# CHAPTER 7: NPV AND CAPITAL BUDGETING

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1 CHAPTER 7: NPV AND CAPITAL BUDGETING I. Introduction Assigned problems are 3, 7, 34, 36, and 41. Read Appendix A. The key to analyzing a new project is to think incrementally. We calculate the incremental cash flows that are associated with the project, i.e., how will the corporation s total after-tax cash flows change if this project is accepted. (1) Don t forget about future inflation when estimating future cash flows! (2) Include all side effects: the project may harm or enhance existing operations. (3) Do not include sunk costs: any money spent in the past is irrelevant. The only cash flows that matter are those that occur now and in the future. (4) Include any opportunity costs. Any asset used for a project might have a higher value in some alternative use. A hypothetical example follows. Example of opportunity cost: Let r=10%. An existing asset belonging to a firm is to be used to support a proposed project. The asset can be sold now for \$100,000 or in five years for \$120,000. There is an opportunity cost to not selling the asset today. NPV of foregone opportunity = -100, ,000/(10.10) 5 = -\$25,489 Perhaps it might be more economical to sell this asset today and to purchase or lease a similar asset. Business investment requires investment in two types of assets: long-term and short-term assets. The long-term assets are the plant, property, and equipment, i.e., assets that will be depreciated over the coming years. The short-term assets include the increased cash, accounts receivable, or inventory that is necessary to support a project. Any increase in short-term assets that is funded or financed with long-term sources is called Net Working Capital and must be included in the calculation of a project s NPV. Example: Let r=10%. A project requires an initial increase of \$10,000 in additional cash and inventory to support it over its 5-year life. This \$10,000 will be recovered at the termination of this project in five years. While this \$10,000 is Page 1

2 tied up for the project, it can t be invested elsewhere in the firm or be distributed to the shareholders, and thus we can estimate the effect of this foregone opportunity. NPV of foregone opportunity = -10,000 10,000/(10.10) 5 = -\$3791 Anything that reduces the amount of resources tied up in cash and inventory would increase the NPV of the project. II. Cash Flow Estimation (1) The initial cost of a project, referred to as Net Investment (NINV) or 0, is calculated as: Equipment Shipping/installation Installed Cost (a) Initial increase in Net Working Capital Proceeds from sale of existing assets Net Investment before taxes Tax on sale of existing assets (b) Net Investment Notes concerning calculation of initial investment: (a) The Installed Cost is the amount that is depreciated over the life of the project. (b) If sale amount > remaining book value: tax paid = [tax rate][sale price book value] If sale amount < remaining book value, then a tax credit exists, using this formula Book Value Installed Cost Cumulative Depreciation (2) The annual net after-tax incremental cash flows 1 through n must be estimated. The general formula follows: i =[ revenue costs depreciation][1 tax rate] depreciation NWC (a) / Salvage or Terminal (b) cash flows Notes concerning calculation of annual net cash flows: (a) NWC represents changes in Net Working Capital. NWC is positive when additional investment in NWC is needed. (b) The Salvage or Terminal cash flows include such items as: sale of assets and taxes on the sale of those assets, and costs associated with the disposal of a project, e.g., environmental cleanup costs. Page 2

3 The net cash flows always ignore the interest costs associated with any debt financing. Thus these cash flows are calculated as if the project were all equity financed. The actual interest cost of any debt financing is reflected in the cost of capital r that is used to calculate the NPV. The cost of capital r is a weighted average of the costs of the debt and equity capital used (covered in Ch. 12 & 17). III. Project Cash Flow and NPV Example A firm has a proposed 5-year project. If accepted today (t=0), the project is expected to generate positive net cash flows for each of the following five years (t=1 through t=5). A new machine will be put in service, and to implement this project, an old machine will be sold. Note the following items: The new machine costs \$1,000,000. Shipping and installation will cost an additional \$500,000. Thus the Installed Cost is \$1,500,000. This new machine will be sold five years from today when this project is completed. We believe that it can be sold for \$100,000 in five years. If this project is accepted, then an old, fully depreciated, machine will be replaced. The old machine can be sold for \$50,000 today. The Installed Cost of this new asset will be depreciated using the IRS 5-year MACRS schedule: year 1, 20%; year 2, 32%; year 3, 19.2%; year 4, 11.52%; year 5, 11.52%; and year 6, 5.76%. Note that these yearly amounts sum to 100%. The project will increase revenues and operating expenses (before depreciation and amortization) by \$800,000 and \$300,000 per year, respectively, for each of the following five years (t=1 to t=5). An initial increase in Net Working Capital of \$50,000 is required today and this amount will be recovered in 5 years when the project is terminated. No other changes in NWC will be required during the project s life. The cost of capital of the project is r=11%. The corporate income tax rate is 40%. Calculation of the project s Initial Investment or 0: The old machine is sold for \$50,000 today and has no remaining book value (fully depreciated). The tax owed on the sale of the old machine is calculated as: tax paid = [tax rate][sale price book value] = [0.40][50,000 0] = \$20,000 Page 3

4 Equipment 1,000,000 Shipping/installation 500,000 Installed Cost 1,500,000 Initial increase in Net Working Capital 50,000 Proceeds from sale of existing assets 50,000 Net Investment before taxes 1,500,000 Tax on sale of existing assets 20,000 Net Investment (an outflow) 1,520,000 If accepted, this project requires an initial outlay of \$1,520,000 cash today. Thus 0 = -\$1,520,000. Calculation of the project s future Net Cash Flows: First, calculate the annual depreciation expense (depreciation is a non-cash expense). Note that an IRS 5-year schedule covers six years. The total amount to be depreciated is the original Installed Cost of \$1,500,000. Year Annual Depreciation Expense 1 [0.20][1,500,000] = \$300,000 2 [0.32][1,500,000] = \$480,000 3 [0.192][1,500,000] = \$288,000 4 [0.1152][1,500,000] = \$172,800 5 [0.1152][1,500,000] = \$172,800 6 [0.0576][1,500,000] = \$86,400 Note that this project is terminated exactly five years from today at t=5, so the last amount to be depreciated is the Year 5 amount of \$172,800. The Year 6 depreciation of \$86,400 is never expensed. Therefore, when the project is terminated at t=5, the remaining accounting Book Value of this asset will be \$86,400. When the project is terminated five years from today (t=5), some salvage or terminal cash flows must be calculated. The machine will be sold for \$100,000 at that time. The remaining book value will be \$86,400. Therefore, the taxes owed at that time on the sale will be: [tax rate][sale price book value] = [0.40][100,000 86,400] = \$5440 Now we calculate annual after-tax incremental cash flows 1, 2, 3, 4, and 5 using the following equation: Page 4

5 i = [ revenue costs depreciation][1 tax rate] depreciation NWC / Salvage or Terminal cash flows 1 = [800, , ,000][1 0.4] 300,000 0 = \$420,000 2 = [800, , ,000][1 0.4] 480,000 0 = \$492,000 3 = [800, , ,000][1 0.4] 288,000 0 = \$415,200 4 = [800, , ,800][1 0.4] 172,800 0 = \$369,120 Note that 5 must include the recovery of the \$50,000 of NWC, sale of machine for \$100,000, and tax payment of \$5440 on the sale of the machine. 5 = [800, , ,800][1 0.4] 172,800 ( 50,000) 100, = \$513,680 Now we have estimates of all the project cash flows 1 through 5. Using the cost of capital r=11%, we can calculate the NPV. NPV = 0 NPV = -1,520, ( 1 r) ( 1 r) ( 1 r) ( 1 r) ( 1 r) 420, , , , ,680 = \$109,282 ( ) ( ) ( ) ( ) ( ) NPV = \$109,282, therefore if this is a stand-alone project, it should be accepted. Also, the IRR= %, which is greater than the hurdle rate of r=11%. IV. Inflation and Capital Budgeting Here, the term nominal refers to the effective annual rate of interest or absolute dollars (not adjusted by inflation into current dollars). Let r=10% per year in capital markets; we call this the nominal rate, as opposed to the real rate. Let the expected rate of inflation be π=5% per year. One widget costs \$1 today. Next year it should cost (\$1)(10.05) = \$1.05. The same \$1 invested in capital markets today grows to (\$1)(10.1) = \$1.10 next year. This year you can buy 1 widget with your \$1. Invest the dollar for one year in the capital markets instead and next year you can buy \$1.10/\$1.05 = widgets. Your purchasing power has increased by 4.76% over the year, and therefore you have earned a real interest rate of 4.76% by investing in capital markets. Page 5

6 For any number of periods, you can always convert nominal cash flows to real (t=0 purchasing power) by discounting by the rate of inflation: (1r real ) n = (1r) n /(1π) n 0, real = n, nominal /(1π) n Any future nominal amount of cash can be converted into current or real dollars by discounting it at the inflation rate. For instance, \$400,000 is expected in t=4 years. r=10% and π=5%. This amount in current or real dollars is 400,000/(10.05) 4 = \$329,081. NPV analysis can be done using one of two techniques: (1) Estimate all future cash flows as nominal and then discount the cash flows back to today at the nominal cost of capital. This is the most common method and the one we have used so far in this Chapter. Remember to calculate the project s future cash flows with inflation in mind! 1 (2) Estimate all future cash flows as real or current dollars and then discount the cash flows back to today at the real cost of capital. If done correctly, the NPV is identical between methods (1) and (2) above. Don t make the mistake of estimating all the cash flows as current dollars, and then discount them at the nominal rate. V. Investments of Unequal Life Equivalent Annual Cost (EAC) method: (1) Find the PV 0 of the costs of a machine, and (2) express this PV as an annuity. This method is easier if we use constant or current dollars and real discount rates. In other words, inflation has been removed from all the dollars and discount rates. Example: A machine costs \$12,000 to purchase and costs \$3000 per year to maintain over its 4-year life. The machine can be sold for \$2000 at the end of year 4. 2 All dollar amounts given are in current dollars. The real cost of capital is r=6% per year. The time line of expected costs is shown below. t=0 t=1 t=2 t=3 t=4-12, = The r=11% cost of capital in the Part III capital budgeting example already accounts for the following: (1) expected inflation, (2) real rate of interest expected on riskless govt. bonds, and (3) the risk premium of the project. 2 The assumption in this example is that this machine can be replaced at t=4 years with an identical machine, having the same annual costs. Page 6

7 Step 1: Calculate the PV of all these cash flows. Since all of these net cash flows are negative, and hence net costs, the minus sign below is dropped for simplicity. PV 0 = /(10.06) 3000/(10.06) /(10.06) /(10.06) 4 PV 0 = 20, Step 2: Express this amount as the PV 0 of a 4-year ordinary annuity and calculate the annual cash flow of the annuity. 1 1 PV0 = C - r n = C - C = r( 1 r) ( ) This machine has an Equivalent Annual Cost or EAC of \$ t=0 t=1 t=2 t=3 t= What if this machine is being compared to another machine having a life of 8 years and an Equivalent Annual Cost of \$6500? The choice is as follows: We can buy either one 8-year machine or two consecutive 4-year machines, and each choice then provides an operating life of 8 years. The best choice is to purchase the machine with the lower EAC: buy the 4-year life with EAC = \$ and replace it with a similar machine in four years. If the 8-year machine were to have an EAC < \$ , then the 8-year machine would be purchased. Decision to replace: An existing machine is expected to have the following annual maintenance and salvage values. The machine has five years remaining in its economic life. This machine achieves the same function as the new 4-year machine having an EAC of \$ in prior example. The real cost of capital is still r=6%. When should this existing machine be replaced with the new 4-year machine having EAC=\$ ? The old machine s estimated cost/salvage schedule is as follows (assume that maintenance costs occur at year end). Page 7

8 Year Maintenance costs Salvage value t=0 (now) t= t= t= t= t= If the new machine is purchased, then it will have an EAC of \$ with the first EAC amount placed exactly one year from now (t=1 years). Now find what will be the actual cost of the old machine if left in operation for the following year. If left in operation for one more year, we forego receiving the salvage sale amount of \$8000 today, and then at t=1 we pay \$4000 maintenance and sell the machine for \$6000. The PV 0 of costs of keeping the old machine operating for one more year is: PV 0 = /(10.06) 6000/(10.06) = \$ The new machine s EAC=\$ occurs at t=1 on the time line shown in prior example. To adequately compare the costs of the old versus new machines, we therefore must convert this old machine s PV 0 =\$ to its FV at t=1: FV 1 = (10.06) = \$6480. Since this cost of \$6480 is greater than the new machine s EAC of \$ , the old machine should be immediately replaced. (What if) Leaving the old machine in service beyond t=1: If actually left in service, rather than replaced today, then what would the old machine cost to operate for one additional year beyond t=1. We calculate the costs of operating the machine from t=1 to t=2 years. The salvage value will fall from \$6000 to \$4000 over this second year. PV 1 = /(10.06) 4000/(10.06) = \$ Now convert this value to t=2. FV 2 = (10.06) = \$6860. This amount is still higher than the \$ EAC of the new machine. We clearly see that the annual cost of operating the old machine only increases during the second year and thus further underscores why the old machine should be replaced today. Page 8

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