Chapter 5 Solutions
Problem 9
Cash Flow:Year 0 -$4,750,000
Year 1 $4,000,000
Year 2 $4,000,000
Year 3-$3,000,000
Plot the NPV Profile…
One way is to first solve the NPV with different discount rates…
NPV = -$4,750,000 + $4,000,000/(1+r) + $4,000,000/(1 + r)2 - $3,000,000/(1+r)3
Discount rate at zero
NPV = -$4,750,000 + $4,000,000 + $4,000,000 -$3,000,000 = $250,000
Discount rate at 5%
NPV = -$4,750,000 + $3,809,524 + $3,628,118 -$2,591,513 = $96,129
Discount rate at 10%
NPV = -$4,750,000 + $3,636,364 + $3,305,785 -$2,253,944 = -$61,795
Discount rate at 15%
NPV = -$4,750,000 + $3,478,261 + $3,024,575 -$1,972,549 = -$219,713
Discount rate at 16%
NPV = -$4,750,000 + $3,448,276 + $2,972,652 -$1,921,973 = -$251,045
Draw Profile with these four points?
What is the IRR…solve for NPV = 0, this is an iterative process or on calculator…
CF0 = -$4,750,000
C01 = $4,000,000, and F01 = 1
C02 = $4,000,000, and F02 = 1
C03 = -$3,000,000, and F03 = 1
Compute IRR = 8.049362 or 8.05% and we reject at 16% cost of equity
Problem 10
What is the MIRR of the above project? We will use the 16% reinvestment rate for the intermediate cash flow…
Year 1: FV = $4,000,000 x (1.16)2 = $5,382,000
Year 2: FV = $4,000,000 x (1.16) = $4,640,000
Year 3: FV = -$3,000,000
Total FV is $7,022,400
MIRR is ($7,022,400/$4,750,000)1/3 – 1 = 13.9% still below the hurdle rate so decision is still to pass on the project.
Problem 11
Two mutual exclusive projects…
- NPV at 10% WACC…
Project A NPV = -$4,000 + $2,000/(1.1) + $1,500/(1.1)2 + $1,250/(1.1)3 + $1,000/(1.1)4 = -$4,000 + $1,818 + $1,240 + $939 + $683 = $680
Project B NPV = -$4,000 + $1,000/(1.1) + $1,500/(1.1)2 + $1,700/(1.1)3 + $2,400/(1.1)4 = -$4,000 + $909 + $1,240 + $1,277 + $1,639 = $1,065
Pick Project B
- IRR is where NPV is zero
Project A 0 = -$4,000 + $2,000/(1+r) + $1,500/(1+r)2 + $1,250/(1+r)3 + $1,000/(1+r)4 and computing r we have…IRR = 18.71%
Project B 0 = -$4,000 + $1,000/(1+r) + $1,500/(1+r)2 + $1,700/(1+r)3 + $2,400/(1+r)4 and computing r we have…IRR = 20.16%
Pick B, again.
- Reinvestment rate for NPV is the WACC or hurdle rate or in this case 10%. Reinvestment rate for IRR is the IRR rate, so for Project A it is 18.71% and for Project B it is 20.16%. Not sure what the author means by showing the effect of each of these rates…one could look at the FV of the cash flows at the IRR rate
For project A we have:
Project A FV = $2,000 (1+0.1871)3 + $1,500 (1+ 0.1871)2 + $1,250 (1+ 0.1871)1 + $1,000 (1+ 0.1871)0 and computing FV we have FV = $7,719.48
Project B FV = $1,000 (1+0.2016)3 + $1,500 (1+ 0.2016)2 + $1,700 (1+ 0.2016)1 + $2,400 (1+ 0.2016)0 and computing FV we have FV = $8,343.41
One could then compare these numbers with the FV if the funds were reinvested at the WACC of the company (hurdle rate of 10%).
Project A FV = $2,000 (1+0.10)3 + $1,500 (1+ 0.10)2 + $1,250 (1+ 0.10)1 + $1,000 (1+ 0.10)0 and computing FV we have FV = $6,852.00
Project B FV = $1,000 (1+0. 10)3 + $1,500 (1+ 0. 10)2 + $1,700 (1+ 0. 10)1 + $2,400 (1+ 0.10)0 and computing FV we have FV = $7,471.00
Project A impact: $7,719.48 vs. $6,852.00 for a delta of $867.48
Project B impact: $8,343.41 vs. $7,471.00 for a delta of $872.41
- MIRR will use 10% for reinvestment rate for the two projects under the IRR model…
Project A future cash flows at 10%
Year 1 = $2,000 x 1.103 = $2,662
Year 2 = $1,500 x 1.102 = $1,815
Year 3 = $1,240 x 1.10 = $1,364
Year 4 = $1,000
FV = $2,662 + $1,815 + $1,364 + $1,000 = $6,841
MIRR = ($6,841 / $4,000)1/4 – 1 = 14.35%
Project B future cash flows at 10%
Year 1 = $1,000 x 1.103 = $1,331
Year 2 = $1,500 x 1.102 = $1,815
Year 3 = $1,700 x 1.10 = $1,870
Year 4 = $2,400
FV = $1,331 + $1,815 + $1,870 + $2,400 = $7,416
MIRR = ($7,416 / $4,000)1/4 – 1 = 16.69%
Problem 12
The IRR must have at least one year with a negative cash flow in order to estimate the discount rate that equates the benefits to the costs. But if a project has only positive cash flow every year…you do not need a model to say yes to the project. The same is not true for the NPV model. If all the future cash flow is positive you can still find the present value of these cash flows…its just that you will always get a positive NPV if you are only adding up positive numbers.
Problem 16
You need to first determine the annual cash flow of the project.
Year zero: -$20,000,000
Years one to thirty: EBIT $3,000,000
Taxes - 1,200,000
Depreciation+ 500,000
OCF$2,300,000
In Year 10 and 20 additional out flow of $5,000,000 so these years have -$2,700,000 cash flow.
In Year 30 you will “salvage” equipment at book value…(no tax issues).
Depreciation over 30 years is $500,000 x 30 = $15,000,000 so remaining book value is $5,000,000. So cash flow is $2,300,000 plus the $5,000,000 or $7,300,000 or do you also count the two extra $5,000,000 so the salvage is $15,000,000?
a. Table of Cash Flows for the Project (in thousands):
Year 01 to 9 10 11 to 19 20 21 to 29 30
-$20,000$2,300-$2,700$2,300-$2,700 $2,300 $7,300
You can short cut this problem by noting that you have an annuity for years 1 through 30…then just add in the other pieces…
-$20,000 - $5,000/(1.125)10 -$5,000/(1.125)20 + $5,000/(1.125)30 + $2,300 x (PVIFA)
-$20,000 -$1,540 -$ 474 + $146 + $17,863 = -$4,005
-$20,000 - $5,000/(1.125)10 -$5,000/(1.125)20 + $15,000/(1.125)30 + $2,300 x (PVIFA)
-$20,000 -$1,540 -$ 474 + $438 + $17,863 = -$3,713
b. Using calculator to solve IRR we have:
CF0 = -20,000
C01 = 2,300 F01 = 9
C02 = -2,700 F02 = 1
C03 = 2,300 F03 = 9
C04 = -2,700 F04 = 1
C05 = 2,300 F05 = 9
C06 = 7,300 F06 = 1
Compute IRR and you get 9.57% so the IRR is less than the 12.5% hurdle rate.
Problem 17
The cash flows are and discount rates are…method one…
In method one I assume that the discount rates are given in annual terms and reflect the average over the period of the discount rate (for example, the 12.5% is the three year discount rate not the discount rate in the third year).
Year Cash flow Discount RatePVCF
0 -$15,000---$15,000
1 $5,000 10.5%$5,000/(1.105) = $4,525
2 $5,000 11.5%$5,000/(1.115)2 = $4,022
3 $10,000 12.5%$10,000/(1.125)3 =$7,023
NPV = -$15,000 + $4,525 +$4,022 + $7,023 = $570
Method two…
In method one I assume that the discount rates are given in annual terms and reflect the rate over that specific year not the entire period (for example, the 12.5% is the thirdyear discount rate not the discount rate for the three years).
Year Cash flow Discount RatePVCF
0 -$15,000---$15,000
1 $5,000 10.5%$5,000/(1.105) = $4,525
2 $5,000 11.5%$5,000/(1.105)(1.115) = $4,058
3 $10,000 12.5%$10,000/(1.105)(1.115)(1.125)= $7,214
NPV = -$15,000 + $4,525 +$4,058 + $7,214 = $797
b. Find the IRR…here the discount rates do not matter…you are not using them to find IRR
CF0 = -15,000
C01 = 5,000, F01 =2
C03 = 10,000, F03 = 1
Compute IRR = 13.94%
Now, the IRR exceeds all of the discount rates so you would accept the project based on IRR > all hurdle rates.
Problem 19
Again, start with the cash flow…
Year(s)Cash Flow
0-$50,000,000
1 – 9 $ 5,000,000
10 -$15,000,000 ($20,000,000 out plus $5,000,000 in)
11-20 $ 5,000,000
NPV = -$50,000 - $20,000/(1.1)10 + $5,000,000 (PVIFA);
NPV = -$50,000,000- $7,710,865 + 42,567,819 = -$15,413,047
Again for an NPV profile you will need to put in various discount rates….
NPV at 0% = $30,000,000
NPV at 2% = $15,350,201
NPV at 4% = $4,440,348
NPV at 6% = -$3,818,289
NPV at 8% = -$10,173,133
NPV at 10% = -$15,413,047
NPV at 12% = -$19,092,247