Solutions to Chapter NPV and Other Investment Criteria NPVA = –100 + 40 × annuity factor(11%, periods) = $24.10 NPVB = –100 + 50 × annuity factor(11%, periods) = $22.19 Both projects are worth pursuing Choose the project with the higher NPV, project A If r = 16%, then NPVA = $11.93 and NPVB = $12.29 Therefore, you should now choose project B IRRA = Discount rate at which 40 × annuity factor(r, periods) = 100 IRRA = 21.86% IRRB = 23.38% No Even though project B has the higher IRR, its NPV is lower than that of project A when the discount rate is lower (as in Problem 1) and higher when the discount rate is higher (as in Problem 3) This example shows that the project with the higher IRR is not necessarily better The IRR of each project is fixed, but as the discount rate increases, project B becomes relatively more attractive compared to project A This is because B’s cash flows come earlier, so their present values fall less rapidly when the discount rate increases The profitability indexes are as follows: Project A Project B 24.10/100 = 2410 22.19/100 = 2219 In this case, with equal initial investments, both the profitability index and NPV will give projects the same ranking This is an unusual case, however, since it is rare for initial investments to be equal Project A has a payback period of 100/40 = 2.5 years Project B has a payback period of years Project A 7-1 Copyright © 2009 McGraw-Hill Ryerson Limited Year Cash Flow ($) -100 40 40 40 40 NPV= Discounted Cash Flow ($) @ 11 percent -100 36.04 32.48 29.24 26.36 24.12 Cumulative Discounted Cash Flow ($) -100 -63.96 -31.48 -2.24 +24.12 Assuming uniform cash flows across time, the fractional year can now be determined Since the discounted cash flows are negative until year and become positive by Year 4, the project pays back sometime in the fourth year Note that out of the total discounted cash flow of $26.36 in Year 4, the first $2.24 comes in by 2.24/26.36 = 0.084 year Therefore, the discounted payback period for Project A is 3.084 years Project B Year Cash Flow ($) -100 50 50 50 NPV= Discounted Cash Flow ($) @ 11 percent -100 45.05 40.60 36.55 22.20 Cumulative Discounted Cash Flow ($) -100 -54.95 -14.35 +22.20 The discounted payback for Project B is years + 14.35/36.55 = 2.39 years No Despite its higher payback, Project A still may be the preferred project, for example, when the discount rate is 11% (as in Problems and 2) Just as in problem 5, you should note that the payback period for each project is fixed, but that NPV changes as the discount rate changes The project with the shorter payback period need not have the higher NPV 10 NPV = −3,000 + 800 × annuity factor(10%, years) = $484.21 At this discount rate, you should accept the project You can solve for IRR by setting the PV of cash flows equal to 3,000 on your calculator and solving for the interest rate: PV = −3000; n = 6; FV = 0; PMT = 800; compute i The IRR is 15.34%, which is the highest discount rate before project NPV turns negative 7-2 Copyright © 2009 McGraw-Hill Ryerson Limited 11 Payback = 2500/600 = 4.167 years, which is less than the cutoff So the firm would accept the project 12 NPV = −10,000 + + + + = $2,378.25 Profitability index = NPV/Investment = 2378 13 Project at percent discount rate Discounted Cash Flow Cumulative Discounted ($) @ percent Cash Flow ($) -3000 -3000 -3000 800 784 -2216 800 768.8 -1447.2 800 753.6 -693.6 800 739.2 45.6 800 724.8 770.4 800 710.4 1480.8 NPV= 1480.8 Since the discounted cash flows become positive by Year 4, the project pays back sometime in the fourth year Note that out of the total discounted cash flow of $739.20 in Year 4, the first $693.60 comes in by 693.60/739.20 = 0.94 year Therefore, the discounted payback for the project is 3.94 years, and thus the project should be pursued Year Cash Flow ($) Project at 12 percent discount rate Year Cash Flow ($) -3000 800 800 800 800 800 800 NPV= Discounted Cash Flow ($) @ 12 percent -3000 714.4 637.6 569.6 508.8 453.6 405.6 289.6 Cumulative Discounted Cash Flow ($) -3000 -2285.6 -1648.0 -1078.4 -569.6 -116.0 289.6 Since the discounted cash flows become positive by Year 6, the project pays back in years + 116/405.6 = 5.28 years Therefore, given the firm’s decision criteria of a discounted payback of years or less, the project should not be pursued As illustrated by the two scenarios above, the firm’s decision will change as the discount rate changes As the discount rate increases, the discounted payback period gets extended 7-3 Copyright © 2009 McGraw-Hill Ryerson Limited 14 NPV = −2.2 + × annuity factor(r, 15 years) − 9/(1 + r)15 When r = 6%, NPV = −2.2 + 2.538 = $0.338 billion When r = 16%, NPV = −2.2 + 1.576 = −$0.624 billion 15 The IRR of project A is 25.69%, and that of B is 20.69% However, project B has the higher NPV and therefore is preferred The incremental cash flows of B over A are –20,000 at time and +12,000 at times and The NPV of the incremental cash flows is $827, which is positive and equal to the difference in project NPVs 16 NPV = 5000 + – = –$197.70 Because NPV is negative, you should reject the offer You should reject the offer despite the fact that IRR exceeds the discount rate This is a “borrowing type” project with positive cash flows followed by negative cash flows A high IRR in these cases is not attractive: You don’t want to borrow at a high interest rate 17 a r = implies NPV = 6,750 + 4,500 + (-18,000) = $-6,750 r = 50% implies NPV = 6750 + + − 18,000 = $1,750 1.5 r = 100% implies NPV = 6750 + + b 18 − 18,000 = $4,500 22 IRR = 33.333%, the discount rate at which NPV = NPV = 10,000 + − 7,500 − 8,500 + = $-2,029.08 1.12 1.12 which is negative So the project is not attractive However, you can note that the IRR of the project is 37.03 % Since the IRR of the project is greater than the required rate of return of 12%, the project should be accepted using this rule On balance, we would use the NPV rule and reject the project 19 NPV9% = –20,000 + 4,000 × annuity factor(9%, periods) = $2139.28 NPV14% = –20,000 + 4,000 × annuity factor(14%, periods) 7-4 Copyright © 2009 McGraw-Hill Ryerson Limited = –$1,444.54 The IRR is 11.81% To confirm this on your calculator, set PV = (−)20,000; PMT = 4000; FV = 0; n = 8, and compute i The project will be rejected for any discount rate above this rate 20 a The present value of the savings is 100/r If r = 08, PV = 1,250 and NPV = –1,000 + 1,250 = $250 If r = 10, PV = 1,000 and NPV = –1,000 + 1,000 = $0 b IRR = 10 or 10% At this discount rate, NPV = $0 c Payback = 10 years d Discounted Payback Year 10 11 12 13 14 15 16 17 18 19 20 21 Cash Flow ($) -1000 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 Discounted Cash Flow ($) @ percent -1000 92.6 85.7 79.4 73.5 68.1 63.0 58.3 54.0 50.0 46.3 42.9 39.7 36.8 34.0 31.5 29.2 27.0 25.0 23.2 21.5 19.9 Cumulative Discounted Cash Flow ($) -1000 -907.4 -821.7 -742.3 -668.8 -600.7 -537.7 -479.4 -425.4 -375.4 -329.1 -286.2 -246.5 -209.7 -175.7 -144.2 -115.0 -88.0 -63.0 -39.8 -18.3 1.6 Discounted payback when cost of capital is percent = 20 years +18.3/19.9 = 20.95 years 7-5 Copyright © 2009 McGraw-Hill Ryerson Limited The NPV=0 when the cost of capital =10% The savings are supposed to last forever Therefore, there is no finite discounted payback period when cost of capital is 10% 21 a NPV of the two projects at various discount rates is tabulated below NPVA = –20,000 + 8,000 × annuity factor(r%, years) = –20,000 + 8,000 [– ] NPVB = –20,000 + 25,000 Discount Rate 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% NPVA 4000 3071 2201 1384 617 –105 –785 –1427 –2033 –2606 –3148 NPVB 5000 3558 2225 990 –154 –1217 –2205 –3126 –3984 –4784 –5532 From the NPV profile, it can be seen that Project A is preferred over Project B if the discount rate is above 4% At 4% and below, Project B has the higher NPV b IRRA = 9.70% [PV = (–)20; PMT = 8; FV = 0; n = 3; compute i] IRRB = 7.72% [PV = (–)20; PMT = 0; FV = 25; n = 3; compute i] 22 We know that the undiscounted project cash flows must sum to the initial investment because payback equals project life Therefore, the discounted cash flows are less than the initial investment, so NPV must be negative 23 NPV = 100 + + = –1.40 Because NPV is negative, you should reject the offer This is so despite the fact that IRR exceeds the discount rate This is a “borrowing type” project with a positive cash flow followed by negative cash flows A high IRR in these cases is not attractive: You don’t want to borrow at a high interest rate 7-6 Copyright © 2009 McGraw-Hill Ryerson Limited 24 a Project A Year Cash Flow ($) -5000 1000 1000 3000 0.00 NPV= Discounted Cash Flow ($) @ 10 percent -5000.00 909.09 826.45 2253.94 0.00 -1010.52 Cumulative Discounted Cash Flow ($) -5000.00 -4090.91 -3264.46 -1010.52 -1010.52 The payback period for Project A is years Project A does not pay back on a discounted basis since cumulative discounted cash flows remain negative until the end of Year Project B Year Cash Flow ($) -1000 1000 2000 3000 NPV= Discounted Cash Flow ($) @ 10 percent -1000.00 826.45 1502.63 2049.04 3378.12 Cumulative Discounted Cash Flow ($) -1000 -1000 -173.55 1329.08 3378.12 The payback period for Project B is years The discounted payback period for Project B is years + 173.55/1502.63 = 2.12 years Project C Year Cash Flow ($) -5000 1000 1000 3000 5000 NPV= Discounted Cash Flow ($) @ 10 percent -5000.00 909.09 826.45 2253.94 3415.07 2404.55 Cumulative Discounted Cash Flow ($) -5000.00 -4090.91 -3264.46 -1010.52 2404.55 The payback period for Project C is years The discounted payback period for Project C is years + 1010.52/3415.07 = 3.3 years b Only B satisfies the 2-year payback criterion 7-7 Copyright © 2009 McGraw-Hill Ryerson Limited c You would accept Project B d Projects B and C Project A B C 25 NPV -1010.52 3378.12 2404.55 e False Payback gives no weight to cash flows after the cutoff date a Year: Sales Costs Depreciation Net income b 100 30 50 20 110 35 50 25 120 40 50 30 130 45 50 35 Cash flow = Net income + Depreciation Year: CF: 70 75 80 85 NPV = –$2.30 NPV is negative even though the book rate of return is greater than the discount rate 26 27 a Cash flow each year = $5,000 – $2,000 = $3,000 NPV = –10,000 + 3,000 × annuity factor(8%, years) = 1,978.13 NPV is positive so you should pursue the project b The accounting change has no effect on project cash flows, and therefore no effect on NPV a The present values of the project cash flows (net of the initial investments) are: NPVA = –2100 + + = $400 NPVB = –2100 + + = $300 The initial investment for each project is 2100 Profitability index (A) = 400/2100 = 0.1905 7-8 Copyright © 2009 McGraw-Hill Ryerson Limited Profitability index (B) = 300/2100 = 0.1429 28 b If you can choose only one project, choose A for its higher profitability index If you can take both projects, you should: Both have positive profitability index a The less–risky projects should have lower discount rates b First, find the profitability index of each project Project A B C D E PV of Cash flow 3.79 4.97 6.62 3.87 4.11 Investment 3 NPV 0.79 0.97 1.62 0.87 1.11 Profitability Index 0.26 0.24 0.32 0.29 0.37 Then select projects with the highest profitability index until the $8 million budget is exhausted Choose, therefore, projects E and C 29 c All the projects have positive NPV All will be chosen if there is no rationing a NPVA = –18 + 10 × annuity factor(10%, periods) =$ 6.87 NPVB = –50 + 25 × annuity factor(10%, periods) =$12.17 b Thus Project B has the higher NPV if the discount rate is 10% Project A has the higher profitability index Project A B c PV Invest ment NPV 24.87 62.17 18 50 6.87 12.17 Profitability Index (= NPV/Investment) 0.38 0.24 A firm with a limited amount of funds available should choose Project A since it has a higher profitability index of 0.38, i.e., a higher “bang for the buck.” For a firm with unlimited funds, the possibilities are: i If the projects are independent projects, then the firm should choose both projects 7-9 Copyright © 2009 McGraw-Hill Ryerson Limited ii However, if the projects are mutually exclusive, then Project B should be selected It has the higher NPV 30 NPV Discount Rate 2% 12% 31 Project A 43.43 16.47 Project B 41.31 17.69 a If r = 2%, choose A b If r = 12%, choose B c The larger cash flows of project A tend to come later, so their present values are more sensitive to increases in the discount rate a Assuming an opportunity cost of capital of 6% (1) Payback Period Projec t I II Cash Flows, Dollars C0 -250000 -25000 C1 12000 15000 C2 18000 8000 C3 18000 6000 C4 30000 6000 C5 250000 500 Payback Period, years 5.69 3.33 Project II provides the lowest payback period, and thus is the project of choice under this decision criterion (2) Discounted Payback Period Project I Year Cash Flow ($) Discounted Cash Flow ($) @ percent -250000.00 12000.00 18000.00 18000.00 30000.00 250000.00 NPV= -250000.00 11320.75 16019.94 15113.15 23762.81 186814.54 3031.19 Cumulative Discounted Cash Flow ($) -250000.00 -238679.25 -222659.31 -207546.16 -183783.35 3031.19 Discounted payback period is years + 183783.35/186814.54 = 4.98 years 7-10 Copyright © 2009 McGraw-Hill Ryerson Limited Discount rate 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 b NPV –2.00 –1.62 –1.28 –0.97 –0.69 –0.44 –0.22 –0.03 0.14 0.29 0.42 0.53 0.62 0.69 0.75 0.79 0.83 0.85 0.85 0.85 0.84 Discount rate 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 NPV 0.82 0.79 0.75 0.71 0.66 0.60 0.54 0.47 0.39 0.32 0.24 0.15 0.06 –0.03 –0.13 –0.22 –0.32 –0.42 –0.53 –0.63 –0.74 At 5% the NPV is: NPV = –22 + + + – = –0.443 Since the NPV is negative the project is not attractive c At 20% the NPV is: NPV = –22 + + + – = 0.840 At 40% the NPV is: NPV = –22 + + + – = –0.634 d At a low discount rate, the positive cash flows ($20 for years) are not discounted much However, the final negative cash flow of $40 does not get discounted very heavily either The net effect is a negative NPV At very high rates, the positive cash flows are discounted very heavily, resulting in a negative NPV For mid-range discount rates, the positive cash flows that occur in the middle of the project dominate and project NPV is positive 7-13 Copyright © 2009 McGraw-Hill Ryerson Limited 35 a Econo-cool costs $300 and lasts for years The annual rental fee with the same PV is $102.53 We solve PMT × annuity factor(21%,5 years) = $300 PMT × 2.92598 = 300 PMT = $102.53 The equivalent annual cost of owning and running Econo-cool is $102.53 + $150 = $252.53 Luxury Air costs $500, and lasts for years Its equivalent annual rental fee is found from PMT × annuity factor(21%,8 years) = $500 PMT = $134.21 The equivalent annual cost of owning and operating Luxury Air is $134.21 + $100 = $234.21 b Luxury Air is more cost effective It has the lower equivalent annual cost c The real interest rate is now 1.21/1.10 – = 10 = 10% Redo (a) and (b) using a 10% discount rate Because energy costs would normally be expected to inflate along with all other costs, we should assume that the real cost of electric bills is either $100 or $150, depending on the model Equiv annual real cost to own Econo-cool = $ 79.14 plus $150 (real operating cost) = 150.00 $229.14 Equiv annual real cost to own Luxury Air = plus $100 (real operating cost) = Luxury Air is still more cost effective 7-14 Copyright © 2009 McGraw-Hill Ryerson Limited $ 93.72 100.00 $193.72 36 Time until purchase Cost 400 320 256 204.80 163.84 131.07 104.86 83.89 NPV at purchase date a – 31.33 48.67 112.67 163.87 204.83 237.60 263.81 284.78 NPV today b –31.33 44.25 93.12 123.12 139.90 147.53 148.91 146.14 Notes: a – Cost + 60 × annuity factor(10%, 10 years) b NPV at purchase date/(1.10)n NPV is maximized when you wait years to purchase the scanner 37 The equivalent annual cost of the new machine is the 4-year annuity with present value equal to $20,000 This is $7005 This can be interpreted as the extra yearly charge that should be attributed to the purchase of the new machine spread over its life It does not yet pay to replace the equipment since the incremental cash flow provided by the new machine, $10,000 – $5000 = $5000, is less than the equivalent annual cost of the machine 38 a The equivalent annual cost (EAC) of the new machine over its 10-year life is found by solving EAC × annuity factor(5%, 10 years) = $20,000 EAC × 7.7217 = $20,000 Therefore, EAC = $2590 Together with maintenance costs of $2000 per year, the equivalent cost of owning and operating is $4590 The old machine costs $5000 a year to operate, and is already paid for (We assume it has no scrap value and therefore no opportunity cost.) The new machine is less costly You should replace b If r = 10%, the equivalent annual cost of the new machine increases to $3255, so the equivalent cost of owning and operating it is now $5255, which is higher than that of the old machine Do not replace Your answer changes because the higher discount rate implies that the opportunity cost of the money tied up in the forklift also is higher 7-15 Copyright © 2009 McGraw-Hill Ryerson Limited 39 For the fourth quarter of 2007 business investment in Machinery and Equipment (M&E) increased by 3.4% to $123.7 billion For all of 2007, business investment in M&E rose by 8.3% For example, the capacity utilization rate for the fourth quarter of 2007 in the food industry was 79 %, paper was 86.6 % and machinery was 82.2 % The capacity utilization rate can be an indicator of the likelihood of future capital spending If the capacity utilization rate is close to 100 %, then it is highly likely that capital spending will increase in order to further increase capacity The information was compiled from ”The Daily”, Economic indicators, Summary tables published by Statistics Canada This information is available through the Statistics Canada website by using the following web link: http://www.statcan.ca/english/daiquo/econind.htm 40 a Present Value = PV = = $100,000 NPV = –$80,000 + $100,000 = $20,000 b Recall that the IRR is the discount rate that makes NPV equal to zero: – Investment + PV of cash flows when discounted at IRR = – 80,000 + = Solving the above equation, we find that: IRR = 5,000/80,000 + 05 = 1125 = 11.25% 41 For harvesting lumber, the value-maximizing rule is to cut the tree when its growth rate equals the discount rate When the tree is young and the growth rate exceeds the discount rate, it pays to wait: the value of the tree is increasing faster than the discount rate When the tree is older and the growth rate is less than r, cutting immediately is better, since the revenue from the tree can be invested to earn a rate of r, which is better than the tree is providing 42 a Time Cash flow − 30 −28 The following graph shows a plot of NPV as a function of the discount rate NPV = when r equals (approximately) either 15.61% or 384% These are the two IRRs 7-16 Copyright © 2009 McGraw-Hill Ryerson Limited b Discount rate 10% 20% 350% 400% NPV Develop? −.868 million 556 284 −.120 No Yes Yes No NPV 0% 50% 100% 150% 200% 250% 300% 350% 400% 450% 500% -2 -4 Discount rate 43 (a) Costs PV 12%, t 50,000 3,000 3,000 3,000 3,000 3,000 60,814.33 Ultra Fast 50,000 7,500 7,500 7,500 7,500 72,780.12 Medium Fast PV of net cost = PV 12%,t – PV of salvage PV of Ultra Fast = 60,814.33 - [9,000/(1.12)5] = $55,707.49 PV of Medium Fast = 72,780.12 - [6,250/(1.12)4] = $68,808.13 EFFECTIVE ANNUAL COST Ultra Fast = $55,707.49/PV factor Medium Fast= $68,808.13/PV factor =$55,707.49 /3.605 = $15,452.84 =$68,808.13/3.037 = $22,656.61 Recommendation – buy Ultra Fast which costs less b) Effective Annual Cost without salvage value Ultra Fast = 60,814.33/3.605 = $16,869.44 Medium Fast = 72,780.12/3.037 = $ 23,964.48 c) With salvage value: Ultra Fast’s effective annual cost at the end of years = $53,392.38/3.037 = $17,580.63 This is lower than Medium Fast (EAC = $22,656.94) Hence, we should purchase Ultra Fast which will be replaced by Hyper 3MM in years Without salvage value: 7-17 Copyright © 2009 McGraw-Hill Ryerson Limited Ultra fast EAC at the end of years =$59,112.05/3.037= $19,463.96 This is still lower than medium fast (EAC = $23,964.48) 44 (a) Payback period: Project A Year cash flow A’s Cumulative cash flow Project B cash flow B’s cumulative cash flow - 10,000 -10,000 -15,000 -15,000 6,000 -4,000 3,000 -12,000 6,000 2,000 5,000 -7,000 6,000 8,000 7,000 6,000 14,000 8,000 8,000 Payback for Project A = + 4,000/6,000 = 1.667 years Payback for Project B = years According to this method, Project A is preferred since it pays back earlier than Project B While the payback method is relatively easy to use, its major disadvantages are: (i) It does not take into consideration the time value of money and (ii) it ignores cash flows beyond the payback period Discounted Payback and NPV: 1.000 -10,000 A’s Cumulative Discount cash flow -10,000 909 5,454 -4,546 2,727 -12,273 826 4,956 410 4,130 -8,143 751 4,506 4,916 5,257 -2,886 683 4,098 9,014 5,464 2,578 Year Discount Factor 10 % NPV A’s discounted cash flow 9,014 B’s discounted cash flow -15,000 B’s cumulative Discount cash flow -15,000 2,578 Based on NPV project A is preferred Discounted payback for Project A = + (4,546/4,956) = 1.92 years Discounted payback for Project B 7-18 Copyright © 2009 McGraw-Hill Ryerson Limited = + (2,886/5464) = 3.53 years Once again, according to the discounted payback method, Project A is preferred since it pays back earlier than Project B The main advantage of the discounted payback period is that this method considers the time value of money, unlike the payback period The main flaw of the discounted payback period is that it does not consider cash flows beyond the payback period and therefore, it may on occasion incorrectly reject positive NPV projects (b) We see from the computations above that the NPV for Project A is $9,014 whereas for Project B it is $2,578 In general, notice also that if projects are able meet the cutoff in terms of a discounted payback, they must have a positive NPV Based on the NPV approach, Project A is preferred to Project B as it has the higher NPV (c) The profitability index for Project A = $9,014/$10,000 = 9014 For Project B, it is $2,578/$15,000 = 172 Once again, Project A is preferable using this method (d) Internal Rate of returnTrial and Error Approach.: Essentially, with this approach we try to select the discount rate at which the IRR for the project =0 This discount rate is, then, also the project’s IRR Project A Let us try a discount rate of 48 % At this rate, NPV = -10,000 + {6,000/ (1+.48)1} + {6,000/ (1+.48)2} + {6,000/ (1+.48)3} + {6,000/ (1+.48)4} NPV = - 105.28 Since NPV is negative at this rate, the IRR should be lower than 48 percent Let us try a discount rate of 46 % At this rate, NPV = -10,000 + {6,000/ (1+.46)1} + {6,000/ (1+.46)2} + {6,000/ (1+.46)3} + {6,000/ (1+.46)4} NPV = 172.82 So, we can assume that the IRR for project A is somewhere between 46% and 48 % 7-19 Copyright © 2009 McGraw-Hill Ryerson Limited Notice that this % difference between the two rates has an “NPV distance” of 278.1 (i.e 105.28 + 172.82) So, by interpolation, NPV will be at a rate which is (2/278.1 x 105.28) below 48%; or 0.76% below 48% = 47.24% It is actually 47.23 % (using a financial calculator) Project B Let us try an initial discount rate of 17 % At this rate, NPV = -15,000 + {3,000/ (1+.17)1} + {5,000/ (1+.17)2} + {7,000/ (1+.17)3} + {8,000/ (1+.17)4} NPV = - 143.54 Since NPV is negative at 17%, the IRR should be lower than this rate Let us try a discount rate of 15 % At this rate, NPV = - 15,000 + {3,000/ (1+.15)1} + {5,000/ (1+.15)2} + {7,000/ (1+.15)3} + {8,000/ (1+.15)4} NPV = 566.04 So, we know that the IRR for project B is some where between 15 % and 17 % Once again, notice that this % difference between the two rates has an “NPV distance” of 709.58 (i.e 143.54 + 566.04) So, by interpolation, NPV will be at a rate which is (2/709.58 x 143.54) below 17%; or 0.4% below 17% = 16.6% It is actually 16.58 % (using a financial calculator) Notice that, for both Projects A and B, we were able to get results for the IRR that were quite close to the actual numbers obtained through a financial calculator Using the IRR rule, Project A (with the higher IRR) is preferred over Project B e) f) Independent projects would be evaluated for acceptance or rejection on a “standalone” basis Mutually exclusive projects would be selected on an “either/or” basis and only those contributing most toward shareholder wealth would be selected In this question we are able to reach the same decision using all the methods (payback, discounted payback, NPV, profitability index, and IRR) However, if we get conflicting decisions using different methods then the NPV method should be used as it is generally considered to be the most robust 7-20 Copyright © 2009 McGraw-Hill Ryerson Limited 45 Assumptions Initial cash flow First years Years -7 Years & Constant thereafter Discount rate Years 100,000 0.15 -0.02 0.08 Cash flow -100,000 0 16,000.00 18,400.00 21,160.00 24,334.00 27,984.10 27,424.42 26,875.93 1.000 0.7940 0.7350 0.6810 0.6300 0.5830 0.5400 0.5001 Discounted cash flow -100000 0 12,704.00 13,524.00 14,409.96 15,330.42 16,314.73 14,809.19 13,440.65 Cumulative discounted cash flow (100,000.00) (100,000.00) (100,000.00) (87,296.00) (73,772.00) (59,362.04) (44,031.62) (27,716.89) (12,907.70) 532.95 NPV after years= $ 532.95 26,875.93 PV of cash flow into foreseeable future= = $335,949.13 0.08 a) Project NPV = 532.95+335,949.13 = $336,482.08 Since the project NPV is positive, we should accept the project b) The technique used in part (a) is the NPV decision rule If a project has a positive NPV that is, the present value of future cash flow is greater than initial cost we accept that project c) Years 1) Cash flow -100,000.00 0 16,000.00 18,400.00 21,160.00 24,334.00 27,984.10 27,424.42 26,875.93 Payback period Cumulative cash flow -100,000.00 -100,000.00 -100,000.00 (84,000.00) (65,600.00) (44,440.00) (20,106.00) 7,878.10 35,302.52 62,178.45 Discounted cash flow -100000.00 0 12,704.00 13,524.00 14,409.96 15,330.42 16,314.73 14,809.19 13,440.65 Cumulative discounted cash flow (100,000.00) (100,000.00) (100,000.00) (87,296.00) (73,772.00) (59,362.04) (44,031.62) (27,716.89) (12,907.70) 532.95 20,106.00 = 6.72 yrs =6+ 27,984.10 7-21 Copyright © 2009 McGraw-Hill Ryerson Limited 2) 12,907.70 = 8.96 yrs Discounted payback = + 13,440.65 When we ignore the time value of money, we use the payback period When we take into account the time value of money, we use the discounted payback period The main advantage of using the payback period is that it is quick and easy to calculate While the payback period is relatively easy to use, its major disadvantages are: (i).It does not take into consideration the time value of money and (ii) it ignores cash flows beyond the payback period The discounted payback period is an improvement over the payback period to the extent that it considers the time value of money However, to the extent that this method ignores cash flows beyond payback, it still suffers from a major flaw d) Assume 150 years of constant cash flow after year IRR= 17.13 % Assume 200 yrs of constant cash flow after year IRR = 17.13 % 46 a) (i) Years Project Alpha cash flow -100,000.00 70,000.00 32,000.00 32,000.00 9,000.00 Project Alpha Project Beta Project Beta cumulative cash cash flow cumulative cash flow flow -100,000.00 -100,000.00 -100,000.00 -30,000.00 40,000.00 -60,000.00 2000.00 40,000.00 -20,000.00 34,000.00 40,000.00 20,000.00 43,000.00 40,000.00 60,000.00 30,000 = 1.94 years Project Alpha payback period = + 32,000 20,000 = 2.5 years Project Beta payback period = + 40,000 Using the payback period, I would select Project Alpha which has the earlier payback period ii) Years Discount factor Alpha discounted C.F Alpha Cum cash flow 7-22 Copyright © 2009 McGraw-Hill Ryerson Limited Beta discounted C.F Beta Cum cash flow 1.000 0.909 0.826 0.751 0.683 100,000.00 63,630.00 26,432.00 24,032.00 6,147.00 -100,000.00 -36,370.00 -9,938.00 14,094.00 20,241.00 100,000.00 36,360.00 33,040.00 30,040.00 27,320.00 -100,000.00 -63,640.00 -30,600.00 -560.00 26,760.00 Notice that we obtained discounted cash flows using the present value tables If you the calculations algebraically you are likely to get a somewhat different result 9,938 = 2.41 yrs Alpha discounted payback =2+ 24,032 Beta discounted payback 560 = 3.02 yrs =3+ 27,320 Once again, using the discounted payback period, Project Alpha is preferred as it pays back earlier on a discounted basis than Project Beta b) Alpha project NPV = $ 20,241.00 Beta project NPV = $ 26,760.00 Undertake Beta project using NPV decision rule since it has the higher NPV c) d) IRR- Alpha = 22.4 % IRR- Beta =21.86 % Select project Alpha using IRR rule (Alpha has the higher IRR) 20,241 = 0.202 Profitability index – Alpha = NPV/ Initial cash flow = 100,000 26,760 = 0.268 Beta = NPV/Initial cash flow = 100,000 Using the profitability index also, we would select Project Beta e) 7-23 Copyright © 2009 McGraw-Hill Ryerson Limited 31,500 27,000 22,500 Cross over point 20.271 % 18,000 IRR project Alpha 22.4 % NPV 13,500 9,000 4,500 -4,500 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 IRR project Beta 21.86 % -9,000 Discount rate Alpha Beta Crossover point – Discount rate NPV = 20.271 % = 3,019.00 Notice that if the cost of capital rate is higher than 20.27%, the NPV rule would prefer project Alpha over project Beta However, if the discount rate is lower than 20.27%, project Beta has the higher NPV and is preferred However, according to the IRR rule, project Alpha (with the higher IRR) will be preferred to project Beta 47 Annual Savings Annual savings on W.C Tax rate Cost of New Machine Number of years (Useful life) Salvage value of New Machine Cost of old Machine Salvage value of old machine in years Depreciable life Salvage value of old machine, if sold today Cost of Capital $12,000.0 $4,000.00 0.35 $75,000 $9,000 $70,000 $6,500 $30,000.0 0.12 Change in capital cost at time t = $75,000.00 - $30,000.00 = $45,000.00 Change in Salvage Value (SV) at time t = $9,000.00 - $6,500.00= $2,500.00 (assuming no tax implications on salvage value) After tax savings in operating cost = $12,000.00(1 - 0.35) = $7,800.00 7-24 Copyright © 2009 McGraw-Hill Ryerson Limited Savings in working capital =$4,000.00 Total Present Value of Change in Depreciation =$ 10,645.25 (see calculation below) Year New machine Depreciation Old machine Depreciation Change in Depreciation Depreciation Tax Shield Discount Factor @ 12 % PV of change in depreciation Total PV of depreciation change 15,000 14,000 1,000 350 0.893 312.55 10,645.25 15,000 14,000 1,000 350 0.797 278.95 15,000 15,000 15,000 15,000 5,250 0.712 3,738 15,000 5,250 0.636 3,339 15,000 5,250 0.567 2,976.75 NPV = [12,000(1-.035)]*(PVIFA 12%,5) + [2,500(1 – 0.35)]*(PVIF 12%,5) + 4,000(PVIFA 12%,5) + 10,645.25 – 45,000 = 7,800 (3.605) + 2,500(.567) + 4,000(3.605) + 10,645.25 – 45,000 = $ 9,601.75 Since the cost savings from installing the new machine and replacing the old machine results in a positive NPV, the new machine should replace the old machine 7-25 Copyright © 2009 McGraw-Hill Ryerson Limited Solution to Minicase for Chapter None of the measures in the summary tables is appropriate for the analysis of this case, although the NPV calculations can be used as the starting point for an appropriate analysis The payback period is not appropriate for the same reasons that it is always inappropriate for analysis of a capital budgeting problem: cash flows after the payback period are ignored; cash flows before the payback period are all assigned equal weight, regardless of timing; the cutoff period is arbitrary The internal rate of return criterion can result in incorrect rankings among mutually exclusive investment projects when there are differences in the size of the projects under consideration and/or when there are differences in the timing of the cash flows In choosing between the two different stamping machines, both of these differences exist The net present value calculations indicate that the Skilboro machines have a greater NPV ($2.56 million) than the Munster machines ($2.40 million) However, since the Munster machines also have a shorter life, it is not clear whether the difference in NPV is simply a matter of longevity In order to adjust for this difference, we can compute the equivalent annual annuity for each: Munster machines: C× − = $2.40 million 0.15 0.15 × (1.15) C × annuity factor(15%, years) = $2.40 million C × 4.16042 = $2.40 million ⇒ C = EAC = $0.57686 million Skilboro machines: C× − = $2.56 million 10 0.15 0.15 × (1.15) C × annuity factor(15%, 10 years) = $2.56 million C × 5.01877 = $2.56 million ⇒ C = EAC = $0.51009 million Therefore, the Munster machines are preferred Another approach to making this comparison is to compute the equivalent annual annuity based on the cost of the two machines The cost of the Munster machine is $8 million, so that the equivalent annual annuity is computed as follows: 7-26 Copyright © 2009 McGraw-Hill Ryerson Limited Munster machines: C× − = $8 million 0.15 0.15 × (1.15) C × annuity factor(15%, years) = $8 million C × 4.16042 = $8 million ⇒ C = EAC = $1.92288 million For the Skilboro machine, we can treat the reduction in operator and material cost as a reduction in the present value of the cost of the machine: PV = $500,000 × − = $2.50938 million 10 0.15 0.15 × (1.15) $12.5 million – $2.50938 million = $9.99062 million C× − = $9.99062 million 10 0.15 0.15 × (1.15) C × annuity factor(15%, 10 years) = $9.99062 million C × 5.01877 = $9.99062 million ⇒ C = EAC = $1.99065 million Here, the equivalent annual cost is less for the Munster machines Note that the differences in the equivalent annual annuities for the two methods are equal (Differences are due to rounding.) 7-27 Copyright © 2009 McGraw-Hill Ryerson Limited ... 25.69%, and that of B is 20.69% However, project B has the higher NPV and therefore is preferred The incremental cash flows of B over A are –20,000 at time and +12,000 at times and The NPV of the... flows, and therefore no effect on NPV a The present values of the project cash flows (net of the initial investments) are: NPVA = –2100 + + = $400 NPVB = –2100 + + = $300 The initial investment. .. $827, which is positive and equal to the difference in project NPVs 16 NPV = 5000 + – = –$197.70 Because NPV is negative, you should reject the offer You should reject the offer despite the fact