Dde 321 - Solutions Exercise 4
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Advanced Corporate Finance
Leonidas Rompolis
EXERCISES -4 (SOLUTIONS) Chapter 10, Practice Questions 2. The spreadsheets show the following results:
Pessimistic -1.2 -10.4 -19.6 -11.9 -2.7
NPV (billions of yen) Expected 3.4 3.4 3.4 3.4 3.4
Optimistic 8.0 17.3 11.1 11.1 9.6
The principal uncertainties are market share, unit price, and unit variable cost. 9. The expected cash flow is: 45 × 0.25 + 35 × 0.5 + 25 × 0.25 − 25 = 10 10 a. NPV = −90 + = −6.66 and the factory should not be built. 0.12 b. At the beginning of year 2 we have the option to continue production or to sell the 10 factory. In the first case the value of the project is = 83.33 . In the second case 0.12 the value is just the market price, i.e. 50. We observe that is more valuable to continue the project, therefore we do not exercise the option to abandon. However, as answer (a) implies neither this possibility should be accepted since the NPV is negative. 1. If Rustic replaces now rather than in one year, several things happen: i. It incurs the equivalent annual cost of the $9 million capital investment. ii. It reduces manufacturing costs. For example, for the “Expected” case, analyzing “Sales” we have (all dollar figures in millions): The economic life of the new machine is expected to be 10 years, so the equivalent annual cost of the new machine is: $9/5.6502 = $1.59 The reduction in manufacturing costs is: 0.5 × $4 = $2.00 Thus, the equivalent annual cost savings is: –$1.59 + $2.00 = $0.41 Continuing the analysis for the other cases, we find:
Sales Manufacturing Cost Economic Life
Equivalent Annual Cost Savings (Millions) Pessimistic Expected Optimistic 0.01 0.41 1.21 -0.59 0.41 0.91 0.03 0.41 0.60
1
Advanced Corporate Finance
Leonidas Rompolis
2. a. Year 0 30 B
Investment 1. Revenue 2. Variable Cost 3. Fixed Cost 4. Depreciation 5. Pre-tax Profit (1-2-3-4) 6. Tax 7. Net Operating Profit (5-6) 8. Operating Cash Flow (4+7) NPV =
Years 1-10 37.500 B 26.000 3.000 3.000 5.500 1.925 3.575 6.575 10.40 B
b. The following table displays the data of the break-even chart. Unit Sales (000’s) 0 100 200
Inflows Revenues Yrs 1-10 0.00 37.50 75.00
Outflows Investment Yr 0 30.00 30.00 30.00
V. Costs Yr 1-10 0.00 26.00 52.00
F. Cost Yr 1-10 3.00 3.00 3.00
Taxes Yr 1-10 -2.10 1.93 5.95
PV Inflows 0.0 230.4 460.8
PV Outflows -35.5 -220.0 -404.5
Break-even chart 80 60
NPV
40 20 0 -20 0
50
100
150
200
250
-40 -60 unit sales
Note that the break-even point can be found algebraically as follows: NPV = -Investment + [(PVA10/10%) × (τc × Depreciation)] + [Quantity × (Price – V.Cost) – F.Cost]×(1 – τc)×(PVA10/10%) Set NPV equal to zero and solve for Q:
Q=
I - (PVA10/10% ×D×τc ) FC + (PVA10/10% )×(P - VC)×(1 - τc ) P - VC
2
NPV -35.5 10.4 56.3
Advanced Corporate Finance
Leonidas Rompolis
=
30,000,000,000 - 6,451,795,461 3,000,000,000 + (6.144567)×(375,000 - 260,000)×(0.65) 375,000 - 260,000
=
23,548,204,539 3,000,000,000 + =51,269+26,087=77,356 459,306 115,000
c. The break-even point is the point where the present value of the cash flows, including the opportunity cost of capital, yields a zero NPV. d. To find the level of costs at which the project would earn zero profit, write the equation for net profit, set net profit equal to zero, and solve for variable costs: Net Profit = (R – VC – FC - D) × (1 – τc) 0 = (37.5 – VC – 3.0 – 1.5) × (0.65) VC = 33.0 This will yield zero profit. Next, find the level of costs at which the project would have zero NPV. Using the data in Table 10.1, the equivalent annual cash flow yielding a zero NPV would be: 15 B/PVA10/10% = 2.4412 B If we rewrite the cash flow equation and solve for the variable cost: NCF = [(R – VC – FC – D) × (1 – τc)] + D 2.4412 = [(37.5 – VC – 3.0 – 1.5) × (0.65)] + 1.5 VC = 31.55 This will yield NPV = 0, assuming the tax credits can be used elsewhere in the company.
3. Year 5: 10000 trials 180 160 140
frequencies
120 100 80 60 40 20 0 cash flows (yen billion)
3
Advanced Corporate Finance
Leonidas Rompolis
Year 10: 10000 trials 140 120
frequencies
100 80 60 40 20 0 cash flows (yen billion)
4.
Expand High demand (60%)
Pilot production and market tests
Quit
Observe demand
Expand
Low demand (40%)
Quit
If the demand is high the NPV is: NPV = −50 +
7.5 = 12.5 , therefore it will be better 0.12
to expand. If the demand is low the NPV is: NPV = −50 +
3 = −25 , and it will be better to 0.12
quit. Therefore, the total NPV of the project is:
4
Advanced Corporate Finance
Leonidas Rompolis NPV = −5 +
0.6 × 12.5 = −1.17 1.42
and the project should not be accepted. 5. We analyze the decision tree by working backwards. So, for example, if we purchase the piston plane and demand is high: The NPV at t = 1 of the ‘Expanded’ branch is:
-150 +
(0.8×800) + (0.2×100) = $461 1.08
The NPV at t = 1 of the ‘Continue’ branch is:
(0.8×410) + (0.2×180) = $337 1.08 Thus, if we purchase the piston plane and demand is high, we should expand further at t = 1. This branch has the highest NPV. Similarly, if we purchase the piston plane and demand is low: The NPV of the ‘Continue’ branch is: (0.4×220) + (0.6×100) = $137 1.08 We can now use these results to calculate the NPV of the ‘Piston’ branch at t = 0:
-180 +
(0.6)×(100 + 461) + (0.4)×(50 + 137) = $201 1.08
Similarly for the ‘Turbo’ branch, if demand is high, the expected cash flow at t = 1 is: (0.8 × 960) + (0.2 × 220) = $812 If demand is low, the expected cash flow is: (0.4 × 930) + (0.6 × 140) = $456 So, for the ‘Turbo’ branch, the combined NPV is: (0.6×150) + (0.4×30) (0.6×812) + (0.4×456) NPV=-350+ + = $319 (1.08) (1.08) 2 Therefore, the company should buy the turbo plane.
5
Advanced Corporate Finance
Leonidas Rompolis
In order to determine the value of the option to expand, we first compute the NPV without the option to expand: (0.6×100) + (0.4×50) NPV=-250+ + (1.08) (0.6)[(0.8×410) + (0.2×180)]+(0.4)[(0.4×220)+(0.6×100)] = $62.07 (1.08) 2 Therefore, the value of the option to expand is: $201 – $62 = $139
Continue Hi demand (.6)
Lo demand (.2) $220
$150
Continue Lo demand (.4) Turbo -$350
Hi demand (.4) $930 Lo demand (.6) $140
$30
Expand -$150 Piston -$180 Hi demand (.6) $100
Hi demand (.8) $960
Continue
Hi demand (.8) $800 Lo demand (.2) $100 Hi demand (.8) $410 Lo demand (.2) $180
Hi demand (.4) Continue Lo demand (.4) $50
$220 Lo demand (.6) $100
6
Advanced Corporate Finance
Leonidas Rompolis
6. First, consider the sequence of events: At t = 0, the investment of $25,000,000 is made. At t = 1, production begins, so the first year of revenue and expenses is recorded at t = 2. At t = 6, the patent expires and competition may enter. Since it takes one year to achieve full production, competition is not a factor until t = 7. (This assumes the competition does not begin construction until the patent expires.) After t = 7, full competition will exist and thus any new entrant into the market will earn the 9% cost of capital. Next, calculate the cash flows: At t = 0: –$25,000,000 At t = 1: $0 At t = 2, 3, 4, 5, 6: Sale of 200,000 units at $100 each, with costs of $65 each, yearly cash flow = $7,000,000. After t = 6, the NPV of new investment must be zero. Hence, to find the selling price per unit (P) solve the following for P: 200,000×(P - 65) 200,000×(P-65) 0 = -25,000,000 + + ...+ 2 1.09 1.0912 Solving, we find P = $85.02 so that, for years t = 7 through t = 12, the yearly cash flow will be: [200,000 × ($85.02 - $65)] = $4,004,000. Finally, the net present value (in millions): 7 7 7 4.004 4.004 NPV = -25 + + + ... + + + ... + 2 3 6 7 1.09 1.09 1.09 1.09 1.0912 NPV = $10.69 or $10,690,000 7. The selling price after t = 6 now changes because the required investment is: [$25,000,000×(1 - 0.03)5] = $21,468,351 After t = 5, the NPV of new investment must be zero, and hence the selling price per unit (P) is found by solving the following equation for P: (200,000)×(P - 65) (200,000)×(P-65) 0 = -21,468,351 + + ... + 2 1.09 1.0912 P = $82.19 Thus, for years t = 7 through t = 12, the yearly cash flow will be: [200,000 × ($82.19 - $65)] = $3,438,000 Finally, the net present value (in millions) is: 7 7 7 3.438 3.438 NPV = -25 + + + ... + + + ... + 2 3 6 7 1.09 1.09 1.09 1.09 1.0912 NPV = $9.18 or $9,180,000
7
Leonidas Rompolis
EXERCISES -4 (SOLUTIONS) Chapter 10, Practice Questions 2. The spreadsheets show the following results:
Pessimistic -1.2 -10.4 -19.6 -11.9 -2.7
NPV (billions of yen) Expected 3.4 3.4 3.4 3.4 3.4
Optimistic 8.0 17.3 11.1 11.1 9.6
The principal uncertainties are market share, unit price, and unit variable cost. 9. The expected cash flow is: 45 × 0.25 + 35 × 0.5 + 25 × 0.25 − 25 = 10 10 a. NPV = −90 + = −6.66 and the factory should not be built. 0.12 b. At the beginning of year 2 we have the option to continue production or to sell the 10 factory. In the first case the value of the project is = 83.33 . In the second case 0.12 the value is just the market price, i.e. 50. We observe that is more valuable to continue the project, therefore we do not exercise the option to abandon. However, as answer (a) implies neither this possibility should be accepted since the NPV is negative. 1. If Rustic replaces now rather than in one year, several things happen: i. It incurs the equivalent annual cost of the $9 million capital investment. ii. It reduces manufacturing costs. For example, for the “Expected” case, analyzing “Sales” we have (all dollar figures in millions): The economic life of the new machine is expected to be 10 years, so the equivalent annual cost of the new machine is: $9/5.6502 = $1.59 The reduction in manufacturing costs is: 0.5 × $4 = $2.00 Thus, the equivalent annual cost savings is: –$1.59 + $2.00 = $0.41 Continuing the analysis for the other cases, we find:
Sales Manufacturing Cost Economic Life
Equivalent Annual Cost Savings (Millions) Pessimistic Expected Optimistic 0.01 0.41 1.21 -0.59 0.41 0.91 0.03 0.41 0.60
1
Advanced Corporate Finance
Leonidas Rompolis
2. a. Year 0 30 B
Investment 1. Revenue 2. Variable Cost 3. Fixed Cost 4. Depreciation 5. Pre-tax Profit (1-2-3-4) 6. Tax 7. Net Operating Profit (5-6) 8. Operating Cash Flow (4+7) NPV =
Years 1-10 37.500 B 26.000 3.000 3.000 5.500 1.925 3.575 6.575 10.40 B
b. The following table displays the data of the break-even chart. Unit Sales (000’s) 0 100 200
Inflows Revenues Yrs 1-10 0.00 37.50 75.00
Outflows Investment Yr 0 30.00 30.00 30.00
V. Costs Yr 1-10 0.00 26.00 52.00
F. Cost Yr 1-10 3.00 3.00 3.00
Taxes Yr 1-10 -2.10 1.93 5.95
PV Inflows 0.0 230.4 460.8
PV Outflows -35.5 -220.0 -404.5
Break-even chart 80 60
NPV
40 20 0 -20 0
50
100
150
200
250
-40 -60 unit sales
Note that the break-even point can be found algebraically as follows: NPV = -Investment + [(PVA10/10%) × (τc × Depreciation)] + [Quantity × (Price – V.Cost) – F.Cost]×(1 – τc)×(PVA10/10%) Set NPV equal to zero and solve for Q:
Q=
I - (PVA10/10% ×D×τc ) FC + (PVA10/10% )×(P - VC)×(1 - τc ) P - VC
2
NPV -35.5 10.4 56.3
Advanced Corporate Finance
Leonidas Rompolis
=
30,000,000,000 - 6,451,795,461 3,000,000,000 + (6.144567)×(375,000 - 260,000)×(0.65) 375,000 - 260,000
=
23,548,204,539 3,000,000,000 + =51,269+26,087=77,356 459,306 115,000
c. The break-even point is the point where the present value of the cash flows, including the opportunity cost of capital, yields a zero NPV. d. To find the level of costs at which the project would earn zero profit, write the equation for net profit, set net profit equal to zero, and solve for variable costs: Net Profit = (R – VC – FC - D) × (1 – τc) 0 = (37.5 – VC – 3.0 – 1.5) × (0.65) VC = 33.0 This will yield zero profit. Next, find the level of costs at which the project would have zero NPV. Using the data in Table 10.1, the equivalent annual cash flow yielding a zero NPV would be: 15 B/PVA10/10% = 2.4412 B If we rewrite the cash flow equation and solve for the variable cost: NCF = [(R – VC – FC – D) × (1 – τc)] + D 2.4412 = [(37.5 – VC – 3.0 – 1.5) × (0.65)] + 1.5 VC = 31.55 This will yield NPV = 0, assuming the tax credits can be used elsewhere in the company.
3. Year 5: 10000 trials 180 160 140
frequencies
120 100 80 60 40 20 0 cash flows (yen billion)
3
Advanced Corporate Finance
Leonidas Rompolis
Year 10: 10000 trials 140 120
frequencies
100 80 60 40 20 0 cash flows (yen billion)
4.
Expand High demand (60%)
Pilot production and market tests
Quit
Observe demand
Expand
Low demand (40%)
Quit
If the demand is high the NPV is: NPV = −50 +
7.5 = 12.5 , therefore it will be better 0.12
to expand. If the demand is low the NPV is: NPV = −50 +
3 = −25 , and it will be better to 0.12
quit. Therefore, the total NPV of the project is:
4
Advanced Corporate Finance
Leonidas Rompolis NPV = −5 +
0.6 × 12.5 = −1.17 1.42
and the project should not be accepted. 5. We analyze the decision tree by working backwards. So, for example, if we purchase the piston plane and demand is high: The NPV at t = 1 of the ‘Expanded’ branch is:
-150 +
(0.8×800) + (0.2×100) = $461 1.08
The NPV at t = 1 of the ‘Continue’ branch is:
(0.8×410) + (0.2×180) = $337 1.08 Thus, if we purchase the piston plane and demand is high, we should expand further at t = 1. This branch has the highest NPV. Similarly, if we purchase the piston plane and demand is low: The NPV of the ‘Continue’ branch is: (0.4×220) + (0.6×100) = $137 1.08 We can now use these results to calculate the NPV of the ‘Piston’ branch at t = 0:
-180 +
(0.6)×(100 + 461) + (0.4)×(50 + 137) = $201 1.08
Similarly for the ‘Turbo’ branch, if demand is high, the expected cash flow at t = 1 is: (0.8 × 960) + (0.2 × 220) = $812 If demand is low, the expected cash flow is: (0.4 × 930) + (0.6 × 140) = $456 So, for the ‘Turbo’ branch, the combined NPV is: (0.6×150) + (0.4×30) (0.6×812) + (0.4×456) NPV=-350+ + = $319 (1.08) (1.08) 2 Therefore, the company should buy the turbo plane.
5
Advanced Corporate Finance
Leonidas Rompolis
In order to determine the value of the option to expand, we first compute the NPV without the option to expand: (0.6×100) + (0.4×50) NPV=-250+ + (1.08) (0.6)[(0.8×410) + (0.2×180)]+(0.4)[(0.4×220)+(0.6×100)] = $62.07 (1.08) 2 Therefore, the value of the option to expand is: $201 – $62 = $139
Continue Hi demand (.6)
Lo demand (.2) $220
$150
Continue Lo demand (.4) Turbo -$350
Hi demand (.4) $930 Lo demand (.6) $140
$30
Expand -$150 Piston -$180 Hi demand (.6) $100
Hi demand (.8) $960
Continue
Hi demand (.8) $800 Lo demand (.2) $100 Hi demand (.8) $410 Lo demand (.2) $180
Hi demand (.4) Continue Lo demand (.4) $50
$220 Lo demand (.6) $100
6
Advanced Corporate Finance
Leonidas Rompolis
6. First, consider the sequence of events: At t = 0, the investment of $25,000,000 is made. At t = 1, production begins, so the first year of revenue and expenses is recorded at t = 2. At t = 6, the patent expires and competition may enter. Since it takes one year to achieve full production, competition is not a factor until t = 7. (This assumes the competition does not begin construction until the patent expires.) After t = 7, full competition will exist and thus any new entrant into the market will earn the 9% cost of capital. Next, calculate the cash flows: At t = 0: –$25,000,000 At t = 1: $0 At t = 2, 3, 4, 5, 6: Sale of 200,000 units at $100 each, with costs of $65 each, yearly cash flow = $7,000,000. After t = 6, the NPV of new investment must be zero. Hence, to find the selling price per unit (P) solve the following for P: 200,000×(P - 65) 200,000×(P-65) 0 = -25,000,000 + + ...+ 2 1.09 1.0912 Solving, we find P = $85.02 so that, for years t = 7 through t = 12, the yearly cash flow will be: [200,000 × ($85.02 - $65)] = $4,004,000. Finally, the net present value (in millions): 7 7 7 4.004 4.004 NPV = -25 + + + ... + + + ... + 2 3 6 7 1.09 1.09 1.09 1.09 1.0912 NPV = $10.69 or $10,690,000 7. The selling price after t = 6 now changes because the required investment is: [$25,000,000×(1 - 0.03)5] = $21,468,351 After t = 5, the NPV of new investment must be zero, and hence the selling price per unit (P) is found by solving the following equation for P: (200,000)×(P - 65) (200,000)×(P-65) 0 = -21,468,351 + + ... + 2 1.09 1.0912 P = $82.19 Thus, for years t = 7 through t = 12, the yearly cash flow will be: [200,000 × ($82.19 - $65)] = $3,438,000 Finally, the net present value (in millions) is: 7 7 7 3.438 3.438 NPV = -25 + + + ... + + + ... + 2 3 6 7 1.09 1.09 1.09 1.09 1.0912 NPV = $9.18 or $9,180,000
7
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