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Chapter contents Overview Future value Present value Net present value The internal rate of return IRR What does IRR mean? Loan tables and investment amortization

In the following spreadsheet. We can apply the formula: NPV requires you to list all the payments. The present value of a constant growth annuity An annuity is a series of annual payments. You have to be careful! The present value of a constant-growth annuity—a series of cash flows with first term CF CF which grows at rate g—that is discounted at rate r is.

The formula gives: The discounted sum of an infinite series of constantly-growing payments only exists when the growth rate g is less than the discount rate r. We use this formula in Chapter 6. All the material is copyright and the rights belong to the author and MIT Press Check with the author before distributing this draft though you will probably get permission The NPV rule for judging investments and projects Capital budgeting principle: Ignore sunk costs and consider only marginal cash flows Capital budgeting and salvage values September Much more material is posted on the PFE website http: Chapter 3: Capital budgeting Accelerated depreciation In-house copying or outsourcing?

A mini-case illustrating foregone opportunity costs The IRR rule for judging investments This issue is dealt with briefly in Section If you have several alternative investments Here are some of the topics covered: Both of these concepts were introduced in Chapter We return to it at greater length in Chapters 4 — There are two standard concepts used in capital budgeting: How should we account for costs incurred in the past?

Suppose PFE. The NPV rule for deciding whether or not a specific project is worthwhile: Suppose we are considering a project which has cash flows CF0. We start off this chapter by summarizing each of these rules— the NPV rule in this section and the IRR rule in the following section. Capital budgeting 4. You will buy either A or B or perhaps neither machine. The NPV rule for deciding between two mutually exclusive projects: Suppose you are trying to decide between two projects A and B..

Project A is preferred to project B if: For example: Project A involves buying expensive machinery which produces a better product at a lower cost. Capital budgeting 5. For the moment we fudge the question of how to choose discount rates. If we have to choose between the projects. Here are the two decision rules for using the IRR in capital budgeting: Hence in cell B It is the compound rate of return which you get from a series of cash flows.

Capital budgeting 6. In order to calculate the finance NPV concept in the spreadsheet. IRR is an interest rate such that: If the appropriate discount rate for a project is r. Suppose project A has cash flows CF0A. The IRR rule for deciding whether a specific investment is worthwhile: Suppose we are considering a project that has cash flows CF Again the logic is clear: Capital budgeting 7.

The IRR rule for deciding between two competing projects: Suppose you are trying to decide between two mutually exclusive projects A and B meaning: Capital budgeting 8. If we have to choose between the two projects by using the IRR rule. In our IRR analysis of the same projects. There are some cases. Project B was chosen because it had the higher IRR. In such cases we should always use the NPV to decide between projects. In many cases your investment decision—to undertake a project or not.

The logic is that if individuals are interested in maximizing their wealth. Capital budgeting 9. The discount rate is 15 percent: Consider the following project: B11 15 IRR B11 14 NPV Thus the project is worthwhile. Should we undertake the project? It is clear that the above project is worthwhile: A general principle We can derive a general principle from this example: For conventional projects.

In the previous section we saw that for conventional projects. This is not to impugn the IRR criterion. Using the NPV criterion may lead us to prefer one project whereas using the IRR criterion may lead us to prefer the other project.

That is. Suppose we have two projects and can choose to invest in only one. To sum up our results before we start: The relevant discount rate is 15 percent. The projects are mutually exclusive: They are both ways to achieve the same end.

C10 Comparing the projects using IRR: If we use the IRR rule to choose between the projects. C10 13 IRR Comparing the projects using NPV: Here the choice is more complicated. Both indicate that Project B should be chosen. As we stated in the introduction to PFE. There is a crossover point in the next subsection you will see that this point is 8. We explore the reasons for this later on. Below we build a table and graph that show the NPV for each project as a function of the discount rate: Indifferent between A B preferred: To do this.

A bit of formula manipulation will show you that the crossover point is the IRR of the differential cash flows. To see this. Which is correct? This is just one example of the general principal discussed in Section 3 that using the NPV is always preferred. D41 What to use? The economic assumption is that consumers maximize their wealth. A friendly local contractor has offered to make the necessary repairs.

Your intention was to sell the house immediately. You recently bought a plot of land and built a house on it.

The following sections discuss some specifics: What should you do? Ignore sunk costs and consider only marginal cash flows This is an important principle of capital budgeting and project evaluation: In the jargon of finance: Ignore sunk costs.

Where is this chapter going? Until this point in the chapter. This attitude is typified in column B below. C7 Of course your father was wrong and your mother right this often happens: Even though you made some disastrous mistakes you never should have built the house in the first place. Sally and Dave—fresh out of business school with a little cash to spare—are considering buying a nifty condo as a rental property.

For the moment we concentrate on a few obvious principles. Here are some additional facts: We postpone a fuller discussion of this topic to Chapters 6 and 7. No one needs to be told that taxes are very important. We use Sally and Dave and their condo to emphasize the place of taxes in the capital budgeting process. In the case of Sally and Dave. When Sally and Dave pay to fix the faucet in their condo.

Since—as we show in the text— depreciation ultimately saves taxes. Sally and Dave get to subtract expenses from their income. The simplest method is straight-line depreciation. This reduces the taxes paid by the owners of the asset and takes account of the fact that the asset has a limited life. The cost of the condo is neither income nor an expense. When Sally and Dave get the rent from their condo. Under United States tax rules. There are many depreciation methods in use.

Notice that this method accelerates the depreciation charges—more than one-sixth of the depreciation is taken annually in years and less in later years.

What is depreciation? In computing the taxes they owe.

A B C Cash flow. A tax shield is a tax saving that results from being able to report an expense for tax purposes. It all has to do with depreciation: Because the depreciation is an expense for tax purposes but not a cash expense.

In general a tax shield just reduces the cash cost of an expense—in the above 4 In Chapter 6 we introduce the concept of free cash flow. If depreciation were not an expense for tax purposes. The result cell B32 gives the cash flow for the condo. Depreciation is a special case of a non-cash expense which generates a tax shield.

In the spreadsheet below we calculate the cash flow in two stages: NPV 1. B15 18 Internal rate of return. IRR B26 expense for tax purposes. B15 PFE. To make a better calculation about their investment. The value of the condo at the end of the year horizon. In Chapter 5 we have our first discussion in this book on how to determine a correct PFE.

Annual cashflow from rental 25 2 B33 36 IRR of investment The tax rules for selling an asset specify that the tax bill is computed on the gain over the book value. Past this point. Sally and Dave are reluctant to make predictions about condo rents and expenses beyond a ten-year horizon. So they write down ten years of cash flows. Book value versus terminal value The book value of an asset is its initial purchase price minus the accumulated depreciation.

If Sally and Dave are right in their terminal value assumption. The table below shows the IRR of the investment as a function of the annual rent and the terminal value: Doing some sensitivity analysis If we really want to be fancy.

At this point the table looks like this: For a given rent. See Chapter 30 for more information. We use the Data Table command and fill in the cell references from the original example: The dialog box tells Excel to repeat the calculation in cell B Pressing OK does the rest. Accountants and the tax authorities compute earnings by subtracting certain kinds of expenses from sales. Here are the facts: In order to compute the cash flow. Highly recommended!

We showed that these non-cash expenses create tax shields—they create cash by saving taxes. Your firm is considering buying a new machine. The NPV is positive. Should the firm buy the machine? Cells F7: F15 show the table of cash flows. In our example. To compute the cash flow produced by this salvage value. Taxable gain The taxable gain on the machine at the time of sale is the difference between the market value and the book value. Book value An accounting concept: The book value of the machine is its initial cost minus the accumulated depreciation the sum of the depreciation taken on the machine since its purchase.

Market value The market value is the price at which the machine can be sold. The machine will still be depreciated on a straight-line basis over 8 years. One more example Suppose we change the example slightly: Machine sold in year 7 2 Cost of the machine 3 Annual anticipated sales 1.

Using this example. Thus even selling an asset at a loss can produce a positive cash flow. Should the value of this equipment be somehow taken into account? The answer to this question has to do with whether the equipment has an alternative use. An example: The cash flows in years have been calculated by your financial analysts: B10 Looks like a fine project!

But now someone remembers that the widget process makes use of some already existing but underused equipment. For example. What if the machine is to occupy space in a building that is currently unused? Should the cost of this space be taken into account?

It all depends on whether there are alternative uses. A mini-case illustrating foregone opportunity costs Your company is trying to decide whether to outsource its photocopying or continue to do it in-house. Here are some details about the two alternatives: We will show you two ways to analyze this decision. The first method values each of the alternatives separately.

The second method looks only at the differential cash flows. Write down the cash flows of each alternative This is often the simplest way to do things. Below we write down the cash flows for each alternative: Note that both NPVs are negative. B20 22 23 Alternative 2: Sell machine and outsource copying 24 Year Cash flow 25 0 9. Alternative 2 2. Fix up machine and do copying in-house 14 Year Cash flow 15 0 This means that Alternative 1 in- house is better than Alternative 2 outsourcing: B30 Method 2: Discounting the differential cash flows In this method we subtract the cash flows of Alternative 2 from those of Alternative 1: B41 The NPV of the differential cash flows is positive.

If you do it in house. Relative to in-house copying. In other words: This is the cost in year 0 of deciding to do the copying in-house.

You have to decide what the NPV of the project is. Accelerated depreciation As you know by now. Here are the relevant calculations: C14 the depreciation taken on the 20 Taxable gain In many cases these two criteria give the same answer to the capital budgeting question.

G30 3. G29 31 32 Net present value 3. NPV is the preferable criterion to use because the NPV is the additional wealth derived from a project. Where they differ.

Calculate the internal rate of return. Give a brief discussion. Your firm is considering two projects with the following cash flows: Calculate the crossover rate. Calculate the project's net present value. Calculate the present values of the future cash flows of the project. Should you undertake the project?

Which project is preferred if you rank by IRR? You are considering a project whose cash flows are given below: Your uncle is a proud owner of an up-market clothing store. Because business is down he is considering replacing the languishing tie department with a new sportswear department.

In order to examine the profitability of such move he hired a financial advisor to estimate the cash flows of the new department.

After six months of hard work the financial advisor came up with the following calculation: Chair Table department department No. Is the financial advisor correct? The machine will be depreciated on a straight line basis over its year life to a salvage value of zero.

Your financial advisor advises you not to take up the offer because the price does not even cover the cost of production. You are the owner of a factory that supplies chairs and tables to schools in Denver.

A factory is considering the purchase of a new machine for one of its units. You are considering the following investment: Calculate the project NPV using straight-line depreciation. A company is considering buying a new machine for one of its factories. Assume straight-line depreciation over the five-year life of the machine.

If you install the air-conditioning system. You are the owner of a factory located in a hot tropical climate. The ABD Company is considering buying a new machine for one of its factories. Calculate the project IRR. In January you get an offer to install an air-conditioning system in your factory. Calculate the project NPV. The company is considering the purchase of a new machine that will top the bar with high quality chocolate.

In his opinion. The "Less Is More" company manufactures swimsuits. The company is considering expanding to the bath robes market. Use a 2-dimensional data table to determine the sensitivity of the profitability to the price and quantity. Depreciation and terminal value: The machine will be depreciated over 10 years to zero salvage value.

The machine can produce up to one million ice cream bars annually. What is the minimum price that the company should charge for each bar if the project is to be profitable? Assume that the price of the bar does not affect sales. The proposed investment plan includes: The company is considering buying a new machine.

The terminal value of the machine is 0. The life span of the machine is 6 years and the machine has no disposal value. The "Car Clean" company operates a car wash business. The new machine is faster then the old one. What is the NPV of replacing the old machine? The Coka company is a soft drink company.

The Coka company is considering whether to start manufacturing cans in its plant. Which photocopy machine should the firm buy?

A company is considering whether to buy a regular or color photocopier for the office. The company has two alternatives to choose from: During the five months of the summer May- September demand is 2. The machines are being depreciated on a straight-line basis to zero salvage value. If the company manufactures 1.

The company has two machines. The Easy Sight company manufactures sunglasses. Plot a graph showing the profitability of investment in each machine type depending on the annual production.

Demand for sunglasses is seasonal. The ZZZ Company is considering investing in a new machine for one of its factories.

The annual costs of the shipping line are estimated at 60 gold coins annually. Use data tables in order to show the profitability of the project as dependent on the annual earnings and the tax rate. Assume that due to insurance and storage costs it is uneconomical to store sunglasses at the factory.

Poseidon is considering opening a shipping line from Athens to Rhodes. In order to open the shipping line Poseidon will have to purchase two ships that cost 1. The life span of each ship is 10 years. What is the maximum tax rate at which the project will be profitable? At the board meeting on Olympus. The Manufacturing project: The loan will have to be paid back in 6 equal annual payments.

What is the minimum loan that the IOC should give in order that the project will be profitable? Kane Running Shoes is considering the manufacturing of a special shoe for race walking which will indicate if an athlete is running i.

Assume that at the end of project that is. The project went ahead. The company is about to launch a new line of products. What is the minimum number of shoes Nike has to sell annually for the project to be profitable in each of the following two cases: The marketing department has to decide whether to use an aggressive or regular campaign.

Aggressive campaign Initial cost. As a result Kane is expecting a large drop in sales of the SWS shoes. The Aphrodite company is a manufacture of perfume. In order to mass-produce the new drug the company needs to purchase new machines. The expected life span of each machine is 5 years. The demand for the new drug is given by the following equation: Monthly growth in profits month Calculate the NPV of each campaign and decide which campaign should the company undertake.

Use data table in order to show the differential NPV as a function of first month payment and growth rate of the aggressive campaign. The Long-Life company has a 10 year monopoly for selling a new vaccine that is capable of curing all known cancers. How many vaccines will the company produce annually? The Gordon dividend model: Weighted average cost of capital page Applying the Gordon cost of equity formula—Courier Corporation The basic principle is that the discount rate for a stream of cash flows should be appropriate to the riskiness of the cash flows May Two uses of the WACC This is a preliminary draft of a chapter of Principles of Finance by Simon Benninga benninga wharton Make sure the material is updated before distributing it Chapter A real estate investment.

Here are two examples: During the rafting season. To complete the NPV analysis. Since the riskiness of the cash flows PFE. The White Water financial analyst has derived the set of anticipated cash flows for the new raft.

By being the first rafting company on the river to have the self-sealing rafts. Chapter 6. Weighted average cost of capital page 2. The raft is more expensive than the existing rafts operated by the company because it is self- sealing—holes in the raft are automatically and permanently fixed by a new technology. In this chapter we discuss the weighted average cost of capital WACC. The WACC is the average rate of return the firm has to pay its shareholders and its lenders.

Weighted average cost of capital page 3. GF has to decide on an appropriate discount rate for the anticipated DC cash flows. The income of a corporation is taxed at the corporate income tax rate. In order to value DC. The shareholders own stock in the firm.

The company is considering buying Dazzling Cascade Water Company. When the firm is profitable. Weighted average cost of capital page 4. The debtholders may be banks. The shareholders in the firm have limited liability. Debtholders are promised a fixed return interest on their lending to the firm.

The cost of equity. They all represent the market-adjusted rate that investors get or demand on various investments or securities. Weighted average cost of capital page 5. United Transport Inc. Another way of putting this is that the WACC is the average return shareholders and debtholders expect to receive from the company. The weighted average cost of capital WACC is the average return that the company has to pay to its equity and debt investors. Weighted average cost of capital page 6.

So be patient! Of the five parameters in the WACC calculation. A model for calculating rE is given in Section 6. This is the question in the computation of the WACC. Most often we calculate TC by computing the average tax rate of the firm. We will usually take E to equal the number of shares of the firm times the market price per share.

Weighted average cost of capital page 7. We will call the formula the Gordon dividend model. An actual example of a calculation for D is given below in Section 6. A detailed example of the calculation of rD for an actual firm is given in Section 6.

Here are some facts about the company and its stock: Weighted average cost of capital page 8. This dividend was paid on March 1. You want value XYZ shares by discounting the stream of future anticipated dividends.

Then the future anticipated dividends per share are: Here is an example which presents most of the logic of our model: It is March 2. In predicting the future dividends of XYZ Corp.

Div1 Div2 Div3 fair share value today.. Using rE to discount the future anticipated dividends. E PFE. Weighted average cost of capital page 9. According to the Gordon dividend model of the previous subsection. Using the Gordon dividend model to calculate the cost of equity rE In the previous subsection we derived the value of a share P0 based on the current dividend per share Div0.

Turning this formula around to solve for the cost of equity rE gives: P0 This is the Gordon dividend model cost of equity formula. In this section we turn this formula around: We derive the cost of equity rE based on the current value of a share P0. In the Gordon dividend model the cost of equity rE —the discount rate to be applied to equity cash flows—is the sum of two terms: D0 10 3 Dividend growth rate.

Suppose you buy the P0 stock today. The term is the P0 anticipated next period dividend return. Div0 3. Applying the Gordon cost of equity formula—Courier Corporation Courier Corporation stock symbol CRRC is a book manufacturer that has experienced rapid growth of sales and profits. This is the anticipated dividend yield of the stock. P0 This the growth rate of all future dividends paid on the stock.

Gordon dividend model cost of equity We derive this below from the dividend series in cells B3: In this case the P0 is the stock price on the date of the calculation 30 September In the example above. Here are two alternative ways to D compute the growth rate g: Use a different time period. Div 0. In choosing as the base year. If—as in the current example—we use as the base year instead of You might want to use this model to project the dividend growth rate.

This gives a cost of equity of You might decide that the past history of Courier dividends is not indicative of its future dividend payouts. Using a different base year Year ended Dividend 2 30 Sept per share 3 0. In this case. Ignore historical dividends altogether. Look at the data below: Courier purchased stock from its shareholders in open- market repurchase transactions. Making up a future growth rate of dividends 2 g. Using the total equity payout instead of per-share data In some of the years — In the examples for Courier Corporation which follow.

Using the figures from column D and using the total market value of the equity on 30 September Column D gives the total equity payout: Most dividend-paying firms think their shareholders want to see a steady pattern of dividend growth. So if they have extra cash. When some of the shares of the firm are repurchased. If you let your stock be repurchased. Firms repurchase stock instead of paying extra dividends for several reasons: So all parties gain.

Why do firms repurchase stock? In recent years share buybacks have exceeded dividends as a form of distribution to shareholders.

When a dividend is paid. Courier had 2. As you can see from the previous spreadsheet. On 30 September Before we can do this. You can see that the debt has been borrowed at various interest rates. In Figure 6. The debt items are marked. In theory rD ought to be the marginal cost of debt— the borrowing rate of the company for additional debt.

A plausible alternative is use information about the current borrowing rate of the company. Figure 6. We use the borrowing rate of 7. We use From Courier Corp. Number of shares times current share price 6 Sept. Computed from total equity payouts 3 Cost of debt. B6 8 9 Percentage of equity. TC We presume that this rate of return reflects the average risk of shareholder and debtholder future cash flows.

There are two important cases where this is often true: The value of the Courier is the discounted value of its future anticipated FCFs. Thus the WACC represents a weighted average of the riskiness of shareholder and debtholder cash flows. Below we define the concept of free cash flow FCF. When a company is considering investing in a project whose risk is comparable to the riskiness of the company as a whole. This is plausible. When the riskiness of a stream of cash flows is similar to the riskiness of the cash flows received by shareholders and debtholders.

The cash flows. This could be something as simple as another printing press to print more books or a warehouse to house them. Suppose the company is thinking of investing in a project whose riskiness is like the riskiness of its current business. Recall that Courier is in the book printing business. In all of these cases. For our purposes in this chapter. The second major use of the WACC is to value companies. If the following table gives you problems.

To accurately define the FCF. For purposes of calculating the FCF. The FCF includes all current liability items related to operations. Capital expenditures CAPEX An increase in fixed assets the long-term productive assets of the company is a use of cash. The sum of the next two items is the change in net working capital. This increase in current liabilities—when related to sales—provides cash to the firm.

We do this by: The FCF definition takes changes in working capital and purchases of new fixed assets into account separately. Since there are 2. This compares favorably with the current share price of Courier. The idea was that because most cash flows occur throughout the year—the appropriate discounting process should discount them as if they occur in mid-year. In terms of the computation just done for Courier. The WACC is defined as: Both of these uses have been illustrated in this chapter.

If we implement mid-year discounting for Courier. For Courier. In many cases professionals do extensive sensitivity analysis on the WACC and the FCF growth to establish a price range—the range of valuations which appears to be reasonable.

Another estimate of its tax rates might thus be A final warning Cost of capital calculations are critical for valuations and controversial. In the table below we summarize how we derived each of the elements of this formula: They involve a mixture of theory and judgment.

Almost every number in the WACC calculation above can be determined in several ways. State taxes are another 2. In practice we usually use either: The most important modification you might want to make to the WACC calculation above involves the cost of equity rE.

In Chapter 14 we will show you how to use this model to calculate the cost of equity. Calculate the cost of equity rE for a company having: Compute the weighted average cost of capital WACC for a company having: Complete the??? Consider the following data regarding Cinema Company.

January 4. Find the cost of equity rE of Cinema using the Gordon dividend model for the total equity payout. You know that tomorrow the company will pay its annual dividend in the amount of 1. After a quick research you have come up with the following data: After a quick search you have found the following data: It is 1 January Using the Gordon model for the total equity payout. As an experienced investor. You are considering a new project to your firm.

What is the company cost of capital? To your judgment. After this time. You just bought a share of ABC Corp. The company does not repurchase any of its shares.

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