AFM 271 Practice Problem Set #2 Spring 2005 Suggested Solutions

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1 AFM 271 Practice Problem Set #2 Spring 2005 Suggested Solutions 1. Text Problems: 6.2 (a) Consider the following table: time cash flow cumulative cash flow 0 -$1,000,000 -$1,000,000 1 $150,000 -$850,000 2 $150,000 -$700,000 3 $150,000 -$550,000 4 $150,000 -$400,000 5 $150,000 -$250,000 6 $150,000 -$100,000 7 $150,000 $50,000 8 $150,000 $200, The payback period is 6 + $100,000/$150,000 = 6.67 years. Since this is less than the 10 year threshold, the project should be accepted. Note that because we have a perpetuity here, an easier way to determine the payback period is to simply divide the cost of the initial investment by the annual cash flows, i.e. $1,000,000/$150,000 = 6.67 years. (b) Consider the following table: time cash flow discounted CF cumulative discounted CF 0 -$1,000,000 -$1,000,000 -$1,000,000 1 $150,000 $135,135 -$864,865 2 $150,000 $121,743 -$743,121 3 $150,000 $109,679 -$633,443 4 $150,000 $98,810 -$534,633 5 $150,000 $89,018 -$445,615 6 $150,000 $80,196 -$365,419 7 $150,000 $72,249 -$293,171 8 $150,000 $65,089 -$228,082 9 $150,000 $58,639 -$169, $150,000 $52,828 -$116, $150,000 $47,592 -$69, $150,000 $42,876 -$26, $150,000 $38,627 $12, Thus the discounted payback period is 12 + $26,147/$38,627 = years. Since this exceeds the threshold of 10 years, the investment should not be made. Note that because we have a perpetuity, a more direct way to solve this is to determine how long an annuity paying $150,000 must last until its present value is the cost of the initial investment, i.e. [ ] 1 1 T $1,000,000 = $150, T ln(1) = ln(0.2667) T = years. 1

2 (c) NPV = $1,000,000 + $150,000/.11 = $363, Average investment: Average net income: $2,000, = $1,000,000. $100,000 + $107,000 + $114,490 + $122, $131, = $115, Note that a quicker way to calculate this is as follows. The last term in the numerator above, $131,079.60, is simply $100, We can think of this as just the future value after 5 years of a payment of $100,000 received after 1 year, at an interest rate of 7%. Similarly, $122, = $100, , so it can be viewed as the future value after 5 years of a payment of $100,000 received after 2 years. The same type of reasoning applies for the other terms. In other words, we can think of the numerator in the expression above for average net income as the future value of a 5 year ordinary annuity paying $100,000 at an interest rate of 7%. Using this idea, average net income is: Therefore: [ $100, ] = $115, AAR = $115, $1,000,000 = So, since AAR < 20% (the firm s cutoff AAR), the machine should not be purchased (a) For project A, we need to solve for r in the following equation. $15,000 + $10,500 (1 + r) + $10,500 (1 + r) 2 = 0. Using a financial calculator, we obtain r = 25.69%. Note that because there are only two periods, this can be solved analytically using the formula for roots of a quadratic equation. In this case, letting x = 1 + r, we have: 15000x x = 0 x = ± (15000)( 10500) 2(15000) = (1.2569, ). This gives an internal rate of return of x 1 = 25.69% (the negative root is meaningless in this context). For project B, we must find the value of r which makes: $300,000 + $195,000 (1 + r) + $195,000 (1 + r) 2 = 0. The solution is 19.43%, using a financial calculator. This can also be solved analytically, again letting x = 1 + r, as follows: x x = 0 x = ± (300000)( ) 2(300000) = (943, ). Discarding the meaningless negative root, we have an internal rate of return of 19.43%. 2

3 (b) The basic IRR rule says to pick the project with the higher IRR, which in this case is A. (c) In this case, the difference in scale has been ignored (project B is much larger than project A). (d) The problem can be remedied by applying either the NPV rule or the incremental IRR rule. (e) Consider B-A. To find the IRR, solve the following equation for r: Using the analytic approach: x x = 0 $285,000 + $184,500 (1 + r) + $184,500 (1 + r) 2 = 0. x = ± (285000)( ) 2(285000) = (909, ), so the incremental IRR is 19.09%. (f) If the opportunity cost of capital is less than 19.09%, B should be selected. Otherwise, pick A. (Note that taking B-A above creates a lending type of project, so we take B rather than A if the IRR is higher than the opportunity cost of capital. If we had used A-B instead, we would have the same IRR but a borrowing type of project, so the rule would be to take A if the IRR is lower than the opportunity cost of capital, i.e. if the opportunity cost of capital is higher than 19.09%). (g) NPV A = $15,000 + $10, $10, = $2, NPV B = $300,000 + $195,000 + $195, = $17, Since it has higher NPV, project B should be selected Generally, the statement is false. If the cash flows of project B occur early and the cash flows of project A occur late, then for a low discount rate the NPV of A can exceed the NPV of B. Examples are easy to construct: Project C 0 C 1 C 2 IRR NPV at 0% NPV at 1% A -$10,000,000 $0 $14,400, $4,400,000 $4,116,263 B -$20,000,000 $24,000, $4,000,000 $3,762,376 In one particular case, the statement is true. If the lives of the two projects are equal and in every time period the cash flows of the project B are twice the cash flows of project A, then the NPV of project B will be twice the NPV of project A for any discount rate between 0% and 20% (though both will equal zero at 20%) Although the profitability index is higher for project B than for project A, the NPV is the increase in the value of the company that will occur if a particular project is undertaken. Thus, the project with the higher NPV should be chosen because it provides the highest increase in the value of the firm. Only in the case of capital rationing could the pension fund manager be correct We have: Since the PI is greater than 1, accept the project. PI = $40,000A7.15 $160,000 = (a) Let the new Sunday early edition be project A, and the new Saturday late edition be project B. For project A, the payback is 2 + (3,600 3,450)/1,350 = 2.11 years. For project B, it is 2 + (6,300 5,700)/2,400 = 2.25 years. Based on the payback period rule, you should choose project A. 3

4 (b) For project A, the average investment is ($3,600 + $0)/2 = $1,800. Depreciation is $1,200 per year. Average net income is: ($1,800 $1,200) + ($1,650 $1,200) + ($1,350 $1,200) 3 = 400. Therefore, AAR is $400/$1,800 = 22.2%. For project B, the average investment is (6,300 + $0)/2 = $3,150. Depreciation is $2,100 per year, so average net income is: ($3,000 $2,100) + ($2,700 $2,100) + ($2,400 $2,100) 3 = 600. This gives an AAR of $600/$3,150 = 19.05%. (c) For project A: $3,600 + $1,800 (1 + r) + $1,650 (1 + r) 2 + $1,350 (1 + r) 3 = 0 r = 16.76%, while for B: $6,300 + $3,000 (1 + r) + $2,700 (1 + r) 2 + $2,400 (1 + r) 3 = 0 r = 14.29%. Therefore, the IRR for A is 16.76% and the IRR for B is 14.29%, and so project A has the higher IRR. (Note that there is no analytic solution in either case, so these values are obtained using a financial calculator.) (d) For the incremental project B-A, we have: $2,700 + $1,200 (1 + r) + $1,050 (1 + r) 2 + $1,050 (1 + r) 3 = 0 r = 11.02%. If the opportunity cost of capital is lower than 11.02%, choose project B, otherwise select project A. (Again, you can reach the same conclusion taking the incremental project as A-B, but that creates a borrowing type of project, so you have to modify the usual IRR rule.) 6.20 (a) The NPV of the original contract is: The NPV of subcontracting is: NPV contract = = $ million. 53 NPV subcontract = = $ million. 52 Since the NPV is higher under the original contract, the firm should not subcontract the work. (b) Consider the incremental IRR from subcontracting. We have to solve: 240 (1 + r) (1 + r) (1 + r) 3 = 0. This can be done analytically as follows (letting x = 1 + r): 240x 2 720x = 0 x = 720 ± (240)(520) 2(240) = (1.7886, ). This means that there are two IRRs: 23% and 78.86%. Note that incremental NPV is positive for discount rates between these two IRRs. Therefore, the firm should subcontract if the discount rate is greater than 23% but lower than 78.86%. 4

5 6.22 (a) To find the IRR, solve the following equation for r: ( (1+r) $600,000 + $100,000 r.08 ) 11 $50,000 = 0. (1 + r) 11 Using a financial calculator, we obtain an internal rate of return of 18.56%. (b) The mine should be opened because its IRR is higher than the opportunity cost of capital of 10%. 7.1 (a) Yes, the reduction in sales of the company s other products is an incremental cash flow. Include these lost sales because they are a cost (a reduction in revenue) which the firm must bear if it chooses to produce the new product. (b) Yes, the expenditures on plant and equipment are incremental cash flows. These are direct costs of the new product line. (c) No, the research and development costs are not included. The costs of research and development undertaken on the product during the last three years are sunk costs and should not be included in the evaluation of the project. The sunk costs must be borne whether or not the firm chooses to produce the new product; thus, they should have no bearing on the acceptability of the project. (d) No, the annual CCA expense is not a cash flow; however, the tax shield generated by the CCA is an incremental cash flow and this should be included. (e) No, dividend payments are not incremental cash flows. Dividend payments are not a cost of the project. The choice of whether or not to pay a dividend is a decision of the firm (as a whole), which is separate from the decision of choosing investment projects. (f) Yes, the resale value is an incremental cash flow and this must be included. Note also that this may generate some tax consequences (e.g. capital gains, recaptured depreciation, terminal loss). (g) Yes, these are incremental costs. The salaries of all personnel connected to the project must be included as costs of that project. Thus, the costs of employees who are on leave for a portion of the project life must be included as costs of that project. 7.8 This can be done either in real terms or in nominal terms. First, consider working in real terms. We have: PV of revenues: (150,000/1.07)/(.10.05) = $2,803,738. PV of expenses: (80,000/1.07)/(.10.03) = $1,068,091. PV of other costs: (40,000/1.07)/(.10 (.01)) = $339,847. PV of lease payments: (20,000/1.07)/(.06 ( )) = $149,031. Note that the growth rate in real terms for the lease payments is not -6%. It is 1/1.07 1, or %. In nominal terms, the discount rate for the risky cash flows is ()(1.07) 1 =.177, and for the risk free cash flows it is (1.06)(1.07) 1 = The nominal growth rate is (1.07)(1.05) 1 =.1235 for the revenues, (1.07)(1.03) 1 =.1021 for the expenses, and (1.07)(0.99) 1 =.0593 for the other costs. Thus we have: PV of revenues: 150,000/( ) = $2,803,738. PV of expenses: 80,000/( ) = $1,068,091. PV of other costs: 40,000/( )) = $339,847. PV of lease payments: 20,000/.1342 = $149,031. Either way, the NPV of the project is $2,803,738 $1,068,091 $339,847 $149,031 = $1,246, Let I denote the cost of the new equipment. We will find the value of I such that NPV is zero. We have: Cost of equipment -I Sale of old equipment $20,000 After-tax savings 10,000 (1.45) A 8.08 $31, Sale of new equipment 5, $2, PV of CCA tax shield (I 20,000)(.20)(.45)/( ) 1.04/1.08 5,000(.20)(.45)/( ) 1.04/ I 7,

6 Therefore, the project NPV is: NPV = I + $47, Setting this to zero gives a value of I = $68, This is the maximum price that should be paid for the equipment. Note that in calculating the PV of the CCA tax shield we have assumed that (i) net acquisitions today are positive (i.e. I > $20,000), and (ii) net acquisitions in year 8 are also positive, so that the half year rule applies at both times The present value of costs over one cycle is: $12,000 + $6,000 A $4, = $30, The equivalent annual cost is $30,402.70/A 4.08 = $9, The present value of such a stream in perpetuity is $114, For Facility I: After-tax maintenance costs $60,000 (1.34) A 7.10 $192, Depreciation tax shield.34 ($2,100,000/7) A 7.10 $496, PV of cash outflow $2,100,000 + $192, $496, $1,796, Therefore, the equivalent annual cost is $1,796,210.67/A 7.10 = $368, For Facility II: After-tax maintenance costs $100,000 (1.34) A $405, Depreciation tax shield.34 ($2,800,000/10) A $584, PV of cash outflow $2,800,000 + $405, $584, $2,620, Therefore, the equivalent annual cost is $2,620,578.64/A = $426, Facility I should be selected because it has lower equivalent annual cost For Mixer X: For Mixer Y: PV after tax benefits = $400,000 + $120,000 A 5.14 = $11, Equiv. annual benefit = $11,969.72/A 5.14 = $3, PV after tax benefits = $600,000 + $130,000 A 8.14 = $3, Equiv. annual benefit = $3,052.31/A 8.14 = $ The firm should choose Mixer X because it provides higher equivalent annual benefits (a) The present value of the CCA tax shield is: Let C be the annual pre-tax cost savings. Then: $59, = $15, $59,400 + $15, C (1.40) A 5.12 = 0 C = $20, Therefore, the annual pre-tax cost savings must be at least $20, to justify the investment. (b) The present value of the CCA tax shield is: $59, $11, = $13,

7 Then: $59,$400 + $13, $11, C (1.40) A 5.12 = 0 C = $18, Therefore, the annual pre-tax cost savings must be at least $18, to justify the investment. 8.4 Let the sales price be x. Depreciation is $120,000 per year. At the accounting break-even point: ($900,000 + $120,000)(1.30) (x $15)(1.30) = 20,000 x = $ Therefore, the sales price should be $66 per unit in order for the firm to have a zero profit (in the first year). 8.6 The break-even point is: ($450,000 + $60,000)(1.40) ($3.00 $1.08)(1.40) = 265,625. Therefore, sales must be 265,625 abalone. At a sales volume of 300,000, the after-tax profit would be: [300,000 $3.00 $450,000 $60, ,000 $1.08](1.40) = $39, Let I be the break-even purchase price. Depreciation is $45,000/15 = $3,000 per year, so the book value of the old harvester is $45,000 5 $3,000 = $30,000. The loss on the sale of the old harvester is $30,000 $20,000 = $10,000. This produces a tax credit of $10, = $3,400. The incremental cost savings (after-tax) are $10,000 (1.40) = $6,000 per year. The incremental depreciation tax shield per year is the new amount of I/10 less the foregone amount of $3,000 (multiplied by the corporate tax rate of 40%). Therefore: NPV = 0 = $6,000 A (I/10 $3,000).40 A $3,400 + $20,000 I I = $57, (a) Base case NPV is $10,000,000 + $750,000 A = $5,762,333. (b) Assuming that there is a 50% chance that the project will be abandoned, the revised NPV is: $10,000,000 + $750, $1,500,000 A $200,000 = $5,673, The option value of abandonment is the increase in the NPV of $89, (a) The profitability index is PI = $15,000/1 + $17,000/12 + $10,000/1 3 A$40,000 5 = $34, $40,000 = Since PI < 1, the investment should not be taken. (b) Payback: The sum of the cash flows during the first three years is $15,000+$17,000+$10,000= $42,000. This exceeds the initial cost of $40,000, implying that the investment should be taken. Discounted payback: The sum of the discounted cash flows during the first three years ($34,623.01) is less than the initial cost ($40,000), and so the investment should not be taken. 7

8 3. (a) The cumulative cash flows over the first 3 years are: $27,000+ $27,000(1.09) + $27,000( ) = $27,000+ $29,430 + $32, = $88, Since this exceeds the initial cost of $80,000, the firm would make the investment. (b) The discounted cumulative cash flows over the first 3 years are: 4. (a) We have $27, $29, $32, = $24, $23, $23, = $71, Since this is less than the initial cost of $80,000, the firm would not make the investment. Discounted CFs: -$5, $2,000/0 = $1, $2,000/0 2 = $1, $4,000/0 3 = $3, Cumulative: -$5, $3, $1, $1, Since the cumulative discounted cash flows are positive after 3 years, the discounted payback period is less than three years and the investment should be taken. (b) The conclusion will not change. The profitability index is the sum of the discounted cash flows (ignoring the initial cost) divided by the initial cost. We know from above that the sum of the discounted cash flows over the three periods exceeds the initial cost, so the profitability index will be greater than one and thus the investment should be taken. (Alternatively, the PI can be calculated directly as 1.30 and the appropriate conclusion drawn.) 5. (a) We have: Since B has a higher NPV, it should be taken. (b) We have: NPV A = $250,000 + $325, = $9,088.01, NPV B = $250,000 + $365, = $9, IRR A : $250,000 = $325,000 (1 + r) 2 r = IRR B : $250,000 = $365,000 (1 + r) 3 r = ( ) $325,000 1/2 1 = 14.02%, $250,000 ( ) $365,000 1/3 1 = 13.44%. $250,000 (c) Since the projects are mutually exclusive, we need to consider incremental cash flows (even though the projects are the same scale, they have different cash flow timing): The IRR on the incremental cash flows is: Period B-A $0 $0 -$325,000 $365,000 $325,000 (1 + r) 2 = $365,000 (1 + r) 3 r = ( ) $365,000 1 = 12.31%. $325,000 Since this is a lending type of investment, it should be made if the IRR is more than the opportunity cost of capital. Since 12.31% > 12%, it is worth taking B instead of A, i.e. B should be selected. (Alternatively, using A-B gives the same IRR, but corresponds to a borrowing type of project, so it should be made if IRR is less than the opportunity cost of capital. It is not in this case, and so it is not worth taking A instead of B.) 8

9 6. (a) We have: ( ) $240,000 1/2 IRR A = 1 = 9.54%, $200,000 ( ) $285,000 1/4 IRR B = 1 = 9.26%, $200,000 PI A = $240,000/1.082 $200,000 PI B = $285,000/1.084 $200,000 = , = (b) Since the projects are mutually exclusive, we need to consider incremental cash flows: Find the internal rate of return r: Period 0 Period 1 Period 2 Period 3 Period 4 B-A $0 $0 -$240,000 $0 $285,000 $240,000 (1 + r) 2 + $285,000 (1 + r) 4 = 0 r = ( ) $285,000 1/2 1 = 8.97%. $240,000 Since this is a lending type of project, we would want to take it (i.e. invest in B instead of A) if the IRR is higher than the opportunity cost of capital. Since the IRR (8.97%) is higher than the opportunity cost of capital (8%), project B should be chosen. (Alternatively, using A-B gives the same IRR, but corresponds to a borrowing type of project, so it should be taken if the IRR is lower than the opportunity cost of capital. It is not here, so again we would conclude that project B should be selected.) 7. (a) We have: $500,000 + $140,000 PI A = 2 = $570,248 $200,000 $200,000 = 2.85, NPV A = $200,000 + $570,248 = $370,248, $120,000 + $350,000 PI B = 2 = $398,347 $100,000 $100,000 = 3.98, NPV B = $100,000 + $398,347 = $298,247, $80,000 + $270,000 PI C = 2 = $295,868 $100,000 $100,000 = 2.96, NPV C = $100,000 + $295,868 = $195,868. (b) All projects are desirable since they have positive NPVs, but with capital rationing, not all can be taken. A simple application of the NPV rule would select A because it has the highest NPV of $370,248. However, in this situation we should use the profitability index criterion. This would involve taking the projects with the highest PIs until the capital budget is spent. In this case, B and C have the highest PIs, and they use up the entire budget, and so B and C should be chosen. Note that the total NPV of B and C is $494,215 (which is higher than that of A). 8. The cost of the report is a sunk cost and is therefore ignored. The real discount rate is 2/ = Then: 9

10 Cost of machine: -$5,000,000 PV after-tax operating revenues: $2,500,000(.60)A $8,721, PV after-tax operating expenses: $1,000,000(.60)[1 (1.05/2) 9 ]/[.12.05] -$3,776, PV salvage value: $750,000/2 9 $270, CCA tax shields: PV perpetual tax shield: [$5,000,000(.25)(.40)/( )](1.06/2) $1,278, UCC in year 9: 5,000,000(1.125)(1.25) 8 = $437,994 UCC in year 9 less salvage: $437,994 $750,000 = $312,006 PV recaptured depreciation: $312,006(.40)/2 9 -$45, PV lost tax shield: $437,994(.25)(.40)/( )]/2 9 -$42, Working capital: $300,000 $75,000/2 + $375,000/2 8 -$215, NPV: $1,191, The cost of the report is a sunk cost and is therefore ignored. The real discount rate is 2/ = Then: Cost of machine: -$5,000,000 PV after-tax operating revenues: $2,500,000(.60)A $8,721, PV after-tax operating expenses: $1,000,000(.60)[1 (1.05/2) 9 ]/[.12.05] -$3,776, PV salvage value: $750,000/2 9 $270, CCA tax shields: UCC now: $4,000,000(1.125)(1.25) 2 = $1,968,750 UCC now less salvage: $1,968,750 $1,250, = $718,750 Net acquisitions now: $5,000,000 $1,250,000 = $3,750,000 PV lost terminal loss: (.40)$718,750 -$287, PV perpetual tax shields: Net acquisitions: [$3,750,000(.25)(.40)/( )](1.06/2) $959, Previous asset: $1,968,750(.25)(.40)/( ) $532, UCC in year 9: 3,750,000(1.125)(1.25) 8 + $1,968,750(1.25) 9 = $476, UCC in year 9 less salvage: $476, $750,000 = $273, PV recaptured depreciation: $273,681.52(.40)/2 9 -$39, PV lost tax shield: $476,318.48(.25)(.40)/( )]/2 9 -$46, Working capital: $300,000 $75,000/2 + $375,000/2 8 -$215, NPV: $1,118, The real discount rate is 1/ = The real growth rate for expenses is 1.07/ = Then: Cost of equipment: -$1,500, PV after-tax operating revenues: $275,000(.64)[1 (1.03/ ) 10 ]/[ ] $1,310, PV after-tax operating expenses: $120,000(.64)[1 ( / ) 10 ]/ [ ] -$604, PV salvage value: $1,800,000/1 10 $633, PV capital gain tax $300,000(.50)(.36)/1 10 -$19, CCA tax shields: PV perpetual tax shield: [$1,500,000(.20)(.36)/( )](1.055/1) $331, UCC in year 10: $1,500,000(1.10)(1.20) 9 = $181, UCC in year 10 less initial cost: $181, $1,500,000 = $1,318, PV recaptured depreciation: $1,318,806.07(.36)/1 10 -$167, PV lost tax shield: $181,193.93(.20)(.36)/( )]/1 10 -$14, Working capital: $150,000 + $75,000/1 5 + $75,000/1 10 -$79, NPV: -$109,

11 Note that the PV of after tax operating revenues can also be worked out in nominal terms. The nominal growth rate is (1.03)(1.025) 1 =.05575, so $275,000(1.025)(.64)[1 ( /1) 10 ][ ] = $1,310, Similarly, the present value of after tax operating expenses can be calculated in nominal terms as follows: $120,000(1.025)(.64)[1 (1.07/1) 10 [.11.07] = $604, The real discount rate is 1/ = The real growth rate for expenses is 1.07/ = Then: Cost of equipment: -$1,500, PV after-tax operating revenues: $275,000(.64)[1 (1.03/ ) 10 ]/[ ] $1,310, PV after-tax operating expenses: $120,000(.64)[1 ( / ) 10 ]/ [ ] -$604, PV salvage value: $100,000/1 10 $35, CCA tax shields: PV perpetual tax shield: [$1,500,000(.20)(.36)/( )](1.055/1) $331, UCC in year 10: $1,500,000(1.10)(1.20) 9 = $181, UCC in year 10 less salvage value: $181, $100,000 = $81, PV terminal loss: $81,193.93(.36)/1 10 $10, PV lost tax shield: $181,193.93(.20)(.36)/( )]/1 10 -$14, Working capital: $150,000 + $75,000/1 5 + $75,000/1 10 -$79, NPV: -$511, The nominal discount rate is (1 +.07)(1 +.03) 1 = 10.21%. The real growth rate for revenues is 1.08/ = %. The real growth rate for expenses is 1.04/ = %. Then: Cost of equipment: -$300, PV after-tax operating revenues: $75,000(.64)[1 ( /1.07) 6 ]/ [ ] $256, PV after-tax operating expenses: $24,000(.64)[1 ( /1.07) 6 ]/ [ ] -$74, PV working capital: $10,000 $5,000/021 $2,000/ $5,000/ $12,000/ $6, PV salvage value: $65,000/021 6 $36, CCA tax shields: PV perpetual CCA tax shield: [$300,000(.20)(.36)/( )]( /021) $68, UCC in year 6: $300,000(.9)(.8 5 ) = $88, Less salvage: $88, $65,000 = $23, PV terminal loss: $23,473.60(.36)/021 6 $4, PV lost tax shield: [$88,473.60(.20)(.36)/( )/ $11, NPV: -$27, The real discount rate is 3/ = %; the real growth rate for revenues is 1.06/ = %; and the nominal growth rate for expenses is (1.01)(1.03) 1 = 4.03%. Then: 11

12 Cost of machinery: -$925, PV after-tax operating revenues: $195,000(.64)(1.03)[1 (1.06/3) 12 ]/ [.13.06] $983, PV after-tax operating expenses: $80,000(.64)[1 (1.0403/3) 12 ]/ [ ] -$359, PV working capital: $10,000 $4,000/3 $2,000/3 2 + $7,000/ $9,000/3 12 -$11, PV salvage value: $30,000/3 12 $6,928 CCA tax shields: PV perpetual CCA tax shield: [$925,000(.20)(.36)/( )](1.065/3) $190, UCC in year 12: $925,000(.9)(.8 11 ) = $71, Less min(c, S): $71, $30,000 = $41, PV terminal loss: $41,511.21(.36)/3 12 $3, PV lost tax shield: [$71,511.21(.20)(.36)/( )/3 12 -$3, NPV: -$114, Note that the PV of after tax operating revenues can also be worked out in real terms, as follows: $195,000(.64)[1 ( / ) 12 ]/[ ] = $983, Similarly, the PV of after tax operating expenses can be calculated as: [$80,000/1.03](.64)[1 (1.01/ ) 12 ]/[ ] = $359, (a) The UCC after two years will be: (b) UCC 2 = $100,000(1.25)(1.5) = $37,500. Since you are selling the asset for more than this, there will be recaptured depreciation of $50,000 $37,500 = $12,500. $100,000(1 d/2)(1 d) = $50,000 (1 d/2)(1 d) = d +.5d 2 =.5.5d 2 1.5d +.5 = 0 Since d must be lower than 50%, the correct value is d = 1.5 ± (.5)(.5) 1 = 1.5 ± 1.25 = { , } (c) If you take CCA at the full rate, the present value of the tax benefits (including the recaptured depreciation) is: $100, $100, $12,500.4 PV = = $10,000 + $10,000 2 = $17,

13 If you take CCA at the lower rate, the present value of the tax benefits is: $100, $100, PV = + 2 = $7, $12, = $17, Therefore, she is not correct: you are better off taking the CCA at the full rate. (d) Note that the total amount of depreciation (including the recaptured depreciation when you claim CCA at the full rate of 50%) claimed is the same in either case: $50,000. Therefore, the decision is affected only by the time value of money. When you take CCA at the full rate, you are getting more of the $50,000 tax deduction earlier, so this alternative has a higher present value after discounting. 15. PV of costs for Machine A: EAC for Machine A: PV of costs for Machine B: EAC for Machine B: Since A has lower EAC, it should be chosen. $30,000 + $8,000 A 3.08 = $50, $50,616.78/A 3.08 = $19, $20,000 + $12,000 A 2.08 = $41, $41,399.18/A 2.08 = $23, Note that we can also use the matching cycle approach to answer this question. Over 6 years, there are two complete cycles for A and three complete cycles for B. The PV of costs for A during the 6 years is: while for B it is: $30,000 + $30,000/ $8,000 A 6.08 = $90,798.00, $20,000 + $20,000/ $20,000/ $12,000 A 6.08 = $107, Since A has lower PV of costs over the 6 year period, it should be chosen. 16. With no inflation, the EAC for the Econo-Cool model is: The EAC for the Luxury-Air model is: $300 + $150 A = $ = EAC A EAC = $ $500 + $100 A = $ = EAC A EAC = $ Hence, since its EAC is lower, we would choose the Luxury-Air model. With 3% inflation, find the real interest rate (using the Fisher equation): The EAC for the Econo-Cool model is: (1 + i) = (1 + r) (1 + π) r = 1.21/ = $300 + $150 A = $ = EAC A EAC = $

14 The EAC for the Luxury-Air model is: $500 + $100 A = $ = EAC A EAC = $ Hence, since its EAC is lower, we would again choose the Luxury-Air model. Thus our answer does not change. 17. Over its three year life, the NPV for A is: $320,000 + $160,000A 3.10 = $77, Therefore, A produces equivalent annual benefits of $77,896.32/A 3.10 = $31, Over its six year life, the NPV for B is: $420,000 + $120,000A 6.10 = $102, Therefore, B produces equivalent annual benefits of $102,631.28/A 6.10 = $23, Since A offers the highest equivalent annual benefits, it should be chosen. 18. We have: Since B has lower EAC, it should be chosen. 19. (a) We have: PV A = $75,000 + $30,000 A 3.12 = $147,054.94, EAC A = $147,054.94/A 3.12 = $61, PV B = $95,000 + $24,000 A 5.12 = $181,514.63, EAC B = $181,514.63/A 5.12 = $50, PV of costs for A: = $40,000 + $10,000A 5.09 = $78, Equivalent annual cost for A: = $78,896.51/A 5.09 = $20, PV of costs for B: = $33,000 + $12,000A 4.09 = $71, Equivalent annual cost for B: = $71,876.64/A 4.09 = $22, Since A has lower equivalent annual costs, it should be chosen. (b) The PV of costs of keeping C for 1 more year and then selling is: The FV of this after 1 year is: $14,000 $8,000 $12,000 + = $17, $17, = $19, As this is lower than the equivalent annual cost of A, C should not be replaced now. The PV of costs (at the end of year 1) of keeping C for the second year and then selling is: The FV of this after year 2 is: $18,000 $5,000 $8,000 + = $19, $19, = $21, As this is higher than the equivalent annual cost of A, C should be replaced after 1 year. 14

15 20. (a) We have: [$72,000 + $42,000](1.40) 6,000 = (P $31)(1.40) $72,000 + $42,000 P = + $31 = $50. 6,000 (b) [9,000($50 $31) $72,000 $42,000](1.40) = $34, (a) We have: 5,400 = 5,400 $80 $70,000 = fixed costs fixed costs = $362,000. (b) [6,750($250 $170) $70,000 $362,000](1.40) = $64, We have: PV break-even point = EAC = $100,000/A 8.12 = $20, [fixed costs + $70,000](1.40) ($250 $170)(1.40) $20, $18,000(.60) ($100,000/8)(.40) ($80 $60)(.60) = 2, ,161 units. 23. Note that since the price of gold is equally likely to rise or fall in each year, the expected price of gold will still be $500 in each year. Specifically: E(price t=1 ) = 0.5 ($500 $50) ($500 + $50) = $500 E(price t=2 ) = 0.25 ($500 2 $50) ($500) ($ $50) = $500 E(price t=3 ) = $ $ $ $650 = $500. Note the intuition for these calculations. After 1 year, there is an equal chance that the price will either rise by $50 or fall by $50 from its current level of $500. After two years, there is a 25% chance that the price will fall by $100 (i.e. $50 in each year), a 50% chance that it will return to $500 (either by first dropping to $450 and then rising by $50 in the second year or by first increasing to $550 and then falling in the second year by $50), and a 25% chance that it will increase by $100 (i.e. $50 in each year). A similar (but more complicated) analysis is needed for the expected price level after 3 years. So, given that the expected gold price is always $500 per ounce, the present value break-even point Q can be calculated as follows: $90,000 + ($500 $460) (0.6) Q A $5,000 (0.6) A $90,000 (0.4) A = 0 Q = 1, ounces. The accounting break-even point can be calculated as follows: Q = total fixed costs ($5,000 + $30,000) (1 0.4) = = 875. contribution margin ($500 $460)(1 0.4) Thus, the accounting break-even point is 875 ounces. 15

16 24. (a) We have: PV of year 1 operating income: = ($300 $250) 10, = $480, PV of years 2-5 operating income, low cost: = ($300 $275) 10,000 A = $872, PV of years 2-5 operating income, high cost: = ($300 $335) 10,000 A = $1,221, NPV of resuming production: = $750,000 + $480, ($872,570.97) +.4( $1,221,599.35) = $234, Since the NPV of resuming production is negative, you should not resume production. (b) We have: NPV of resuming production: = $750,000 + $480, ($872,570.97) +.4( $200,000/1.04) = $177, Since the NPV is positive, you should resume production. The value of the abandonment option is $177, ( $234,327.93) = $411, (Note that this can also be calculated as.4[ $200,000/1.04 ( $1,221,599.35)] = $411, ) 25. (a) We have:.60($40,000) +.40($12,000) NPV = $300, [.80($40,000/.10) +.20($12,000/.10)] +.40[.25($40,000/.10) +.75($12,000/.10)] + = $300,000 + $26, $256, = $17, (b) If demand is high in the first year, the choice is between.80($40,000) +.20($12,000).10 = $344,000 and the abandonment value of $275,000, so the business will not be abandoned. If demand is low in the first year, the choice is between.25($40,000) +.75($12,000).10 = $190,000 and the abandonment value of $275,000, so the busineses will be abandoned in this case. Then:.60($40,000) +.40($12,000) NPV = $300, ($344,000) +.40($275,000) + = $300,000 + $26, $287, = $13, Option value = $13, ( $17,090.91) = $30,

17 26. (a) The NPV is: NPV = $3,500 + $ = $500. [.50($250) +.50($150).05 ] (b) If the market turns out to be strong, the NPV after one year of buying the machine is: NPV = $3,500 + $250/.05 = $1,500. If the market turns out to be weak, the NPV after one year of buying the machine is: NPV = $3,500 + $150/.05 = $500. Therefore the firm will buy the machine only if the market is strong. The NPV of this decision today is: NPV =.50($1,500) +.50($0) 1.05 The value of the option to wait is then $ $500 = $ = $ (c) The NPV calculation in part (a) is unaffected. This is because the expected future cash flow of $200 per year forever has not changed. However, the value of the option to wait in part (b) will be higher under this scenario. This is because the firm will not invest if the market turns out to be weaker than in (b) (we already know it won t invest if after-tax cash flows are $150 per year, so it will not at $100 per year), but it will invest if the market turns out to be strong. The NPV of the project will increase relative to (b) because it will be more profitable in the strong market state and remain at zero NPV in the weak market state. As a result, the value of the option to wait will increase. 27. (a) We have: NPV = $1,000, $300, $100, = $1,000,000 + $900,000 $200,000 = $300,000. (b) If demand is high after 1 year, the present value at that time of the net cash flows is $300,000/.20 = $1,500,000. Alternatively, if demand is low after 1 year, the present value at that time of the net cash flows is $100,000/.20 = $500,000. Since $500,000 < $340,000 < $1,500,000, the franchise should be sold if demand is low after 1 year. Thus: $340,000 $100,000 NPV = $1,000, $1,500, = $1,000,000 + $900,000 + $80,000 = $20,000. The value of the option to sell the franchise is $20,000 ( $300,000) = $280,000. (c) We have: $340,000 $100,000 NPV = $1,150, $1,500, = $1,150,000 + $1,200,000 + $40,000 = $90,000. The value of the option to undertake additional promotion expenses is $90,000 ( $20,000) = $110,000. (d) The value of the option of selling the team would be reduced. This option has value because CSE can sell the team for a positive cash flow after 1 year, rather than losing money forever if demand is low. Under the scenario described, the only factor which changes is the probability that demand is low is reduced. Consequently, it is less likely that the option to sell the team will be exercised, and so the value of that option must decline. 17

18 28. Consider the following decision tree: today foreign branch, t=0 cost = -$200 million foreign subsidiary, t=0 cost = -$300 million 1st yr inc = $30 million 44 clients at $1 million per client, yr 2 onward, 50% 11 clients at $1 million per client, yr 2 onward, 50% joint venture, t=1 income of $30 million forever own service network, t=1 cost = $-200 million, loss of $5.5 million at t=2 income of $80 million each year thereafter do not invest Calculate the NPV of opening a foreign bank branch: [ ] (44)($1,000,000) (0.5) (11)($1,000,000) 0.1 NPV branch = $200,000,000 + = $50,000, Use the Fisher relation to find the appropriate nominal discount rate for the other alternative: (1 + i) = () (1.03) i = Now calculate the PVs of staying in the joint venture or developing own service network at t = 1: PV joint venture = $30,000, = $225,563,909.77, PV service network = $200,000,000 $5,500,000 + $80,000, (0.133)(33) = $326,040, Thus, choose development of own service network. Now find NPV at time 0 of choosing the subsidiary option with own service network: $30,000,000 + $326,040, NPV subsidiary = $300,000,000 + = $14,245, Therefore, our choice at t = 0 is between the subsidiary option ($ million), the branch option ($50 million), and the do nothing option ($0). Clearly, the branch option should be chosen. 29. Consider the following decision tree: 18

19 70% income(a): $100,000/yr income(b): $300,000/yr A: t = 1 cost: $0 income: $100,000/yr CONTINUE t = 2 20% income(a): $100,000/yr income(b): $100,000/yr INVEST t = 0 cost: $500,000 cost: $30,000 income $100,000/yr NODE 2 A or B or C B: t = 1 cost: $900,000 income(a): $100,000/yr income(b): $100,000 for 1 yr NODE 3 stop or continue 10% income(a): $100,000/yr income(b): $50,000/yr NODE 1 today C: t = 1 sell for $600,000 STOP t = 2 salvage: $0 income(a): $100,000/yr DO NOT INVEST Nodes represent decision points. We work backwards, so we start at node 3: Node 3: decide whether to continue with MRI/CT scan option after year 2. If we continue: PV t=2 (continue) = $300, $100, $50, = $1,175,000. If we do not continue, we get salvage of $0, so we choose to continue (note that we can consider this decision without incorporating the base case operating income since it is the same in all cases, i.e. whether or not we continue, and in all possible outcomes if we do continue). Node 2: decide whether to take A, B, or C: So, choose B. A : PV t=1 = $100, B : PV t=1 = $100, C : PV t=1 = $600,000. = $666,667, + $100, $1,175,000 $900,000 = $829,167, 1.2 Node 1: decide whether or not to invest: [ ] $100, $1,175, $900,000 invest: PV t=0 = + $100,000 $30,000 $500, = $272,083. So, the net present value of the project today is $272,083. Since this is greater than the NPV of not investing ($0), we would choose to invest in the clinic. 30. (a) Uncertain. The statement will be true if we are considering normal projects (i.e. investing/lending type projects with a single IRR) and there is no capital rationing. In this case, because the projects are independent, we can treat each project as a separate decision. If NPV is positive, then IRR will be higher 19

20 than the opportunity cost of capital (think of the NPV profile). The converse also holds (IRR higher than opportunity cost of capital implies positive NPV). The same argument also works for negative NPV (or IRR less than opportunity cost of capital). However, the statement could be false in other circumstances, such as cases where there are multiple IRRs. (b) False. This is like borrowing money, so one would be better off paying a lower interest rate than the opportunity cost of capital, i.e. being able to borrow on favourable terms is valuable. (c) Uncertain. Because the project is acceptable under the discounted payback rule, we know that the discounted future cash flows before the cutoff date more than cover the initial cost of the investment. Provided that any future cash flows after the cutoff date are positive, the initial cost will obviously be more than covered by the discounted future cash flows. According to the profitability index rule, an investment is acceptable if all of the discounted future cash flows exceed the initial cost of the investment. Therefore, the two rules will be in agreement as long as there are no large negative cash flows after the cutoff date used for the discounted payback rule. Such cash flows would be ignored in the discounted payback calculation, but could make the profitability index less than one. (d) False. The average accounting return rule is based on net income, not cash flows. The payback rule gives no weight to cash flows after the cutoff period. (It does give too much weight to cash flows which come relatively late but before the cutoff period as a result of not discounting these cash flows.) (e) False. Other things being equal, firms would invest more but this is due to receiving the CCA tax shields earlier. Because of the time value of money, this means that more projects would have positive NPVs, and so there would be an increase in equipment purchases by firms. (f) Uncertain. If this particular asset is the only asset that the firm has in its CCA class, then the statement is true. This is because the firm will have depreciated the asset down to some level below its initial cost, but then it will sell it for an amount that is higher than the initial cost. As a result, the undepreciated capital cost in the asset class less the adjusted cost of disposal must be negative, leading to recaptured depreciation. However, if the firm has other assets in the same CCA class, then the pooling of assets means that the undepreciated capital cost in the asset class less the adjusted cost of disposal of this asset may not be negative, so there might not be recaptured depreciation (i.e. if the total balance in the CCA pool is still positive after this particular asset is sold, then there is no recapture). (g) True. The contribution margin gives the amount each unit contributes towards the company s fixed costs. If the number of units sold exceeds the accounting break-even point, the net profit will equal the excess of units sold over the break-even point multiplied by the contribution margin. If the number of units sold is lower than the accounting break-even point, the net loss will equal the shortfall of units sold compared to the break-even point, multiplied by the contribution margin. (h) True, assuming that the opportunity cost of capital is greater than zero. Let I be the initial cost of the firm s investment and let T denote its economic lifetime. Then the accounting break-even point is given by fixed costs(1 T c ) + (I/T)(1 T c ) contribution margin = (I/T ) + fixed costs(1 T c) (I/T)T c contribution margin On the other hand, the present value break-even point (given the assumption about constant variables) is EAC + fixed costs(1 T c ) (I/T)T c contribution margin The only difference between these two expressions is the first term in the numerator, i.e. (I/T) (which is the depreciation expense) for the accounting break-even point vs. the EAC for the present value break-even point. Note that EAC = I/A T r. At a discount rate of r = 0%, the present value of an annuity paying $1 for T periods is simply $T, so A T r = T and the accounting and present value break-even points will be the same. For any r > 0, the present value of a T -period annuity will be less than T, i.e. A T r < T, implying that EAC > I/T and so the present value break-even point will be higher than the accounting break-even point. 20

21 (i) False. The accounting break-even sales point is Q = [fixed costs + depreciation] (1 T c ) [unit sales price - unit variable cost] (1 T c ), which is easily seen to be independent of the corporate tax rate T c. However, the present value break-even sales point is Q = [EAC + fixed costs (1 T c) depreciation T c ], [unit sales price - unit variable cost] (1 T c ) which is not (in general) independent of the corporate tax rate. The only case where it would be independent of T c is if the EAC were equal to the annual straight line depreciation since in this case Q = [depreciation+ fixed costs (1 T c) depreciation T c ] [unit sales price - unit variable cost] (1 T c ) [fixed costs + depreciation] (1 T c ) = [unit sales price - unit variable cost] (1 T c ), so the accounting and present value break-even points are the same (and thus both are independent of T c ). However, if C is the initial cost of the investment, and T is its economic lifetime, then the annual depreciation is C/T, while the EAC is C/A T r. These are only equal if AT r = T, which is not true for any r > 0 (though it is true in the economically unrealistic case that r = 0). (j) False. If neither of the projects had an option, their NPVs would be identical (since the projects are identical except for the option). The value of an option is at least 0 (since its value is positive if the state of the world is such that exercising the option is favourable, and its value is zero if it is unfavourable to exercise the option). Hence, NPV A = NPV B + value of abandonment option NPV B. 21