2 -Year 0: -78,000 -28,800 -Year 1: 28,300 9,600 -Year 2: 34,800 17,400 -Year 3: 43,700 15,600 1. If the required return is 11 percent and the company applies the profitability index decision rule, which project should the firm accept? 2. If the company applies the NPV decision rule, which project should it take? 3. Explain why your answers in parts (a) and (b) are different.
1. Steps: Cash Flow 1 -PV of Cash Flows = npv(11,0,{28300,34800,43700}) = 85,693 -PI = PV of Cash Flows / Cost of Investment -PI = 85,693/78,000 = 1.099 Cash Flow 2 -PV of Cash Flows = npv(11,0,{9600,17400,15600}) = 34,177.50 -PI = 34,177.50/28,800 = 1.187 -We should accept project 2 since it has a higher PI 2. Steps Cash Flow 1 -npv(11,-78000,{28300,34800,43700}) -NPV = \$7,693.02 Cash Flow 2 -npv(11,-28800,{9600,17400,15600}) -NPV = \$5,377.46 -We should accept project 1 because the NPV is higher 3. Explanation: -Profitability Index, like IRR, is a relative measure. For standard independent projects NPV, IRR and PI should give the same accept/reject decision. For mutually exclusive projects the relative rankings of IRR and PI may differ from the ranking given by NPV. -Using the profitability index to compare mutually exclusive projects can be ambiguous when the magnitudes of the cash flows for the two projects are of different scale. In this problem, Project I is roughly 2.5 times as large as Project II and produces a larger NPV, yet the profitability index criterion implies that Project II is more acceptable.
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NPV and Profitability Index: The Required Return is 12% Project Y Project Z -Year 0: -43,400 -78,000 -Year 1: 19,800 32,000 -Year 2: 17,500 30,100 -Year 3: 20,700 29,500 -Year 4: 14,600 27,300 1. What is the profitability index for each project? 2. What is the NPV for each project? 3. Which, if either, of the projects should the company accept?
1. Steps: Project Y -Present Value of Cash Flows = npv(12,0,{19800,17500,20700,14600}) = 55,641.90 -PI = 55,641.90/43400 = 1.282 Project Z -Present Value of Cash Flows = npv(12,0,{32000,30100,29500,27300}) = 90,914.10 -PI = 90,914.10/78,000 = 1.166 2. Steps: Project Y -npv(12,-43400,{19800,17500,20700,14600}) -NPV = 12,241.90 Project Z -npv(12,-78000,{32000,30100,29500,27300}) -NPV = 12,914.10 3. The company should accept project Z because it has a greater NPV
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An engineer is considering upgrading four production-lines. She has determined that upgrading all four lines is economically justifiable and proposes to invest the \$64,000 necessary to make these improvements. Her boss declines her request for all of the funds and states she may spend only \$40,000 for improvements this year. Given her discount rate of 8%, what is the highest NPV she can obtain from her total investment? Project Initial NPV Investment A \$12,000 \$1,250 B \$15,000 \$1,150 C \$12,000 \$1,640 D \$25,000 \$2,840
Steps: 1. Project A PI = 13,250/12,000 = 1.10 2. Project B PI = 16,150/15,000 = 1.08 3. Project C PI = 13,640/12,000 = 1.14 4. Project D PI = 27,840/25,000 = 1.11 5. Project A, C, and D have the highest PI 6. Project C+D = \$37,000 leaving \$3,000 for Project A 7. 3,000/12,000 = 25% of project A 8. (1,250*.25) + 1,640 + 2,840 = \$4,792.50
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Our engineer is now considering three new production lines. While each of these lines appear to have a positive NPV, their total cost would be \$39 million. Her capital budget cannot exceed \$30 million. The designs on each line have been optimally engineered, so no further design changes will be possible: these projects are not divisible. Given her discount rate of 8%, what projects should she invest in? Project Initial NPV Investment A \$12 \$1.250 B \$15 \$1.150 C \$12 \$1.640 A. A B. B C. C D. A and B E. A and C F. B and C G. A and B and C
E. A and C Steps: 1. Project A PI = 13,250,000/12,000,000 = 1.10 2. Project B PI = 16,150,000/15,000,000 = 1.08 3. Project C PI = 13,640,000/12,000,000 = 1.14 4. Not Divisible so you should only invest in projects A and C since they have the higher PI's
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Textiles Unlimited has gathered projected cash flows for two projects. Cash Flow A Cash Flow B -Year 0: -105,000 -105,000 -Year 1: 42,200 52,600 -Year 2: 34,600 39,400 -Year 3: 38,700 35,500 -Year 4: 40,500 30,100 1. At what interest rate would the company be indifferent between the two projects? 2. Which project is better if the required return is 12 percent?
1. -4.44% 2. Project B
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The Steel Factory is considering a project that will produce annual cash flows of \$43,800, \$40,200, \$46,200, and \$41,800 over the next four years, respectively. What is the internal rate of return if the initial cost of the project is \$127,900?
13.00%
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You are considering an investment for which you require a rate of return of 8.5 percent. The investment costs \$67,400 and will produce cash inflows of \$25,720 for three years. 1. Should you accept this project based on its internal rate of return? 2. Why or why not?
1. No 2. Because the IRR is 7.08 percent
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The Tool Box needs to purchase a new machine costing \$1.46 million. Management is estimating the machine will generate cash inflows of \$223,000 the first year and \$600,000 for the following three years. 1. If management requires a minimum 12 percent rate of return, should the firm purchase this particular machine based on its IRR? 2. Why or why not?
1. Yes 2. Because the IRR is 12.74 percent
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What is the IRR of the following set of cash flows? -Year 0: -61,300 -Year 1: 18,900 -Year 2: 64,500 -Year 3: 7,600 A. 14.90 percent B. 17.78 percent C. 23.86 percent D. 12.93 percent E. 16.33 percent
C. 23.86 percent
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Consider the following cash flows: -Year 0: -32,500 -Year 1: 14,800 -Year 2: 16,900 -Year 3: 12,200 1. What is the NPV at a discount rate of zero percent? 2. What is the NPV at a discount rate of 11 percent? 3. What is the NPV at a discount rate of 21 percent? 4. What is the NPV at a discount rate of 31 percent?
1. \$11,400 2. \$3,470.29 3. -\$1,839.09 4. -\$5,927.55
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Consider the following two mutually exclusive projects: Cash Flow (X) Cash Flow (Y) -Year 0: -15,900 -15,900 -Year 1: 6,710 7,290 -Year 2: 7,290 7,730 -Year 3: 4,810 3,630 1. What is the IRR of Project X? 2. What is the IRR of Project Y? 3. What is the crossover rate for these two projects?