Finance Chapter 9 Solutions

1. Calculating Payback What is the payback period for the following set of cash flows? To calculate the payback period, we need to find the time that the project requires to recover its initial investment. After three years, the project has created: $2,100 + 3,000 + 2,300 = $7,400 in cash flows. The project still needs to create another: $8,300 – 7,400 = $900 in cash flows. During the fourth year, the cash flows from the project will be $1,700. So, the payback period will be three years, plus what we still need to make divided by what we will make during the fourth year. The payback period is: Payback = 3 + ($900/$1,700) Payback = 3.53 years 2. Calculating Payback An investment project provides cash inflows of $745 per year for eight years. What is the project payback period if the initial cost is $1,700? What if the initial cost is $3,300? What if it is $6,100? To calculate the payback period, we need to find the time that the project requires to recover its initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial cost is $1,700, the payback period is: Payback = 2 + ($210/$745) Payback = 2.28 years There is a shortcut to calculate payback period when the project cash flows are an annuity. Just divide the initial cost by the annual cash flow. For the $3,300 cost, the payback period is: Payback = $3,300/$745 Payback = 4.43 years The payback period for an initial cost of $6,100 is a little trickier. Notice that the total cash inflows after eight years will be: Total cash inflows = 8($745) Total cash inflows = $5,960 If the initial cost is $6,100, the project never pays back. Notice that if you use the shortcut for annuity cash flows, you get: Payback = $6,100/$745 Payback = 8.19 years This answer does not make sense since the cash flows stop after eight years, so again, we must conclude the payback period is never. 5. Calculating Discounted Payback An investment project costs $17,000 and has annual cash flows of $4,700 for six years. What is the discounted payback period if the discount rate is zero percent? What if the discount rate is 5 percent? If it is 19 percent? R = 0%: 3 + ($2,900/$4,700) = 3.62 years Discounted payback = Regular payback = 3.62 years R = 5%: $4,700/1.05 + $4,700/1.052 + $4,700/1.053 + $4,700/1.054 = $16,665.97 $4,700/1.055 = $3,682.57 Discounted payback = 4 + ($17,000 – 16,665.97)/$3,682.57 Discounted payback = 4.09 years R = 19%: $4,700(PVIFA19%,6) = $16,025.95 The project never pays back. 6. Calculating AAR You’re trying to determine whether to expand your business by building a new manufacturing plant. The plant has an installation cost of $13.5 million, which will be depreciated straight-line to zero over its four-year life. If the plant has projected net income of $1,570,000, $1,684,200, $1,716,300, and $1,097,400 over these four years, respectively, what is the project’s average accounting return (AAR)? Our definition of AAR is the average net income divided by the average book value. The average net income for this project is: Average net income = ($1,570,000 + 1,684,200 + 1,716,300 + 1,097,400)/4 Average net income = $1,516,975 And the average book value is: Average book value = ($13,500,000 + 0)/2 Average book value = $6,750,000 So, the AAR for this project is: AAR = Average net income/Average book value AAR = $1,516,975/$6,750,000 AAR = .2247, or 22.47% 7. Calculating IRR [LO5] A firm evaluates all of its projects by applying the IRR rule. If the required return is 14 percent, should the firm accept the following project? The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = –$34,000 + $15,000/(1+IRR) + $17,000/(1+IRR)2 + $13,000/(1+IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 15.80% Since the IRR is greater than the required return, we would accept the project. 9. CF0 =−34,00 0CF1=15,000CF2 =17 , 000CF3 =13 , 00 0 IRR=15.80 % Calculating NPV and IRR [LO1, 5] A project that provides annual cash flows of $11,700 for nine years costs $63,000 today. Is this a good project if the required return is 8 percent? What if it’s 20 percent? At what discount rate would you be indifferent between accepting the project and rejecting it? The NPV of a project is the PV of the inflows minus the PV of the outflows. Since the cash inflows are an annuity, the equation for the NPV of this project at an 8 percent required return is: NPV = –$63,000 + $11,700(PVIFA8%, 9) NPV = $10,088.59 CF0 =−63 , 00 0CF1=11 , 7 00 F 01: 9I=8 NPV =$ 10,088.59 At an 8 percent required return, the NPV is positive, so we would accept the project. The equation for the NPV of the project at a 20 percent required return is: NPV = –$63,000 + $11,700(PVIFA20%, 9) NPV = –$15,837.69 CF0 =−63,00 0CF1=11,700 F 01: 9 I=20 NPV =−$ 15,837.69 At a 20 percent required return, the NPV is negative, so we would reject the project. We would be indifferent to the project if the required return was equal to the IRR of the project, since at that required return the NPV is zero. The IRR of the project is: 0 = –$63,000 + $11,700(PVIFAIRR, 9) IRR = 11.72% CF0 =−63,00 0CF1=11,700 F 01: 9 IRR=11.72 10. Calculating IRR What is the IRR of the following set of cash flows? The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = –$15,400 + $7,300/(1 + IRR) + $9,100/(1 + IRR)2 + $5,900/(1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 21.77% CF0 =−15,40 0CF1=7,300CF2 =9,100CF3 =5,900IRR=21.77 % 11. Calculating NPV For the cash flows in the previous problem, what is the NPV at a discount rate of zero percent? What if the discount rate is 10 percent? If it is 20 percent? If it is 30 percent? The NPV of a project is the PV of the inflows minus the PV of the outflows. At a zero discount rate (and only at a zero discount rate), the cash flows can be added together across time. So, the NPV of the project at a zero percent required return is: NPV = –$15,400 + 7,300 + 9,100 + 5,900 NPV = $6,900 CF0 =−15,40 0CF1=7,300CF2 =9,100CF3 =5,900I =0NPV =6,900 The NPV at a 10 percent required return is: NPV = –$15,400 + $7,300/1.1 + $9,100/1.12 + $5,900/1.13 NPV = $3,189.78 CF0 =−15,40 0CF1=7,300CF2 =9,100CF3 =5,900I =10NPV =3,189.78 The NPV at a 20 percent required return is: NPV = –$15,400 + $7,300/1.2 + $9,100/1.22 + $5,900/1.23 NPV = $417.13 CF0 =−15,40 0CF1=7,300CF2 =9,100CF3 =5,900I =20NPV =417.13 And the NPV at a 30 percent required return is: NPV = –$15,400 + $7,300/1.3 + $9,100/1.32 + $5,900/1.33 NPV = –$1,714.52 CF0 =−15,40 0CF1=7,300CF2 =9,100CF3 =5,900I =30NPV =−1,714.52 Notice that as the required return increases, the NPV of the project decreases. This will always be true for projects with conventional cash flows. Conventional cash flows are negative at the beginning of the project and positive throughout the rest of the project. 12. NPV versus IRR Bruin, Inc., has identified the following two mutually exclusive project a. What is the IRR for each of these projects? Using the IRR decision rule, which project should the company accept? Is this decision necessarily correct? b. If the required return is 11 percent, what is the NPV for each of these projects? Which project will the company choose if it applies the NPV decision rule? c. Over what range of discount rates would the company choose Project A? Project B? At what discount rate would the company be indifferent between these two projects? Explain. a. The IRR is the interest rate that makes the NPV of the project equal to zero. The equation for the IRR of Project A is: 0 = –$37,500 + $17,300/(1+IRR) + $16,200/(1+IRR)2 + $13,800/(1+IRR)3 + $7,600/(1+IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 19.71% CF0 =−37,50 0CF1=17,300CF2 =16,200CF3 =13,800CF 4=7,600IRR=19.71% The equation for the IRR of Project B is: 0 = –$37,500 + $5,700/(1+IRR) + $12,900/(1+IRR)2 + $16,300/(1+IRR)3 + $27,500/(1+IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 18.76% CF0 =−37,50 0CF1=5,7 00CF2 =12,900CF3 =16,300CF 4=27,500IRR=18.76 % Examining the IRRs of the projects, we see that IRR A is greater than IRRB, so the IRR decision rule implies accepting Project A. This may not be a correct decision however, because the IRR criterion has a ranking problem for mutually exclusive projects. To see if the IRR decision rule is correct or not, we need to evaluate the project NPVs. b. The NPV of Project A is: NPVA = –$37,500 + $17,300/1.11+ $16,200/1.112 + $13,800/1.113 + $7,600/1.114 NPVA = $6,330.67 CF0 =−37,50 0CF1=17,300CF2 =16,200CF3 =13,800CF 4=7,600I =11NPV =6,330.67 And the NPV of Project B is: NPVB = –$37,500 + $5,700/1.11 + $12,900/1.112 + $16,300/1.113 + $27,500/1.114 NPVB = $8,138.59 CF0 =−37,50 0CF1=5,7 00CF2 =12,900CF3 =16,300CF 4=27,500IRR=11NPV =8,138.59 The NPVB is greater than the NPVA, so we should accept Project B. c. To find the crossover rate, we subtract the cash flows from one project from the cash flows of the other project. Here, we will subtract the cash flows for Project B from the cash flows of Project A. Once we find these differential cash flows, we find the IRR. The equation for the crossover rate is: Crossover rate: 0 = $11,600/(1 + R) + $3,300/(1 + R)2 – $2,500/(1 + R)3 – $19,900/(1 + R)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: R = 16.48% ∆ CF 0=0∆ CF 1=11,600∆ CF 2=3,300∆ CF 3=2,500∆ CF 4=19,900IRR=16.48(Crossover Rate) At discount rates above 16.48 percent choose Project A; for discount rates below 16.48 percent choose Project B; indifferent between A and B at a discount rate of 16.48 percent. 16. Problems with Profitability Index [LO1, 7] The Sloan Corporation is trying to choose between the following two mutually exclusive design projects: a. If the required return is 10 percent and the company applies the profitability index decision rule, which project should the firm accept? b. If the company applies the NPV decision rule, which project should it take? c. Explain why your answers in (a) and (b) are different. a. The profitability index is the PV of the future cash flows divided by the initial investment. The cash flows for both projects are an annuity, so: PII = $28,900(PVIFA10%,3)/$63,000 PII = 1.141 CF0 =0CF1=28,900 F 01:3 I =10NPV =71,883 P.I. = 71,833 / 63,000 = 1.141 PIII = $7,900(PVIFA10%,3)/$15,500 PIII = 1.267 CF0 =0CF1=7,900 F 01: 3 I =10 NPV =19,639 P.I. = 19,639 / 15,500 = 1.267 The profitability index decision rule implies that we accept Project II, since PI II is greater than PII. b. The NPV of each project is: NPVI = –$63,000 + $28,900(PVIFA10%,3) NPVI = $8,870.02 CF0 =−63,00 0CF1=28,900 F 01: 3 I =10NPV =8,870.02 NPVII = –$15,500 + $7,900(PVIFA10%,3) NPVII = $4,146.13 CF0 =−15,50 0CF1=28,900 F 01:3 I =10NPV =4,146.13 The NPV decision rule implies accepting Project I, since the NPVI is greater than the NPVII. c. Using the profitability index to compare mutually exclusive projects can be ambiguous when the magnitude of the cash flows for the two projects are of different scale. In this problem, Project I is roughly four times as large as Project II and produces a larger NPV, yet the profitability index criterion implies that Project II is more acceptable. 19. MIRR [LO6] Solo Corp. is evaluating a project with the following cash flows: **I use the combination approach in the video. I would strongly suggest watching the video in terms of computing the MIRR using the combination approach. I would suggest using the combination approach in working out a possible exam question***. The MIRRs for the project with all three approaches is: Discounting approach: In the discounting approach, we find the value of all negative cash outflows at Time 0, while any positive cash inflows remain at the time at which they occur. So, discounting the cash outflows to Time 0, we find: Time 0 cash flow = –$47,000 – $9,500/1.105 Time 0 cash flow = –$52,898.75 So, the MIRR using the discounting approach is: 0 = –$52,898.75 + $16,900/(1 + MIRR) + $20,300/(1 + MIRR)2 + $25,800/(1 + MIRR)3 + $19,600/(1 + MIRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find: MIRR = 19.68% Reinvestment approach: In the reinvestment approach, we find the future value of all cash except the initial cash flow at the end of the project. So, reinvesting the cash flows to Time 5, we find: Time 5 cash flow = $16,900(1.104) + $20,300(1.103) + $25,800(1.102) + $19,600(1.10) – $9,500 Time 5 cash flow = $95,040.59 So, the MIRR using the reinvestment approach is: 0 = –$47,000 + $95,040.59/(1 + MIRR)5 $95,040.59/$47,000 = (1 + MIRR)5 MIRR = ($95,040.59/$47,000)1/5 – 1 MIRR = .1512, or 15.12% Combination approach: In the combination approach, we find the value of all cash outflows at Time 0, and the value of all cash inflows at the end of the project. So, the value of the cash flows is: Time 0 cash flow = –$47,000 – $9,500/1.105 Time 0 cash flow = –$52,898.75 Time 5 cash flow = $16,900(1.104) + $20,300(1.103) + $25,800(1.102) + $19,600(1.10) Time 5 cash flow = $104,540.59 So, the MIRR using the combination approach is: 0 = –$52,898.75 + $104,540.59/(1 + MIRR)5 $104,540.59/$52,898.75 = (1 + MIRR)5 MIRR = ($104,540.59/$52,898.75)1/5 – 1 MIRR = .1460, or 14.60% 24. Multiple IRRs [LO5] This problem is useful for testing the ability of financial calculators and spreadsheets. Consider the following cash flows. How many different IRRs are there? (Hint: Search between 20 percent and 70 percent.) When should we take this project? The equation for the IRR of the project is: 0 = –$3,024 + $17,172/(1 + IRR) – $36,420/(1 + IRR)2 + $34,200/(1 + IRR)3 – $12,000/(1 + IRR)4 Using Descartes rule of signs, from looking at the cash flows we know there are four IRRs for this project. Even with most computer spreadsheets, we have to do some trial and error. From trial and error, IRRs of 25 percent, 33.33 percent, 42.86 percent, and 66.67 percent are found. We would accept the project when the NPV is greater than zero. See for yourself that the NPV is greater than zero for required returns between 25 percent and 33.33 percent or between 42.86 percent and 66.67 percent.