12.2 Based On NPV
К оглавлению1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97
Problem. Suppose a firm is considering the following project, where all of the dollar figures are in
thousands of dollars. In year 0, the project requires $11,350 investment in plant and equipment, is
depreciated using the straight-line method over seven years, and there is a salvage value of $1,400 in year
7. The project is forecast to generate sales of 2,100 units in year 1 and grow at a sales growth rate of
55.0% in year 2. The sales growth rate is forecast to decline by 15.0% in years 3 and 4, to decline by
20.0% in year 5, to decline by 25.0% in year 6, to decline by 30.0% in year 7. Unit sales will drop to zero
in year 8. The inflation rate is forecast to be 2.0% in year 1, rising to 4.0% in year 5, and then leveling off.
The real cost of capital is forecast to be 11.0% in year 1, rising to 12.2% in year 5, and then leveling off.
The tax rate is forecast to be a constant 35.0%. Sales revenue per unit is forecast to be $9.70 in year 1 and
then grow with inflation. Variable cost per unit is forecast to be $7.40 in year 1 and then grow with
inflation. Cash fixed costs are forecast to be $5,280 in year 1 and then grow with inflation. What is the
project NPV? What is the NPV Break-Even Point in Year 1 Unit Sales, where NPV equals zero? What is
the NPV Break-Even Point in the Year 2 Sales Growth Rate, where NPV equals zero? What is the NPV
Break-Even Contour in the two-dimensional space of Year 1 Unit Sales and Year 2 Sales Growth Rate?
Solution Strategy. Start with the Project NPV - Basics spreadsheet. Move the Unit Sales line out of the
Key Assumptions area, since that is what we are going to solve for. Restructure the Unit Sales forecast to
depend on the Sales Growth Rate, which we be a key variable. Structure the Sales Grow Rate forecast
over the entire to period to depend on how fast the growth rate is initially. This will make it easy to use
Solver and to create a Data Table later on. Project the cash flows of the project and calculate the NPV.
Use Solver to determine the amount of year 1 unit sales that will cause the NPV to equal zero, when the
sales growth rate is at the base case level of 5% per year. Use Solver to determine the sales growth rate
that will cause the NPV to equal zero, when the year 1 unit sales is at the base case level of 39,000. Create
a two-variable data table using two input variables (year 1 unit sales and sales growth rate) and the output
variable: NPV. Use the data table to create a three-dimensional graph showing the NPV Break-Even
Contour.
FIGURE 12.4 Spreadsheet for Break-Even Analysis Based On Net Present Value.
How To Build This Spreadsheet Model.
1. Open the Basics Spreadsheet and Add Rows. Open the spreadsheet that you created for Project
NPV - Basics and immediately save the spreadsheet under a new name using the File | Save As
command. Select the cell A15 and click on Insert | Row. Select the range A5:I5, click on Edit |
Cut, select the cell A15, and click on Edit | Paste.
2. Inputs. Enter the year 2 sales growth rate into the cell D5, the change in the sales growth rate into
the range E6:I6, and the year 1 unit sales of 2,100 in cell C16.
3. Sales Growth Rate on date t = (Sales Growth Rate on date t-1) + (Change in Sales Growth Rate
on date t). Enter =D5+E6 in cell E5 and copy it across.
4. Unit Sales on date t = (Unit Sales on date t-1) * (1 + Unit Sales Growth Rate). Enter
=C16*(1+D5) in cell D16 and copy it across.
The project NPV is $3,217 and should be accepted. But how sure are you of this result? How sensitive is
this result to small changes in the assumptions? The Break-Even Point gives you an idea of the robustness
of this result.
5. NPV Break-Even Point in Year 1 Unit Sales. Use the built-in Solver tool to numerically solve
for the NPV Break-Even Point in Year 1 Unit Sales. From Excel’s main menu, click on Tools and
then Solver. (If Solver does not appear on the Tools menu, then click on Add-Ins, check Solver
Add-In, and click on OK.) In the Solver dialog box, enter the Net Present Value cell B40 as the
Set Target Cell. In the Equal To row, click on the option button for Value of and enter 0 in the
adjacent box. Enter the Year 1 Unit Sales cell C16 as the By Changing Cell. See figure below.
Click on the Solve button.
FIGURE 12.5 Solver dialog box.
By trial and error, the Solver adjusts the value of the Year 1 Unit Sales in cell C16 until the Net
Present Value in cell B40 equals zero (within a very small error tolerance). This results in a NPV
Break-Even Point in Year 1 Unit Sales (shown in cell C16) of 1,853.
6. NPV Break-Even Point in Sales Growth Rate. Repeat the NPV Break-Even Analysis only
using Year 2 Sales Growth Rate as the changing cell. Enter 2100 in cell C16 in order to restore
the default assumption for Year 1 Unit Sales. From Excel’s main menu, click on Tools and then
Solver. In the Solver dialog box, enter Year 2 Sales Growth Rate cell D5 as the By Changing
Cell. Click on the Solve button. By trial and error, the Solver adjusts the value of the Sales
Growth Rate in cell D5 until the Net Present Value in cell B40 equals zero. This results in a NPV
Break-Even Point in Sales Growth Rate (shown in cell D5) of 49.5%.
FIGURE 12.6 Two Way Data Table and 3D Graph.
7. Create A List of Input Values and An Output Formula. Create a list of input values for the
Year 1 Unit Sales (1,700, 1,900, 2,100, etc.) in the range C45:F45. Similarly, create a list of input
values for the Year 2 Sales Growth Rate (45.0%, 50.0%, 55.0%, etc.) in the range B46:B50.
Create an output formula that references the Net Present Value by entering the formula =B40 in
cell B45.
8. Two-Variable Data Table. Select the range B45:F50 for the Two-Variable Data Table. This
range includes both the list of input values at the top and side of the data table and the output
formula in the upper left corner. Then choose Data | Table from the main menu and a Table
dialog box pops up. Enter the cell address C16 (Year 1 Unit Sales) in the Row Input Cell, enter
the cell address D5 (Year 2 Sales Growth Rate) in the Column Input Cell, and click on OK. The
data table shows what combinations of Year 1 Unit Sales and Year 2 Sales Growth yield a
positive NPV. Thus, you can assess how optimistic vs. pessimistic your assumptions have to be in
order to a get a positive NPV. Thus, you see how robust your conclusions are to variations in the
inputs.
9. 3-D Graph. Highlight the interior of the data table (excluding the top or side) in the range
C46:F50 and then choose Insert | Chart from the main menu. Select a Surface chart type and
make other selections to complete the Chart Wizard.
The 3-D Graph shows the Net Present Value of the project for combinations of Year 1 Unit Sales and
Year 2 Sales Growth Rate. The multi-color surface illustrates various ranges of NPV. In the top corner,
the dark blue color is for NPV > $15,000. Below it, a light red section is for a NPV of $10,000 to
$15,000. And so on. At the intersection of the Light Green section ($0 to $5,000) and the Light Yellow
section (-$5,000 to $0) is a contour highlighted by the arrow. This is the NPV Break-Even Contour,
where NPV = 0. Every point on this contour represents a combination of Year 1 Unit Sales and Year 2
Sales Growth Rate for which the NPV = 0. The 3-D Graph shows that project's positive NPV is very
sensitive. If the Year 1 Unit Sales are a little bit lower than assumed or if the year 2 Sale Growth Rate is a
little bit lower than assumed, then the whole project could have a negative NPV.
Problem. Suppose a firm is considering the following project, where all of the dollar figures are in
thousands of dollars. In year 0, the project requires $11,350 investment in plant and equipment, is
depreciated using the straight-line method over seven years, and there is a salvage value of $1,400 in year
7. The project is forecast to generate sales of 2,100 units in year 1 and grow at a sales growth rate of
55.0% in year 2. The sales growth rate is forecast to decline by 15.0% in years 3 and 4, to decline by
20.0% in year 5, to decline by 25.0% in year 6, to decline by 30.0% in year 7. Unit sales will drop to zero
in year 8. The inflation rate is forecast to be 2.0% in year 1, rising to 4.0% in year 5, and then leveling off.
The real cost of capital is forecast to be 11.0% in year 1, rising to 12.2% in year 5, and then leveling off.
The tax rate is forecast to be a constant 35.0%. Sales revenue per unit is forecast to be $9.70 in year 1 and
then grow with inflation. Variable cost per unit is forecast to be $7.40 in year 1 and then grow with
inflation. Cash fixed costs are forecast to be $5,280 in year 1 and then grow with inflation. What is the
project NPV? What is the NPV Break-Even Point in Year 1 Unit Sales, where NPV equals zero? What is
the NPV Break-Even Point in the Year 2 Sales Growth Rate, where NPV equals zero? What is the NPV
Break-Even Contour in the two-dimensional space of Year 1 Unit Sales and Year 2 Sales Growth Rate?
Solution Strategy. Start with the Project NPV - Basics spreadsheet. Move the Unit Sales line out of the
Key Assumptions area, since that is what we are going to solve for. Restructure the Unit Sales forecast to
depend on the Sales Growth Rate, which we be a key variable. Structure the Sales Grow Rate forecast
over the entire to period to depend on how fast the growth rate is initially. This will make it easy to use
Solver and to create a Data Table later on. Project the cash flows of the project and calculate the NPV.
Use Solver to determine the amount of year 1 unit sales that will cause the NPV to equal zero, when the
sales growth rate is at the base case level of 5% per year. Use Solver to determine the sales growth rate
that will cause the NPV to equal zero, when the year 1 unit sales is at the base case level of 39,000. Create
a two-variable data table using two input variables (year 1 unit sales and sales growth rate) and the output
variable: NPV. Use the data table to create a three-dimensional graph showing the NPV Break-Even
Contour.
FIGURE 12.4 Spreadsheet for Break-Even Analysis Based On Net Present Value.
How To Build This Spreadsheet Model.
1. Open the Basics Spreadsheet and Add Rows. Open the spreadsheet that you created for Project
NPV - Basics and immediately save the spreadsheet under a new name using the File | Save As
command. Select the cell A15 and click on Insert | Row. Select the range A5:I5, click on Edit |
Cut, select the cell A15, and click on Edit | Paste.
2. Inputs. Enter the year 2 sales growth rate into the cell D5, the change in the sales growth rate into
the range E6:I6, and the year 1 unit sales of 2,100 in cell C16.
3. Sales Growth Rate on date t = (Sales Growth Rate on date t-1) + (Change in Sales Growth Rate
on date t). Enter =D5+E6 in cell E5 and copy it across.
4. Unit Sales on date t = (Unit Sales on date t-1) * (1 + Unit Sales Growth Rate). Enter
=C16*(1+D5) in cell D16 and copy it across.
The project NPV is $3,217 and should be accepted. But how sure are you of this result? How sensitive is
this result to small changes in the assumptions? The Break-Even Point gives you an idea of the robustness
of this result.
5. NPV Break-Even Point in Year 1 Unit Sales. Use the built-in Solver tool to numerically solve
for the NPV Break-Even Point in Year 1 Unit Sales. From Excel’s main menu, click on Tools and
then Solver. (If Solver does not appear on the Tools menu, then click on Add-Ins, check Solver
Add-In, and click on OK.) In the Solver dialog box, enter the Net Present Value cell B40 as the
Set Target Cell. In the Equal To row, click on the option button for Value of and enter 0 in the
adjacent box. Enter the Year 1 Unit Sales cell C16 as the By Changing Cell. See figure below.
Click on the Solve button.
FIGURE 12.5 Solver dialog box.
By trial and error, the Solver adjusts the value of the Year 1 Unit Sales in cell C16 until the Net
Present Value in cell B40 equals zero (within a very small error tolerance). This results in a NPV
Break-Even Point in Year 1 Unit Sales (shown in cell C16) of 1,853.
6. NPV Break-Even Point in Sales Growth Rate. Repeat the NPV Break-Even Analysis only
using Year 2 Sales Growth Rate as the changing cell. Enter 2100 in cell C16 in order to restore
the default assumption for Year 1 Unit Sales. From Excel’s main menu, click on Tools and then
Solver. In the Solver dialog box, enter Year 2 Sales Growth Rate cell D5 as the By Changing
Cell. Click on the Solve button. By trial and error, the Solver adjusts the value of the Sales
Growth Rate in cell D5 until the Net Present Value in cell B40 equals zero. This results in a NPV
Break-Even Point in Sales Growth Rate (shown in cell D5) of 49.5%.
FIGURE 12.6 Two Way Data Table and 3D Graph.
7. Create A List of Input Values and An Output Formula. Create a list of input values for the
Year 1 Unit Sales (1,700, 1,900, 2,100, etc.) in the range C45:F45. Similarly, create a list of input
values for the Year 2 Sales Growth Rate (45.0%, 50.0%, 55.0%, etc.) in the range B46:B50.
Create an output formula that references the Net Present Value by entering the formula =B40 in
cell B45.
8. Two-Variable Data Table. Select the range B45:F50 for the Two-Variable Data Table. This
range includes both the list of input values at the top and side of the data table and the output
formula in the upper left corner. Then choose Data | Table from the main menu and a Table
dialog box pops up. Enter the cell address C16 (Year 1 Unit Sales) in the Row Input Cell, enter
the cell address D5 (Year 2 Sales Growth Rate) in the Column Input Cell, and click on OK. The
data table shows what combinations of Year 1 Unit Sales and Year 2 Sales Growth yield a
positive NPV. Thus, you can assess how optimistic vs. pessimistic your assumptions have to be in
order to a get a positive NPV. Thus, you see how robust your conclusions are to variations in the
inputs.
9. 3-D Graph. Highlight the interior of the data table (excluding the top or side) in the range
C46:F50 and then choose Insert | Chart from the main menu. Select a Surface chart type and
make other selections to complete the Chart Wizard.
The 3-D Graph shows the Net Present Value of the project for combinations of Year 1 Unit Sales and
Year 2 Sales Growth Rate. The multi-color surface illustrates various ranges of NPV. In the top corner,
the dark blue color is for NPV > $15,000. Below it, a light red section is for a NPV of $10,000 to
$15,000. And so on. At the intersection of the Light Green section ($0 to $5,000) and the Light Yellow
section (-$5,000 to $0) is a contour highlighted by the arrow. This is the NPV Break-Even Contour,
where NPV = 0. Every point on this contour represents a combination of Year 1 Unit Sales and Year 2
Sales Growth Rate for which the NPV = 0. The 3-D Graph shows that project's positive NPV is very
sensitive. If the Year 1 Unit Sales are a little bit lower than assumed or if the year 2 Sale Growth Rate is a
little bit lower than assumed, then the whole project could have a negative NPV.