UNIT 10 Chapter 9 10S.pdf

UNIT 10?Chapter 9?Problem Solutions 8. The NPV of a project is the PV of the outflows minus the PV of the inflows. The equation for the NPV of this project at an 11 percent required return is: NPV = ? $34,000 + $16,000/1.11 + $18,000/1.11 2 + $15,000/1.11 3 = $5,991.49 At an 11 percent required return, the NPV is positive, so we would accept the project. The equation for the NPV of the project at a 30 percent required return is: NPV = ? $34,000 + $16,000/1.30 + $18,000/1.30 2 + $15,000/1.30 3 = ?$4,213.93 At a 30 percent required return, the NPV is negative, so we would reject the project. 8. CFo ?$34,000 CFo ?$34,000 C01 $16,000 C01 $16,000 C02 $18,000 C02 $18,000 C03 $15,000 C03 $15,000 I = 11% I = 30% Shift NPV Shift NPV $5,991.49 ?$4,213.93 11. The NPV of a project is the PV of the outflows minus the PV of the inflows. At a zero discount rate (and only at a zero discount rate), the cash flows can be added together across time. So, the NPV of the project at a zero percent required return is: NPV = ?$19,500 + 9,800 + 10,300 + 8,600 = $9,200 The NPV at a 10 percent required return is: NPV = ?$19,500 + $9,800/1.1 + $10,300/1.1 2 + $8,600/1.1 3 = $4,382.79 The NPV at a 20 percent required return is: NPV = ?$19,500 + $9,800/1.2 + $10,300/1.2 2 + $8,600/1.2 3 = $796.30 And the NPV at a 30 percent required return is: NPV = ?$19,500 + $9,800/1.3 + $10,300/1.3 2 + $8,600/1.3 3 = ?$1,952.44 Notice that as the required return increases, the NPV of the project decreases. This will always be true for projects with conventional cash flows. Conventional cash flows are negative at the beginning of the project and positive throughout the rest of the project. 11. CFo ?$19,500 CFo ?$19,500 C01 $9,800 C01 $9,800 C02 $10,300 C02 $10,300 C03 $8,600 C03 $8,600 I = 0% I = 10% Shift NPV Shift NPV $9,200 $4.382.79 CFo ?$19,500 CFo ?$19,500 C01 $9,800 C01 $9,800 C02 $10,300 C02 $10,300 C03 $8,600 C03 $8,600 I = 20% I = 30% Shift NPV Shift NPV $796.30 ?$1,952.44 18. At a zero discount rate (and only at a zero discount rate), the cash flows can be added together across time. So, the NPV of the project at a zero percent required return is: NPV = ?$684,680 + 263,279 + 294,060 + 227,604 + 174,356 = $274,619 If the required return is infinite, future cash flows have no value. Even if the cash flow in one year is $1 trillion, at an infinite rate of interest, the value of this cash flow today is zero. So, if the future cash flows have no value today, the NPV of the project is simply the cash flow today, so at an infinite interest rate: NPV = ?$684,680 The interest rate that makes the NPV of a project equal to zero is the IRR. The equation for the IRR of this project is: 0 = ?$684,680 + $263,279/(1+IRR) + $294,060/(1+IRR) 2 + $227,604/(1+IRR) 3 + 174,356/(1+IRR) 4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 16.23% 18. CFo ?$684,680 CFo ?$684,680 C01 $263,279 C01 $263,279 C02 $294,060 C02 $294,060 C03 $227,604 C03 $227,604 C04 $174,356 C04 $174,356 I = 0% Shift IRR Shift NPV 16.23% $274,619 25. a. Here the cash inflows of the project go on forever, which is a perpetuity. Unlike ordinary perpetuity cash flows, the cash flows here grow at a constant rate forever, which is a growing perpetuity. If you remember back to the chapter on stock valuation, we presented a formula for valuing a stock with constant growth in dividends. This formula is actually the formula for a growing perpetuity, so we can use it here. The PV of the future cash flows from the project is: PV of cash inflows = C 1 /(R ? g) PV of cash inflows = $85,000/(.13 ? .06) = $1,214,285.71 NPV is the PV of the outflows minus the PV of the inflows, so the NPV is: NPV of the project = ?$1,400,000 + 1,214,285.71 = ?$185,714.29 The NPV is negative, so we would reject the project. b. Here we want to know the minimum growth rate in cash flows necessary to accept the project. The minimum growth rate is the growth rate at which we would have a zero NPV. The equation for a zero NPV, using the equation for the PV of a growing perpetuity is: 0 = ?$1,400,000 + $85,000/(.13 ? g) Solving for g, we get: g = .0693 or 6.93% 1. To calculate the payback period, we need to find the time that the project has recovered its initial investment. After three years, the project has created: $1,600 + 1,900 + 2,300 = $5,800 in cash flows. The project still needs to create another: $6,400 ? 5,800 = $600 in cash flows. During the fourth year, the cash flows from the project will be $1,400. So, the payback period will be 3 years, plus what we still need to make divided by what we will make during the fourth year. The payback period is: Payback = 3 + ($600 / $1,400) = 3.43 years 2. To calculate the payback period, we need to find the time that the project has recovered its initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial cost is $2,400, the payback period is: Payback = 3 + ($105 / $765) = 3.14 years There is a shortcut to calculate the future cash flows are an annuity. Just divide the initial cost by he annual cash flow. For the $2,400 cost, the payback period is: Payback = $2,400 / $765 = 3.14 years For an initial cost of $3,600, the payback period is: Payback = $3,600 / $765 = 4.71 years The payback period for an initial cost of $6,500 is a little trickier. Notice that the total cash inflows after eight years will be: Total cash inflows = 8($765) = $6,120 If the initial cost is $6,500, the project never pays back. Notice that if you use the shortcut for annuity cash flows, you get: Payback = $6,500 / $765 = 8.50 years This answer does not make sense since the cash flows stop after eight years, so again, we must conclude the payback period is never. 3. Project A has cash flows of $19,000 in Year 1, so the cash flows are short by $21,000 of recapturing the initial investment, so the payback for Project A is: Payback = 1 + ($21,000 / $25,000) = 1.84 years Project B has cash flows of: Cash flows = $14,000 + 17,000 + 24,000 = $55,000 during this first three years. The cash flows are still short by $5,000 of recapturing the initial investment, so the payback for Project B is: B: Payback = 3 + ($5,000 / $270,000) = 3.019 years Using the payback criterion and a cutoff of 3 years, accept project A and reject project B. 4. When we use discounted payback, we need to find the value of all cash flows today. The value today of the project cash flows for the first four years is: Value today of Year 1 cash flow = $4,200/1.14 = $3,684.21 Value today of Year 2 cash flow = $5,300/1.14 2 = $4,078.18 Value today of Year 3 cash flow = $6,100/1.14 3 = $4,117.33 Value today of Year 4 cash flow = $7,400/1.14 4 = $4,381.39 To find the discounted payback, we use these values to find the payback period. The discounted first year cash flow is $3,684.21, so the discounted payback for a $7,000 initial cost is: Discounted payback = 1 + ($7,000 ? 3,684.21)/$4,078.18 = 1.81 years For an initial cost of $10,000, the discounted payback is: Discounted payback = 2 + ($10,000 ? 3,684.21 ? 4,078.18)/$4,117.33 = 2.54 years Notice the calculation of discounted payback. We know the payback period is between two and three years, so we subtract the discounted values of the Year 1 and Year 2 cash flows from the initial cost. This is the numerator, which is the discounted amount we still need to make to recover our initial investment. We divide this amount by the discounted amount we will earn in Year 3 to get the fractional portion of the discounted payback. If the initial cost is $13,000, the discounted payback is: Discounted payback = 3 + ($13,000 ? 3,684.21 ? 4,078.18 ? 4,117.33) / $4,381.39 = 3.26 years 5. R = 0%: 3 + ($2,100 / $4,300) = 3.49 years discounted payback = regular payback = 3.49 years R = 5%: $4,300/1.05 + $4,300/1.05 2 + $4,300/1.05 3 = $11,709.97 $4,300/1.05 4 = $3,537.62 discounted payback = 3 + ($15,000 ? 11,709.97) / $3,537.62 = 3.93 years R = 19%: $4,300(PVIFA 19%,6 ) = $14,662.04 The project never pays back. 7. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = ? $34,000 + $16,000/(1+IRR) + $18,000/(1+IRR) 2 + $15,000/(1+IRR) 3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 20.97% Since the IRR is greater than the required return we would accept the project. 7. CFo ?$34,000 C01 $16,000 C02 $18,000 C03 $15,000 Shift IRR 20.97% 9. The NPV of a project is the PV of the outflows minus the PV of the inflows. Since the cash inflows are an annuity, the equation for the NPV of this project at an 8 percent required return is: NPV = ?$138,000 + $28,500(PVIFA 8%, 9 ) = $40,036.31 At an 8 percent required return, the NPV is positive, so we would accept the project. The equation for the NPV of the project at a 20 percent required return is: NPV = ?$138,000 + $28,500(PVIFA 20%, 9 ) = ?$23,117.45 At a 20 percent required return, the NPV is negative, so we would reject the project. We would be indifferent to the project if the required return was equal to the IRR of the project, since at that required return the NPV is zero. The IRR of the project is: 0 = ?$138,000 + $28,500(PVIFA IRR, 9 ) IRR = 14.59% 9. CFo ?$138,000 CFo ?$138,000 CFo ?$138,000 C01-9 $28,500 C01-9 $28,500 C01-9 $28,500 I = 8% I = 20% Shift IRR Shift NPV Shift NPV 14.59% $40,036.31 ?$23,117.45 10. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = ?$19,500 + $9,800/(1+IRR) + $10,300/(1+IRR) 2 + $8,600/(1+IRR) 3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 22.64% 10. CFo ?$19,500 C01 $9,800 C02 $10,300 C03 $8,600 Shift IRR 22.64% 26. The IRR of the project is: $58,000 = $34,000/(1+IRR) + $45,000/(1+IRR) 2 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 22.14% At an interest rate of 12 percent, the NPV is: NPV = $58,000 ? $34,000/1.12 ? $45,000/1.12 2 NPV = ?$8,230.87 At an interest rate of zero percent, we can add cash flows, so the NPV is: NPV = $58,000 ? $34,000 ? $45,000 NPV = ?$21,000.00 And at an interest rate of 24 percent, the NPV is: NPV = $58,000 ? $34,000/1.24 ? $45,000/1.24 2 NPV = +$1,314.26 The cash flows for the project are unconventional. Since the initial cash flow is positive and the remaining cash flows are negative, the decision rule for IRR in invalid in this case. The NPV profile is upward sloping, indicating that the project is more valuable when the interest rate increases. 27. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the IRR of the project is: 0 = $20,000 ? $26,000 / (1 + IRR) + $13,000 / (1 + IRR) 2 Even though it appears there are two IRRs, a spreadsheet, financial calculator, or trial and error will not give an answer. The reason is that there is no real IRR for this set of cash flows. If you examine the IRR equation, what we are really doing is solving for the roots of the equation. Going back to high school algebra, in this problem we are solving a quadratic equation. In case you don?t remember, the quadratic equation is: x = a acbb 2 4 2 ?±? In this case, the equation is: x = )00026(2 )00013)(00020(4)00026()00026( 2 , ,,,, ??±?? The square root term works out to be: 676,000,000 ? 1,040,000,000 = ?364,000,000 The square root of a negative number is a complex number, so there is no real number solution, meaning the project has no real IRR. 28. First, we need to find the future value of the cash flows for the one year in which they are blocked by the government. So, reinvesting each cash inflow for one year, we find: Year 2 cash flow = $205,000(1.04) = $213,200 Year 3 cash flow = $265,000(1.04) = $275,600 Year 4 cash flow = $346,000(1.04) = $359,840 Year 5 cash flow = $220,000(1.04) = $228,800 So, the NPV of the project is: NPV = ?$450,000 + $213,200/1.11 2 + $275,600/1.11 3 + $359,840/1.11 4 + $228,800/1.11 5 NPV = ?$2,626.33 And the IRR of the project is: 0 = ?$450,000 + $213,200/(1 + IRR) 2 + $275,600/(1 + IRR) 3 + $359,840/(1 + IRR) 4 + $228,800/(1 + IRR) 5 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 10.89% While this may look like a MIRR calculation, it is not an MIRR, rather it is a standard IRR calculation. Since the cash inflows are blocked by the government, they are not available to the company for a period of one year. Thus, all we are doing is calculating the IRR based on when the cash flows actually occur for the company. 19. The MIRR for the project with all three approaches is: Discounting approach: In the discounting approach, we find the value of all cash outflows to time 0, while any cash inflows remain at the time at which they occur. So, the discounting the cash outflows to time 0, we find: Time 0 cash flow = ?$16,000 ? $5,100 / 1.10 5 Time 0 cash flow = ?$19,166.70 So, the MIRR using the discounting approach is: 0 = ?$19,166.70 + $6,100/(1+MIRR) + $7,800/(1+MIRR) 2 + $8,400/(1+MIRR) 3 + 6,500/(1+MIRR) 4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: MIRR = 18.18% Reinvestment approach: In the reinvestment approach, we find the future value of all cash except the initial cash flow at the end of the project. So, reinvesting the cash flows to time 5, we find: Time 5 cash flow = $6,100(1.10 4 ) + $7,800(1.10 3 ) + $8,400(1.10 2 ) + $6,500(1.10) ? $5,100 Time 5 cash flow = $31,526.81 So, the MIRR using the discounting approach is: 0 = ?$16,000 + $31,526.81/(1+MIRR) 5 $31,526.81 / $16,000 = (1+MIRR) 5 MIRR = ($31,526.81 / $16,000) 1/5 ? 1 MIRR = .1453 or 14.53% Combination approach: In the combination approach, we find the value of all cash outflows at time 0, and the value of all cash inflows at the end of the project. So, the value of the cash flows is: Time 0 cash flow = ?$16,000 ? $5,100 / 1.10 5 Time 0 cash flow = ?$19,166.70 Time 5 cash flow = $6,100(1.10 4 ) + $7,800(1.10 3 ) + $8,400(1.10 2 ) + $6,500(1.10) Time 5 cash flow = $36,626.81 So, the MIRR using the discounting approach is: 0 = ?$19,166.70 + $36,626.81/(1+MIRR) 5 $36,626.81 / $19,166.70 = (1+MIRR) 5 MIRR = ($36,626.81 / $19,166.70) 1/5 ? 1 MIRR = .1383 or 13.83% 20. With different discounting and reinvestment rates, we need to make sure to use the appropriate interest rate. The MIRR for the project with all three approaches is: Discounting approach: In the discounting approach, we find the value of all cash outflows to time 0 at the discount rate, while any cash inflows remain at the time at which they occur. So, the discounting the cash outflows to time 0, we find: Time 0 cash flow = ?$16,000 ? $5,100 / 1.11 5 Time 0 cash flow = ?$19,026.60 So, the MIRR using the discounting approach is: 0 = ?$19,026.60 + $6,100/(1+MIRR) + $7,800/(1+MIRR) 2 + $8,400/(1+MIRR) 3 + 6,500/(1+MIRR) 4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: MIRR = 18.55% Reinvestment approach: In the reinvestment approach, we find the future value of all cash except the initial cash flow at the end of the project using the reinvestment rate. So, the reinvesting the cash flows to time 5, we find: Time 5 cash flow = $6,100(1.08 4 ) + $7,800(1.08 3 ) + $8,400(1.08 2 ) + $6,500(1.08) ? $5,100 Time 5 cash flow = $29,842.50 So, the MIRR using the discounting approach is: 0 = ?$16,000 + $29,842.50/(1+MIRR) 5 $29,842.50 / $16,000 = (1+MIRR) 5 MIRR = ($29,842.50 / $16,000) 1/5 ? 1 MIRR = .1328 or 13.28% Combination approach: In the combination approach, we find the value of all cash outflows at time 0 using the discount rate, and the value of all cash inflows at the end of the project using the reinvestment rate. So, the value of the cash flows is: Time 0 cash flow = ?$16,000 ? $5,100 / 1.11 5 Time 0 cash flow = ?$19,026.60 Time 5 cash flow = $6,100(1.08 4 ) + $7,800(1.08 3 ) + $8,400(1.08 2 ) + $6,500(1.08) Time 5 cash flow = $34,942.50 So, the MIRR using the discounting approach is: 0 = ?$19,026.60 + $34,942.50/(1+MIRR) 5 $34,942.50 / $19,026.60 = (1+MIRR) 5 MIRR = ($34,942.50 / $19,026.60) 1/5 ? 1 MIRR = .1293 or 12.93% 16. a. The profitability index is the PV of the future cash flows divided by the initial investment. The cash flows for both projects are an annuity, so: PI I = $27,000(PVIFA 10%,3 ) / $53,000 = 1.267 PI II = $9,100(PVIFA 10%,3 ) / $16,000 = 1.414 The profitability index decision rule implies that we accept project II, since PI II is greater than the PI I . b. The NPV of each project is: NPV I = ?$53,000 + $27,000(PVIFA 10%,3 ) = $14,145.00 NPV II = ?$16,000 + $9,100(PVIFA 10%,3 ) = $6,630.35 The NPV decision rule implies accepting Project I, since the NPV I is greater than the NPV II . c. Using the profitability index to compare mutually exclusive projects can be ambiguous when the magnitude of the cash flows for the two projects are of different scale. In this problem, project I is roughly 3 times as large as project II and produces a larger NPV, yet the profitability index criterion implies that project II is more acceptable. 16. Project I CFo $0 CFo ?$53,000 C01-3 $27,000 C01-3 $27,000 I = 10% I = 10% Shift NPV Shift NPV $67,145.00 $14,145.00 PI = $67,145.00 / $53,000 = 1.267 Project II CFo $0 CFo ?$16,000 C01-3 $9,100 C01-3 $9,100 I = 10% I = 10% Shift NPV Shift NPV $22,630.35 $6,630.35 PI = $22,630.35 / $16,000 = 1.414 21. Since the NPV index has the cost subtracted in the numerator, NPV index = PI ? 1. mihajltp