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**Managerial Accounting by Whitecotton, Libby, and Phillips, second edition.**

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**Chapter 11 Capital Budgeting Chapter 11: Capital Budgeting**

In this chapter, you will learn about a number of methods or tools that managers can use to help them make capital investment decisions.

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**Capital Budgeting Process**

Plant expansion Equipment selection Equipment replacement Capital budgeting is a decision-making approach aimed at helping managers make decisions about investments in major capital assets, such as new facilities, equipment, new products, and research and development projects. Capital budgeting is a decision-making approach aimed at helping managers make decisions about investments in major capital assets, such as new facilities, equipment, new products, and research and development projects. The amounts of money involved are usually large and the investment period is typically many years. Capital budgeting decisions often involve plant expansion, equipment selection, equipment replacement, lease or buy alternatives, and cost reduction issues, such as should new equipment be purchased to reduce costs. Lease or buy Cost reduction

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**Capital Investment Decisions**

There are two main types of capital investment decisions. Screening Decisions Preference Decisions There are two main types of capital budgeting decisions: Screening decisions relate to whether a proposed project passes a preset hurdle. For example, a company may have a policy of accepting projects only if they promise a return of 20% on the investment. Preference decisions relate to selecting among several competing courses of action. For example, a company may be considering several different machines to replace an existing machine on the assembly line. Pertain to whether or not some proposed investment is acceptable; these decisions come first. Attempt to rank acceptable alternatives from the most to least appealing.

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**Capital Investment Decisions**

Capital budgeting decisions can be placed into two categories. Independent Projects Mutually Exclusive Projects Capital budgeting decisions can be placed into two categories: Independent projects are unrelated, so investing in one does not preclude or eliminate investing in the other projects. Mutually exclusive projects involve a choice between competing alternatives, where selection of one of the projects implies rejection of all of the other alternatives. Projects are unrelated, so investing in one does not preclude or eliminate investing in the other projects. Investment choices are competing alternatives, so accepting one leads to rejection of the others.

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**Capital Budgeting Methods**

There are five different methods that managers can use to evaluate capital investment alternatives. Companies often use either the accounting rate of return method or the payback method as an initial screening tool for evaluating capital investments. However, both methods suffer from an important limitation: they are non-discounting methods, which means they ignore the time value of money. The time value of money is the principle that money is more valuable today than it will be in the future. Net present value, internal rate of return, and profitability index are referred to as discounted cash flow methods. These methods are generally superior to non-discounting methods because they incorporate the time value of money.

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**Capital Budgeting Methods**

To illustrate how the five capital budgeting methods work, assume that managers in Apple’s iPod division are considering producing a special version of the iPod Touch that would be marketed to children and their parents. The new device, called the iKids Touch, would be designed to appeal to children, with durable components, “kid friendly” controls, and bright colors. The basic question that managers must answer is whether this proposed project is worth the $1 million up-front investment. To illustrate how the five capital budgeting methods work, assume that managers in Apple’s iPod division are considering producing a special version of the iPod Touch that would be marketed to children and their parents. The new device, called the iKids Touch, would be designed to appeal to children, with durable components, “kid friendly” controls, and bright colors. It would also allow parents to restrict the types of content that can be downloaded onto the device to age-appropriate music, books, and educational games. Managers are evaluating the iKids Touch as an independent project, or one that would not preclude Apple from investing in alternative projects. Some projections related to this hypothetical project are illustrated on this slide. The basic question that managers must answer is whether this proposed project is worth the $1 million up-front investment. We conduct our analysis by using each of the capital budgeting methods.

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Learning Objective 11-1 Calculate the accounting rate of return and describe its major weaknesses. Learning objective 11-1 is to calculate the accounting rate of return and describe its major weaknesses.

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**Accounting Rate of Return**

Annual Net Income Initial Investment Accounting Rate of Return ÷ = The accounting rate of return, sometimes called the annual rate of return, is an estimate of the average annual return on investment that a project is expected to generate over its life. We calculate the annual rate of return by dividing annual after-tax net income by the initial investment in assets used to generate the annual after-tax net income. The annual rate of return for iKids Touch’s project is 10.8 percent. If 10.8 percent is above iKids Touch’s required rate of return (sometimes called the hurdle rate), the project would be acceptable (a screening decision). iKids Touch might also compare the 10.8 percent return on this project with returns on other projects to rank the projects for a preference decision. $108,000 $1,000,000 10.8% ÷ =

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**Accounting Rate of Return**

Shortcoming and Criticisms The time value of money is ignored. The accounting rate of return is based on accounting net income instead of cash flow. Depreciation may be calculated several ways and, in addition, other accounting method alternatives may have an impact on reported net income. The accounting rate of return is a simple and intuitive approach that is often used as a preliminary screening tool to evaluate capital investments. However, it suffers from two major limitations. First, it does not incorporate the time value of money. Second, it is based on accounting net income rather than cash flow. For years, accounting and finance experts have debated the relative merits of net income versus cash flow for making investment decisions. While research suggests that both are useful, the bottom line is that cash is king. Cash is more objective than net income, is not as easily manipulated, and is not influenced by accounting choices such as the method used to depreciate the capital assets.

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**Net Cash Flow versus Net Income**

To convert from net income to net cash flow, we must add back the depreciation that was deducted in the computation of net income, as shown below. In this chapter, we assume that the only difference between net income and net cash flow is depreciation expense. Other differences between net income and net cash flow are discussed in more detail in Chapter 12. To keep things simple, we also ignore the effect of income taxes, which are based on another source of accounting rules set by the IRS. In our Apple example, the original cost of the investment was $1 million. For accounting purposes, this cost will be fully depreciated over the five-year life of the assets, resulting in an annual depreciation expense of $200,000. This amount is subtracted as an expense on the income statement, as shown on this slide. But the company doesn’t actually pay $200,000 in cash for depreciation. The $1,000,000 was paid up-front and is simply allocated across the 5-year life of the project. To convert from net income to net cash flow, we must add back the depreciation that was deducted in the computation of net income, as shown on this slide. The capital budgeting techniques illustrated throughout the remainder of this chapter use annual net cash flow of $308,000 to evaluate the capital investment rather than $108,000 in net income.

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**Calculate the payback period and describe its major weaknesses.**

Learning Objective 11-2 Calculate the payback period and describe its major weaknesses. Learning objective 11-2 is to calculate the payback period and describe its major weaknesses.

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**Net Income + Depreciation**

Payback Period Initial Investment Annual Net Cash Flow Payback Period ÷ = Net Income + Depreciation The payback period is the length of time it takes a project to recover its initial cost. We can calculate the payback period by dividing the initial investment required, by the annual net cash flow. Because depreciation is a non-cash expense that was deducted in the calculation of net income, we add this amount back to net income to get to the annual cash flow. In general a shorter payback period is better than a longer payback period. The payback period for iKids Touch’s project is 3.25 years. iKids Touch may have an investment decision rule such as: invest only in projects with a payback period of five years or less. If so, they would invest in the iKids Touch project because its payback period is less than five years. 3.25 years $1,000,000 $308,000 ÷ = $108,000 + $200,000

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**The payback period is somewhere between 3 and 4 years.**

When annual cash flows are unequal, the payback period must be computed on a year by year basis by subtracting the net cash flow from the unpaid investment balance each year. When annual cash flows are unequal, the payback period must be computed on a year by year basis by subtracting the net cash flow from the unpaid investment balance each year. Given the information shown for iKids Touch, the payback period is somewhere between 3 and 4 years. The payback period is somewhere between 3 and 4 years.

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**Payback Period Shortcoming and Criticisms**

The time value of money is ignored. The payback period ignores cash flows after the payback period. The payback method is relatively simple and is commonly used to determine how long it will take for a project to recoup its initial investment. However, as with the accounting rate of return method, it does not incorporate the time value of money. In addition, this method does not take into account anything that happens after the payback period. It ignores any benefits (and costs) that occur after the project has paid for itself.

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Time Value of Money One dollar received today is worth more than one dollar received a year from now because the dollar can be invested to earn interest. The time value of money is the simple idea that the value of a dollar changes over time because it can be invested to earn interest. Given a choice between receiving $1,000 today or $1,000 a year from now, which would you prefer? You should have answered today because you could invest the money and have even more in the future. For example, if you invest $1,000 today at 10 percent interest, you would have $1,100 [$1,000 × (1.10%)] a year from now. If you reinvest the $1,100 at the same 10 percent rate, you will earn even more interest in the following year due to a force called compounding, where interest is earned on top of interest. At the end of three years, you would have $1,331, as shown on this slide. This is an example of a future value problem. We know how much money we have today ($1,000), but need to know how much it will be worth at some point in the future.

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Time Value of Money Discounting is exactly the opposite of compounding. Just as interest builds up over time through compounding, discounting involves backing out the interest to determine the equivalent value in today’s present value dollars. The capital budgeting techniques we discuss in the next section involve present value problems. In a present value problem, we know how much money we will receive in the future, but we need to know what that money is worth in today’s dollars. Instead of compounding (adding interest on top of interest), present value problems involve discounting future cash flows back to their equivalent value in today’s (present value) dollars. For example, if the interest rate is 10 percent, receiving $1,000 in three years is equivalent to receiving $751 today, as shown on this slide. As you can see, discounting is exactly the opposite of compounding. Just as interest builds up over time through compounding, discounting involves backing out the interest to determine the equivalent value in today’s present value dollars.

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**Time Value of Money Discounted Cash Flow Methods Net Present Value**

Internal Rate of Return Profitability Index Assumptions: All future cash flows happen at the end of the year. Cash inflows are immediately reinvested in another project. All cash flows can be projected with 100 percent certainty. The capital budgeting methods described in the remainder of the chapter will require you to solve two types of present value problems. The first involves calculating the present value of a single cash flow that will happen at some point in the future, similar to the example above. The second involves computing the present value of an annuity, or a stream of cash flows that occur uniformly across time. If you need a refresher on these topics, you should review the supplement, paying particular attention to the present value calculations. These present value concepts are at the heart of the capital budgeting techniques presented in the remainder of this chapter: • Net present value • Internal rate of return • Profitability index All three are discounted cash flow methods , which means they explicitly recognize the time value of money by stating future cash flows in terms of their present value. As with any quantitative analysis, we must make some assumptions to use these methods. These assumptions are: All future cash flows happen at the end of the year. In reality, companies receive and pay cash throughout the year. But for simplicity, we assume that all cash flows happen at the end of the year. Cash inflows are immediately reinvested in another project. The net present value method assumes that cash inflows are reinvested at the minimum or required rate of return. The internal rate of return method assumes that cash flows are reinvested at the project’s internal rate of return. This difference in assumption will be discussed later in the chapter when comparing the two methods. All cash flows can be projected with 100 percent certainty. In general, the further the cash flow extends into the future, the more uncertain it is. But we assume that all cash flows are known with certainty.

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Learning Objective 11-3 Calculate net present value and describe why it is superior to the other capital budgeting techniques. Learning objective 11-3 is to calculate net present value and describe why it is superior to the other capital budgeting techniques.

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**Net Present Value (NPV)**

The net present value (NPV) method compares the present value (PV) of a project’s future cash inflows to the PV of the cash outflows. The reason is that accounting net income is based on accruals that ignore the timing of cash flows into and out of an organization. The net present value (NPV) method compares the present value (PV) of a project’s future cash inflows to the PV of the cash outflows. The difference between the present value of the cash inflows and outflows is called the net present value. To compute the present value of the cash inflows and outflows, we need to know the appropriate discount rate , or the rate used to discount future cash flows to reflect the time value of money. The discount rate should reflect the company’s cost of capital, also called the weighted-average cost of capital. The cost of capital is computed by averaging the after-tax cost of debt and the cost of equity, weighted according to the relative mix of these two sources of capital funding. The after-tax cost of debt reflects the interest paid on debt (which is tax deductible). The cost of equity is the return that investors require in exchange for the risk they take by investing in the company’s stock.

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**Net Present Value (NPV)**

Chose a discount rate – the minimum required rate of return. Calculate the present value of cash inflows. Calculate the present value of cash outflows. NPV = – To find the net present value of an investment, we first chose an interest rate, referred to as the required rate of return. It is the minimum acceptable rate of return on investment alternatives. Second, we use the required rate of return to calculate the present value of cash inflows over the life of the investment. Third, we repeat step two for cash outflows. In many net present value applications, the only cash outflow is the initial cost, which is already a present value. That is the case in our iKids Touch example. Finally, net present value is the present value of cash inflows less the present value of cash outflows.

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**Net Present Value (NPV)**

Relationship Between NPV and the Required Rate of Return As a general rule, if the net present value is positive or zero, the project is acceptable since the promised return is equal to or greater than the required rate of return. When we have a negative net present value, the project is not acceptable.

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**Net Present Value (NPV)**

Let’s return to iKids Touch’s proposal. Recall that the up-front investment is $1,000,000, and the product’s estimated life is 5 years. iKids Touch’s required rate of return is 12%. iKids Touch estimates the new product will generate $308,000 in cash flow for each of the next five years. For our iKids Touch example, we will begin with the simple scenario in which cash flows are the same each year. Recall that the project requires an initial investment of $1,000,000 and will generate $308,000 in cash flow for each of the next five years. Assume that Apple’s cost of capital is 12 percent. Because the cash flows are the same each year, we can use the present value of annuity table shown in Table 11.4A. This table shows that the present value of a five-year annuity at a 12 percent discount rate is To compute the present value of the cash inflows for years one through five, we multiply by the annual cash flow of $308,000. We then subtract the $1,000,000 cash outflow that happens at the start of the project (time zero). The resulting net present value of $110,278 indicates that the present value of the project’s cash inflows is $110,278 greater than the cost of the original investment using a discount rate of 12 percent. Note that the $1,000,000 cash outflow for the initial investment is already a present value as it occurs at the inception of the project. In general, a positive or zero NPV indicates that a project is acceptable; a negative NPV indicates that it is not acceptable. Since the NPV is positive, we know the rate of return is greater than the 12 percent discount rate.

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**Net Present Value (NPV)**

Assume that the expected cash flows for the iKids Touch project for years 1 to 5 are $250,000, $300,000, $340,000, $375,000, and $300,000, respectively. The project will still require an investment of $1,000,000 and the cost of capital is still 12 percent. If the cash flows from the project are not the same each year, we cannot use the PV of an annuity table. Instead, we must use the present value of $1 table (Table 11.2A). This table provides the present value factors on a year-by-year basis, rather than aggregating the values across all five years as the annuity table does. To illustrate, assume that the expected cash flows for the iKids Touch project for years 1 to 5 are $250,000, $300,000, $340,000, $375,000, and $300,000, respectively. The project will still require an investment of $1,000,000 and the cost of capital is still 12 percent. To compute the present value of an uneven stream of cash flows, we multiply the cash flow for each year by the present value of $1 factor discounted at a rate of 12 percent, as shown in Table 11.2A. We then sum the present value of the future cash flows, and, finally, subtract the cash outflow for the original investment to determine the net present value, as shown on this slide.

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Computing NPV in Excel Using Excel to compute the net present value provides a value of $112,925. The value is slightly different because of rounding in the present value tables used previously. Here are the steps we use with Excel: Input future cash flows (years 1-n) in spreadsheet. Use the NPV function to compute the PV of the future cash flows. Subtract the initial investment to get the NPV of the project.

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**Relationship Between NPV and Discount Rates**

This graph shows how the NPV will change when the discount rate, or cost of capital, changes. Can you predict from this graph the rate at which the NPV will be zero? The discount rate that yields a zero NPV is also called the internal rate of return.

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Learning Objective 11-4 Predict the internal rate of return and describe its relationship to net present value. Learning objective 11-4 is to predict the internal rate of return and describe its relationship to net present value.

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**Internal Rate of Return (IRR)**

The internal rate of return is the interest rate that makes . . . Present value of cash inflows Present value of cash outflows = The internal rate of return (IRR) is the rate of return that yields a zero net present value. Mathematically, it is computed as the discount rate that would make the present value of the cash inflows equal to the cash outflows. The net present value equal zero.

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Computing IRR in Excel Solving for the internal rate of return mathematically is extremely challenging, but you can get pretty close using a trial-and-error approach. In practice, most managers would use Excel or a financial calculator to solve for the internal rate of return. This slide illustrates the following two steps to compute the IRR in Excel. Input all of the future cash flows and the initial investment as a cash outflow. Use the IRR function to compute the internal rate of return. One important note about the IRR function is that you must include the original cash outflow in the calculation. Otherwise it is impossible to find the rate at which the present value of the cash inflows equals the present value of the cash outflows. In contrast, the NPV function in Excel excludes the original cash outflow because it happens at time zero and thus does not need to be discounted. One important note about the IRR function is that you must include the original cash outflow in the calculation.

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**Internal Rate of Return (IRR)**

The IRR is closely related to the NPV method. A positive NPV occurs when the internal rate of return is greater than the required rate of return. A negative NPV occurs when the internal rate of return is less than the required rate of return. When the NPV is zero, the internal rate of return and the required rate of return are equal. These relationships are summarized in the equations on this slide. The internal rate of return and the net present value methods are generally preferred over the accounting rate of return and the payback methods because they incorporate the time value of money. But which of these two methods is better? IRR and NPV will provide consistent information to managers for deciding whether a single project is acceptable or unacceptable (screening decisions). However, the NPV method is preferred over the IRR when managers must choose between competing alternatives (preference decisions). NPV is preferred for these decisions because of a key assumption about how the future cash flows are reinvested into other projects. The NPV method assumes that cash flows are reinvested at the cost of capital, while the IRR assumes that the cash flows are reinvested in projects with the same internal rate of return. In general, assuming that funds will be reinvested at the cost of capital is more realistic and conservative, particularly if the IRR of the project is very high. Managers may not be able to reinvest the funds in an equally attractive project. The NPV method ensures that managers invest in projects that meet or exceed the cost of capital. However, one advantage of IRR compared to NPV is that it is stated on a relative or percentage basis. It is easier for managers to interpret a relative measure (such as a 20% internal rate of return) than it is an absolute dollar value, particularly if they don’t have anything to compare it to.

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Profitability Index The profitability index is the ratio of a project’s benefits (measured by the present value of the future cash flows) to its costs (or required investment). The profitability index is the ratio of a project’s benefits (measured by the present value of the future cash flows) to its costs (or required investment). It is a derivative of the NPV method, but it provides a relative measure of the project’s performance rather than an absolute dollar amount. When we computed NPV, we first calculated the present value of the future cash inflows, and then subtracted the original investment, or cash outflow. The profitability index is stated as a ratio of these two items instead. For example, in our Apple scenario, the NPV of the cash flows in years 1 to 5 was $1,112,925 and the original investment (at time zero) was $1,000,000. The profitability index would be computed as shown on this slide. A profitability index greater than one indicates that the present value of the future cash flows (numerator) is greater than the initial investment (denominator); thus, the project is acceptable. A profitability index of less than one means that the present value of the future cash flows (numerator) is less than the initial investment (denominator), indicating that the project is unacceptable. In this case, the profitability index indicates that Apple should move ahead with the iKids Touch project. Profitability Index > 1 = Project Acceptable Profitability Index < 1 = Project Unacceptable

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**Comparing Capital Budgeting Methods**

This slide provides a summary of the five capital budgeting methods. Any of these methods can be used for screening decisions or to evaluate a single project to determine whether it is acceptable. The discounted cash flow methods (NPV, IRR, and profitability index) are superior to the accounting rate of return and payback methods, especially when it comes to preference decisions, because they incorporate the time value of money. The NPV method is best for evaluating mutually exclusive projects, or deciding between two competing alternatives. The profitability index is best for prioritizing independent projects when limited investment funds are available.

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Learning Objective 11-5 Use the net present value method to analyze mutually exclusive capital investments. Learning objective 11-5 is to use the net present value method to analyze mutually exclusive capital investments.

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**Case 1: Lease or Buy Equipment**

iKids Touch is trying to decide whether to buy a new copier or lease it from a copier company, and has gathered the following information about the two options: iKids Touch is trying to decide whether to buy a new copier or lease it from a copier company, and has gathered the information shown about the two options. iKids Touch uses net present value to evaluate investment options. If the discount rate is 10%, should iKids Touch lease or buy? iKids Touch uses net present value to evaluate investment options. If the discount rate is 10%, should the company lease or buy?

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**Case 1: Lease or Buy Equipment**

To analyze this decision, we can use the NPV method to compare the relevant costs (in present dollar values) of each option. To analyze this decision, we can use the NPV method to compare the relevant costs (in present dollar values) of each option. The annual cash operating costs and lease payments are the same amount in each of the four years, so we can use the PV of annuity table (Table 11.4A) to determine the present value of these cash payments. For the purchase option, there is an additional cash flow from the sale of the copier for its salvage value at the end of four years. Because this is a one-time cash flow, we will need to use the PV of $1 table (Table 11.2A) to discount the amount. These calculations are shown on this slide. The NPV analysis shows that it is less costly to lease the copier than pay for it outright. The lease option will save the company about $1,419, measured in today’s dollars. Lease option costs $1,419 less.

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**Case 2: Investing in Automation**

iKids Touch is thinking of spending $10,000,000 to automate a production facility. The investment is expected to have the following effects: Automation will increase the capacity of the plant and allow it to boost production and sales by 20 percent. The company will be able to reduce packaging labor cost per unit by 30 percent. Factory supervision costs will increase by $500,000 per year. The estimated useful life of the equipment is six years, at which point it will have a residual value of $1,000,000. Straight-line depreciation of the assets will be $1,500,000 per year [($10,000,000 – 1,000,000) ÷ 6 years = $1,500,000]. Assume that Apple is thinking of renovating one of the facilities where its final products are packaged and prepared for shipment to customers. The renovation would include the installation of robots to assemble the various items that must be included in each package (e.g., phone, charger, instruction manual, earphones), pack them into larger boxes, and prepare them for shipping. The renovation would require a $10 million investment that would have the following benefits: Automation will increase the capacity of the plant and allow it to boost production and sales by 20 percent. The company will be able to reduce packaging labor cost per unit by 30 percent. Factory supervision costs will increase by $500,000 per year. The estimated useful life of the equipment is six years, at which point it will have a residual value of $1,000,000. Straight-line depreciation of the assets will be $1,500,000 per year [($10,000,000 – 1,000,000) ÷ 6 years = $1,500,000].

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**Case 2: Investing in Automation**

This table summarizes the effects on net income. The per-unit costs in the first column are assumed. Note that automation increases net income by $1,100,000. The table on this slide summarizes the effects on net income. The per-unit costs in the first column are assumed. Note that automation increases net income by $1,100,000. iKids Touch uses net present value to evaluate investments. If the discount rate is 12%, should the company make the $10,000,000 investment? iKids Touch uses net present value to evaluate investments. If the discount rate is 12%, should the company make the $10,000,000 investment?

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**Case 2: Investing in Automation**

Remember that the net present value method is based on cash flow rather than net income. So, we need to add back the depreciation (a noncash expense) to net income to get net cash flow. We also need to incorporate the initial investment (at time zero) and the salvage value of the machinery at the end of six years. Remember that the net present value method is based on cash flow rather than net income. So, we need to add back the depreciation (a noncash expense) to net income to get net cash flow. We also need to incorporate the initial investment (at time zero) and the salvage value of the machinery at the end of six years. This analysis is summarized in the table shown on your screen. The positive net present value of $1,196,240 means that the proposed investment in automation will generate a return in excess of the 12% cost of capital. Any return in excess of the cost of capital is acceptable and will create economic wealth for the firm. But once again, this analysis is contingent on a number of assumptions and ignores the effect of taxes. The positive net present value of $1,196,240 means that the proposed investment in automation will generate a return in excess of the 12% cost of capital.

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Learning Objective 11-6 Use the profitability index to prioritize independent capital investment projects. Learning objective 11-6 is to use the profitability index to prioritize independent capital investment projects.

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**Prioritizing Independent Projects**

The profitability index is used to prioritize capital investment projects. Present Value of Future Cash flows Profitability Index Initial Investment = ÷ Managers usually have limited funds to invest and must prioritize their investment alternatives. The net present value is stated as an absolute dollar value, which makes it difficult to compare projects of different size. Managers should prioritize capital investment projects based on a factor called the profitability index, which is the ratio of the present value of future cash flows divided by the initial investment. A profitability index greater than one means that a project has a positive NPV, since the present value of the future cash flows is greater than the initial investment. When using the profitability index to prioritize projects, the preference rule is: the higher the profitability index, the more desirable the project. When using the profitability index to prioritize projects, the preference rule is: the higher the profitability index, the more desirable the project.

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**Prioritizing Independent Projects**

iKids Touch is trying to decide how to prioritize their limited research and development budget. They are considering these three independent projects. iKids Touch is trying to decide how to prioritize their limited research and development budget. They are considering the three independent projects shown. How should iKids Touch prioritize these three projects? Based on the profitability index, Project A should be first, followed by Project B, then C. The profitability index takes into account both the present value of the future cash flows (benefits) and the initial investment (cost). This is the appropriate way to prioritize the projects because it focuses on how much value (in terms of future cash flows) is generated for every dollar of investment. If the decision is based net present value, Project C would be first, followed by B, and then C. Project C generates the most net present value ($400,000), but also requires the largest investment. How should iKids Touch prioritize these three projects? A, then B, then C

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**Supplement 11A Time Value of Money Supplement 11A: Time Value of Money**

This supplement provides additional details about how to use present value and future value tables to incorporate the time value of money. Although you can use a calculator or computer to do the computations, you need to understand where the numbers come from.

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Learning Objective 11-S1 Use present value and future value tables to incorporate the time value of money. Learning objective 11-S1 is to use present value and future value tables to incorporate the time value of money.

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Time Value of Money Future Value of a Single Payment Present Value of a Single Payment Future Value of an Annuity Present Value of an Annuity Present and future value problems may involve two types of cash flow: a single payment or an annuity (a fancy word for a series of equal cash payments) The table on this slide illustrates the basic difference between present value and future value problems. In some business situations, you will know the dollar amount of a cash flow that occurs in the future and will need to determine its value now. This type of situation is known as a present value problem. The opposite occurs when you know the dollar amount of a cash flow that occurs today and need to determine its value at some point in the future, which requires solving a future value problem. Present and future value problems may involve two types of cash flow: a single payment or an annuity (a fancy word for a series of equal cash payments). Thus, you need to learn how to deal with four different situations related to the time value of money: 1. Future value of a single payment. 2. Present value of a single payment. 3. Future value of an annuity. 4. Present value of an annuity.

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**Future Value of a Single Amount**

To solve a future value problem, you need to know three things: Amount to be invested. Interest rate (i) the amount will earn. Number of periods (n) in which the amount will earn interest. In problems involving the future value of a single amount, you will be asked to calculate how much money you will have in the future as the result of investing a certain amount right now. If you were to receive a gift of $10,000, for instance, you might decide to put it in a savings account and use the money as a down payment on a house after you graduate. The future value computation would tell you how much money you will have from this gift when you graduate. To solve a future value problem, you need to know three things: 1. Amount to be invested. 2. Interest rate (i) the amount will earn. 3. Number of periods (n) in which the amount will earn interest. You can avoid the detailed computations by using Table 11.1A. and finding the factor for 3 years at 10% (1.3310) and multiplying it by the present value of $1,000. Using Table 11.1A.: $1,000 × = $1,331

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**Present Value of a Single Amount**

The present value of a single amount is the value to you today of receiving some amount of money in the future. To compute the present value of an amount to be received in the future, we must discount (a procedure that is the opposite of compounding) at i interest rate for n periods. The present value of a single amount is the value to you today of receiving some amount of money in the future. For instance, you might be offered an opportunity to invest in a financial instrument that would pay you $1,000 in three years. Before you decide whether to invest, you would want to determine the present value of the instrument. To compute the present value of an amount to be received in the future, we must discount (a procedure that is the opposite of compounding) at i interest rate for n periods. In discounting, the interest is subtracted rather than added as it is in compounding. Graphically, the present value of $1 due at the end of the third period with an interest rate of 10 percent can be represented as shown on this slide. Assume that today is January 1, 2013, and you have the opportunity to receive $1,000 cash on December 31, 2015 (three years from today). At an interest rate of 10 percent per year, how much is the $1,000 payment worth to you on January 1, 2013 (today)? You could discount the amount year by year, but it is easier to use Table 11.2A , Present Value of $1. Using Table 11.2A.: $1,000 × = $751.30

47
**Future Value of an Annuity**

The future value of an annuity includes compound interest on each payment from the date of payment to the end of the term of the annuity. Each new payment accumulates less interest than prior payments because the number of periods in which to accumulate interest decreases. An annuity is a series of consecutive payments characterized by: 1. An equal dollar amount each interest period. 2. Interest periods of equal length (year, half a year, quarter, or month). 3. An equal interest rate each interest period. Examples of annuities include monthly payments on a car or house, yearly contributions to a savings account, and monthly pension benefits. The future value of an annuity includes compound interest on each payment from the date of payment to the end of the term of the annuity. Each new payment accumulates less interest than prior payments because the number of periods in which to accumulate interest decreases. The future value of an annuity of $1 for three periods at 10 percent may be represented graphically as shown on this slide. Assume that each year for three years, you deposit $1,000 cash into a savings account that earns 10 percent interest per year. You make the first $1,000 deposit on December 31, 2013, the second one on December 31, 2014, and the third and last one on December 31, The first $1,000 deposit earns compound interest for two years (for total principal and interest of $1,210); the second deposit earns interest for one year (for total principal and interest of $1,100). The third deposit earns no interest because it was made on the day that the balance is computed. Thus, the total amount in the savings account at the end of three years is $3,310 ($1,210 + $1,100 + $1,000). To calculate the future value of this annuity, we could compute the interest on each deposit. However, a faster way is to refer to Table 11.3 A, Future Value of an Annuity of $1. Assume that each year for three years, you deposit $1,000 cash into a savings account that earns 10 percent interest per year. You make the first $1,000 deposit on December 31, 2013, the second one on December 31, 2014, and the third and last one on December 31, To calculate the future value of this annuity, use Table 11.3A , Future Value of an Annuity of $1. Using Table 11.3A.: $1,000 × = $3,310

48
**Present Value of an Annuity**

The present value of an annuity is the value now of a series of equal amounts to be received (or paid out) for some specified number of periods in the future. Assume you are to receive $1,000 cash on each December 31 for three years: 2013, 2014, and How much would the sum of these three $1,000 future amounts be worth on January 1, 2013, assuming an interest rate of 10 percent per year? To calculate the present value of this annuity, use Table 11.4A , Present Value of an Annuity of $1. The present value of an annuity is the value now of a series of equal amounts to be received (or paid out) for some specified number of periods in the future. A good example is a retirement program that offers employees a monthly income after retirement. The present value of an annuity of $1 for three periods at 10 percent can be represented graphically as shown on this slide. Assume you are to receive $1,000 cash on each December 31 for three years: 2013, 2014, and How much would the sum of these three $1,000 future amounts be worth on January 1, 2013, assuming an interest rate of 10 percent per year? Using Table 11.4A., find the factor for 10 percent for 3 years. It is Now, multiply $1,000 times to get a present value of $2,487 (rounded). What this means is that you would be indifferent between getting $2,487 now or $1,000 at the end of each of the next three years. Assume you are to receive $1,000 cash on each December 31 for three years: 2013, 2014, and How much would the sum of these three $1,000 future amounts be worth on January 1, 2013, assuming an interest rate of 10 percent per year? The answer is $2,487 (rounded). Using Table 11.4A.: $1,000 × = $2,487 (rounded)

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End of Chapter 11 End of chapter 11.

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