Solutions to Tutorial week 2: CQ5.4; QP5.12; QP5.16; QP6.10; QP6.14; QP6.18 Note: You might see small differences for some solutions, it is because I’m using excel spread sheet and PVIFA formula to do the calculations. And you may have a little bit different answers when you used calculator. Chapter 5 CQ 5.4. A project has perpetual cash flows of C per period, a cost of I, and a required return of R. What is the relationship between the project’s payback and its IRR? What implications does your answer have for longlived projects with relatively constant cash flow? Solution: For a project with future cash flows that are an annuity: Payback = I / C And the IRR is: 0 = – I + C / IRR Solving the IRR equation for IRR, we get: IRR = C / I Notice this is just the reciprocal of the payback. So: IRR = 1 / PB For long-lived projects with relatively constant cash flows, the sooner the project pays back, the greater is the IRR, and the IRR is approximately equal to the reciprocal of the payback period.
QP5.12. Problems with profitability index. The Robb Computer Corporation is trying to choose between the following two mutually exclusive design projects: Year Cash Flow (I) Cash Flow (II) 0 -$30,000 -$12,000 1 18,000 7,500 2 18,000 7,500 3 18,000 7,500 a. If the required return is 10 percent and Robb Computer 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 our answer in (a) and (b) are different? Solution: 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 = $18,000(PVIFA10%,3 ) / $30,000 = 1.493 PIII = $7,500(PVIFA10%,3) / $12,000 = 1.555 The profitability index decision rule implies that we accept project II, since PIII is greater than the PII. b. The NPV of each project is: NPVI = – $30,000 + $18,000(PVIFA10%,3) = $14,777.16 NPVII = – $12,000 + $7,500(PVIFA10%,3) = $6,657.15 The NPV decision rule implies accepting Project I, since the NPV I is greater than the NPVII. 2
c. 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 scales. In this problem, project I is roughly 3 times as large as project II and produces a larger NPV, yet the profit-ability index criterion implies that project II is more acceptable. QP5.16. Comparing investment criteria. Consider the following cash flows if two mutually exclusive projects for AZ-Motorcars. Assume the discount rate for AZ-Motorcars is 10 percent. Year AZM Mini-SUV AZF Full-SUV 0 -$450,000 -$800,000 1 320,000 350,000 2 180,000 420,000 3 150,000 290,000 a. b. c. d.
Based on the payback period, which project should be accepted? Based on the NPV, which project should be accepted? Based on the IRR, which project should be accepted? Based on this analysis, is incremental IRR analysis necessary? If yes, please conduct the analysis.
Solution: a. The payback period is the time that it takes for the cumulative undiscounted cash inflows to equal the initial investment. AZM Mini-SUV: Cumulative cash flows Year 1 = $320,000 = $320,000 Cumulative cash flows Year 2 = $320,000 + 180,000 = $500,000 Payback period = 1+ $130,000 / $180,000 = 1.72 years AZF Full-SUV: Cumulative cash flows Year 1 = $350,000 = $350,000 Cumulative cash flows Year 2 = $350,000 + 420,000 = $770,000 Cumulative cash flows Year 3=$350,000+420,000+290,000=$1,060,000 Payback period = 2+ $30,000 / $290,000 = 2.1 years 3
Since the AZM has a shorter payback period than the AZF, the company should choose the AZM. Remember the payback period does not necessarily rank projects correctly. b. The NPV of each project is: NPVAZM = –$450,000 + $320,000 / 1.10 + $180,000 / 1.102 + $150,000 / 1.103 NPVAZM = $102,366 NPVAZF = –$800,000 + $350,000 / 1.10 + $420,000 / 1.102 + $290,000 / 1.103 NPVAZF = $83,170 The NPV criteria implies we accept the AZM because it has the highest NPV. c. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the AZM is: 0 = –$450,000 + $320,000 / (1 + IRR) + $180,000 / (1 + IRR)2 + $150,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRAZM = 24.65% And the IRR of the AZF is: 0 = –$800,000 + $350,000 / (1 + IRR) + $420,000 / (1 + IRR)2 + $290,000 / (1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRRAZF = 15.97% 4
The IRR criteria implies we accept the AZM because it has the highest IRR. Remember the IRR does not necessarily rank projects correctly. d. Incremental IRR analysis is not necessary. The AZM has the smallest initial investment, and the largest NPV, so it should be accepted.
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Chapter 6 QP 6.10. Calculating EAC. You are evaluating two different silicon wafer milling machines. The Techron I costs $215,000, has a threeyear life, and has pretax operating costs of $35,000 per year. The Techron II costs $270,000, has a five-year life, and has pretax operating costs of $44,000 per year. For both milling machines, use straight-line depreciation to zero over the project’s life and assume a salvage value of $20,000. If your tax rate is 35 percent and your discount rate is 12 percent, compute the EAC for both machine. Which do you prefer? Why Solution: We will need the aftertax salvage value of the equipment to compute the EAC. Even though the equipment for each product has a different initial cost, both have the same salvage value. The aftertax salvage value for both is: Both cases: aftertax salvage value = $20,000(1 – 0.35) = $13,000 To calculate the EAC, we first need the OCF and NPV of each option. The OCF and NPV for Techron I is: OCF = – $35,000(1 22,750+25,083=$2,333
–
0.35)
+
0.35($215,000/3)
=
$-
NPV = –$215,000 + $2,333(PVIFA12%,3) + ($13,000/1.123) = -215,000+5,603+9,253=-$200,144 EAC = –$200,144 / (PVIFA12%,3) = –$83,329 And the OCF and NPV for Techron II is: OCF = – $44,000(1 – 0.35) + 0.35($270,000/5) = –$28,600+18,900=$9,700 NPV = –$270,000 – $9,700(PVIFA12%,5) + ($13,000/1.125) = -270,000-34,966+7377=-$297,589 EAC = –$297,589 / (PVIFA12%,5) = –$82,554 6
The two milling machines have unequal lives, so they can only be compared by expressing both on an equivalent annual basis, which is what the EAC method does. Thus, you prefer the Techron II because it has the lower (less negative) annual cost. QP6.14. Comparing mutually exclusive projects. Vandalay Industries is considering the purchase of a new machine for the production of latex. Machine A costs $2,900,000 and will last for six years. Variable costs are 35 percent of sales, and fixed costs are $195,000 per year. Machine B costs $5,700,000 and will last for nine years. Variable costs for this machine are 30 percent and fixed costs are $165,000 per year. The sales for each machine will be $12 million per year. The required return is 10 percent and the tax rate is 35 percent. Both machines will be depreciated on a straight-line basis. If the company plans to replace the machine when it wears out on a perpetual basis, which machine should you choose? Solution: Since we need to calculate the EAC for each machine, sales are irrelevant. EAC only calculates the costs of operating the equipment, not the sales. Using the bottom up approach, or net income plus depreciation, method to calculate OCF, we get: Variable costs Fixed costs Depreciation EBT Tax Net income + Depreciation OCF
Machine A –$4,200,000 –195,000 –483,333 –$4,878,333 1,707,417 –$3,170,916 483,333 –$2,687,583
Machine B –$3,600,000 –165,000 –633,333 –$4,398,333 1,539,417 –$2,858,916 633,333 –$2,225,583
The NPV and EAC for Machine A is: NPVA = –$2,900,000 – $2,687,583(PVIFA10%,6) NPVA = –$14,605,123 EACA = – $14,605,123 / (PVIFA10%,6) EACA = –$3,353,445 And the NPV and EAC for Machine B is: 7
NPVB = –$5,700,000 – 2,225,583(PVIFA10%,9) NPVB = –$18,517,133 EACB = – $18,517,133 / (PVIFA10%,9) EACB = –$3,215,338 You should choose Machine B since it has a less negative EAC.
QP6.18. Cash flow valuation. Phillips Industries runs a small manufacturing operation. For this fiscal year, it expects real net cash flows of $190,000. Phillips is an ongoing operation but it expects competitive pressures to erode its real net cash flows at 4 percent per year in perpetuity. The appropriate real discount rate for Phillips is 11 percent. All net cash flows are received at year-end. What is the present value of the net cash flows from Phillips’s operations? Solution: The present value of the company is the present value of the future cash flows generated by the company. Here we have real cash flows, a real interest rate, and a real growth rate. The cash flows are a growing perpetuity, with a negative growth rate. Using the growing perpetuity equation, the present value of the cash flows are: PV = C1 / (R – g) PV = $190,000 / [.11 – (–.04)] PV = $1,266,667
Chapter 7: QP7.1, QP7.4, QP7.8. Chapter 10: QP10.18, QP10.20, QP10.23. Chapter 7
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QP 7.1. Sensitivity analysis and break-even point. We are evaluating a project that costs $644,000, has an eight-year life, and has no salvage value. Assume that depreciation is straight-line to zero over the life of the project. Sales are projected at 70,000 units per year. Price per unit is $37, variable cost per unit is $21, and fixed costs are $725,000 per year. The tax rate is 35 percent, and we require a 15 percent return on this project. a. Calculate the accounting break-even point. b. Calculate the base-case cash flow and NPV. What is the sensitivity of NPV to changes in the sales figure? Explain what your answer tells you about a 500-unit decrease in projected sales. c. What is the sensitivity of OCF to changes in the variable cost figure? Explain what your answer tells you about a $1 decrease in estimated variable costs. a. To calculate the accounting breakeven (when Net Income=0), we first need to find the depreciation for each year. The depreciation is: Depreciation = $644,000/8 Depreciation = $80,500 per year And the accounting breakeven is: (P-v) QA-FC-D=NI=0 QA = ($725,000 + 80,500)/($37 – 21) QA = 50,344 units b. We will use the tax shield approach to calculate the OCF. The OCF is: OCFbase = [(P – v)Q – FC](1 – tc) + tcD OCFbase = [($37 – 21)(70,000) – $725,000](0.65) + 0.35($80,500) OCFbase = $284,925 Now we can calculate the NPV using our base-case projections. There is no salvage value or NWC, so the NPV is: NPVbase = –$644,000 + $284,925(PVIFA15%,8) NPVbase = $634,550.08 To calculate the sensitivity of the NPV to changes in the quantity sold, we will calculate the NPV at a different quantity. We will use sales of 71,000 units (you can create any new level as you 9
like, e.g., 80,000 units, 69,000 units or whatever, it won’t affect the sensitivity analysis). The OCF at this sales level is: OCFnew = [($37 – 21)(71,000) – $725,000](0.65) + 0.35($80,500) OCFnew = $295,325 And the NPV is: NPVnew = –$644,000 + $295,325(PVIFA15%,8) NPVnew = $681,218.22 So, the change in NPV for every unit change in sales is: NPV/S = ($634,550.08 – 681,218.22)/(70,000 – 71,000) NPV/S = +$46.668 If sales were to drop by 500 units, then NPV would drop by: NPV drop = $46.668(500) = $23,334.07 You may wonder why we chose 71,000 units. Because it doesn’t matter! Whatever sales number we use, when we calculate the change in NPV per unit sold, the ratio will be the same. c. To find out how sensitive OCF is to a change in variable costs, we will compute the OCF at a variable cost of $22. Again, the number we choose to use here is irrelevant: We will get the same ratio of OCF to a one dollar change in variable cost no matter what variable cost we use. So, using the tax shield approach, the OCF at a variable cost of $22 is: OCFnew = [($37 – 22)(70,000) – 725,000](0.65) + 0.35($80,500) OCFnew = $239,425 So, the change in OCF for a $1 change in variable costs is: OCF/v = ($284,925 – 239,425)/($21 – 22) OCF/v = –$45,500 If variable costs decrease by $1 then, OCF would increase by $45,500
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QP 7.4. Financial break-even. L.J.’s Toys Inc. just purchased a $390,000 machine to produce toy cars. The machine will be fully depreciated by the straight-line method over its five-year economic life. Each toy sells for $25. The variable cost per toy is $11, and the firm incurs fixed costs of $280,000 each year. The corporate tax rate for the company is 34 percent. The appropriate discount rate is 12 percent. What is the financial break-even point for the project? When calculating the financial breakeven point (when NPV=0), we express the initial investment as an equivalent annual cost (EAC). Dividing the initial investment by the five-year annuity factor, discounted at 12 percent, the EAC of the initial investment is: EAC = Initial Investment / PVIFA12%,5 EAC = $390,000 / 3.60478 EAC = $108,189.80 Note that this calculation solves for the annuity payment with the initial investment as the present value of the annuity. In other words: PVA = C({1 – [1/(1 + R)]t } / R) $390,000 = C{[1 – (1/1.12)5 ] / .12} C = $108,189.80 The annual depreciation is the cost of the equipment divided by the economic life, or: Annual depreciation = $390,000 / 5 Annual depreciation = $78,000 Now we can calculate the financial breakeven point. The financial breakeven point for this project is: QF = [EAC + FC(1 – tc) – D(tc)] / [(P – VC)(1 – tc)] QF = [$108,189.80 + $280,000(1 – 0.34) – $78,000(0.34)] / [($25 – 11)(1 – 0.34)] QF = 28,838.72 or about 28,839 units QP 7.8. Decision trees. B&B has a new baby powder ready to market. If the firm goes directly to the market with the product, there is only a 55 percent chance of success. However, the firm can conduct customer 11
segment research, which will take a year and cost $1.2 million. By going through research, B&B will be able to better target potential customers and will increase the probability of success to 70 percent. If successful, the baby powder will bring a present value profit (at time of initial selling) of $19 million. If unsuccessful, the present value payoff is only $6 million. Should the firm conduct customer segment research or go directly to market? The appropriate discount rate is 15 percent. The company should analyze both options, and choose the option with the greatest NPV. So, if the company goes to market immediately, the NPV is: NPV = CSuccess (Prob. of Success) + CFailure (Prob. of Failure) NPV = $19,000,000(.55) + $6,000,000(.45) NPV = $13,150,000 Customer segment research requires a $1.2 million cash outlay. Choosing the research option will also delay the launch of the product by one year. Thus, the expected payoff is delayed by one year and must be discounted back to Year 0. So, the NPV of the customer segment research is: NPV= C0 + {[CSuccess (Prob. of Success)] + [CFailure (Prob. of Failure)]} / (1 + R)t NPV = –$1,200,000 + {[$19,000,000 (0.70)] + [$6,000,000 (0.30)]} / 1.15 NPV = $11,930,434.78 The company should go to market now since it has the largest NPV Graphically, the decision tree for the project is:
Research $11.93 million at t = 0 Start
Success $19 million at t = 1 Failure $6 million at t = 1 Success
No Research
$19 million at t = 0
$13.15 million at t = 0 Failure $6 million at t = 0 12
Chapter 10
QP 10. 18. Refer back to Table 10.2. What range of returns would you expect to see 68 percent of the time for large-company stocks? What about 95 percent of the time? Looking at the large-company stock return history in Table 10.2, we see that the mean return was 11.8 percent, with a standard deviation of 20.3 percent. The range of returns you would expect to see 68 percent of the time is the mean plus or minus 1 standard deviation, or: RÎ m ± 1s = 11.8% ± 20.3% = –8.50% to 32.10% The range of returns you would expect to see 95 percent of the time is the mean plus or minus 2 standard deviations, or: RÎ m ± 2s = 11.8% ± 2(20.3%) = –28.80% to 52.40%
QP 10. 20. A stock has had returns of 27 percent, 12 percent, 32 percent, -12 percent, 19 percent, and -31 percent over the last six years. What are the arithmetic and geometric returns for the stock? The arithmetic average return is the sum of the known returns divided by the number of returns, so: Arithmetic average return = (.27 + .12 + .32 –.12 + .19 –.31) / 6 Arithmetic average return = .0783, or 7.83% Using the equation for the geometric return, we find: Geometric average return = [(1 + R1) × (1 + R2) × … × (1 + RT)]1/T – 1 Geometric average return = [(1 + .27)(1 + .12)(1 + .32)(1 – .12)(1 + . 19)(1 – .31)](1/6) – 1 Geometric average return = .0522, or 5.22% Remember, the geometric average return will always be less than the arithmetic average return if the returns have any variation. If the 13
returns are constant for each period the geometric average will be equalled to arithmetic average.
QP 10. 23. You bought one of Bergen Manufacturing Co.’s 7 percent coupon bonds one year ago for $1,080.50. These bonds make annual payments and mature six years from now. Suppose you decide to sell your bonds today when the required return on the bonds is 5.5 percent. If the inflation rate was 3.2 percent over the past year, what would be your total real return on the investment? To find the return on the coupon bond, we first need to find the price of the bond today. Since one year has elapsed, the bond now has six years to maturity, so the price today is: P1 = $70(PVIFA5.5%,6) + $1,000/1.0556 P1 = $1,074.93 You received the coupon payments on the bond, so the nominal return was: R = ($1,074.93 – 1,080.50 + 70) / $1,080.50 R = .0596, or 5.96% And using the Fisher effect equation to find the real return, we get: r = (1.0596 / 1.032) – 1 r = .0268, or 2.68%
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