Fundamentals of Corporate Finance Phần 7 pot

In row 13 we discount cash flow at a 12 percent discount rate and in cell B14 we add the present value of each cash flow to find project NPV.. www-ec.njit.edu/~mathis/interactive/FCCalcB

386 SECTION FOUR

inflation rate of 3 percent The firm believes that it will remain in the building for 4years What is the present value of its rental costs if the discount rate is 10 percent?The present value can be obtained by discounting the nominal cash flows at the 10percent discount rate as follows:

Notice the real cash flow is a constant, since the lease payment increases at the rate of

inflation The present value of each cash flow is the same regardless of the method used

to discount The sum of the present values is, of course, also identical

䉴 Self-Test 3 Nasty Industries is closing down an outmoded factory and throwing all of its workers

out on the street Nasty’s CEO, Cruella DeLuxe, is enraged to learn that it must tinue to pay for workers’ health insurance for 4 years The cost per worker next year will

con-be $2,400 per year, but the inflation rate is 4 percent, and health costs have con-been creasing at three percentage points faster than inflation What is the present value of thisobligation? The (nominal) discount rate is 10 percent

in-Separate Investment and Financing Decisions

When we calculate the cash flows from a project, we ignore how that project is financed

The company may decide to finance partly by debt but, even if it did, we would neither subtract the debt proceeds from the required investment nor recognize the interest and

principal payments as cash outflows Regardless of the actual financing, we should viewthe project as if it were all equity-financed, treating all cash outflows required for theproject as coming from stockholders and all cash inflows as going to them

We do this to separate the analysis of the investment decision from the financing cision We first measure whether the project has a positive net present value, assumingall-equity financing Then we can undertake a separate analysis of the financing deci-sion We discuss financing decisions later.

de-Calculating Cash Flow

A project cash flow is the sum of three components: investment in fixed assets such asplant and equipment, investment in working capital, and cash flow from operations:

Total cash flow = cash flow from investment in plant and equipment

+ cash flow from investments in working capital + cash flow from operations

Let’s examine each of these in turn

CAPITAL INVESTMENT

To get a project off the ground, a company will typically need to make considerable front investments in plant, equipment, research, marketing, and so on For example,Gillette spent about $750 million to develop and build the production line for its Mach3razor cartridge and an additional $300 million in its initial marketing campaign, largelybefore a single razor was sold These expenditures are negative cash flows—negativebecause they represent a cash outflow from the firm

up-Conversely, if a piece of machinery can be sold when the project winds down, thesales price (net of any taxes on the sale) represents a positive cash flow to the firm

Gillette’s competitor, Slick, invests $800 million to develop the Mock4 razor blade Thespecialized blade factory will run for 7 years, until it is replaced by a more advancedtechnology At that point, the machinery will be sold for scrap metal, for a price of $50million Taxes of $10 million will be assessed on the sale

Therefore, the initial cash flow from investment is –$800 million, and the cash flow

in 7 years from the disinvestment in the production line will be $50 million – $10 lion = $40 million

mil-INVESTMENT IN WORKING CAPITAL

We pointed out earlier that when a company builds up inventories of raw materials orfinished product, the company’s cash is reduced; the reduction in cash reflects the firm’sinvestment in inventories Similarly, cash is reduced when customers are slow to paytheir bills—in this case, the firm makes an investment in accounts receivable Invest-ment in working capital, just like investment in plant and equipment, represents a neg-ative cash flow On the other hand, later in the life of a project, when inventories are sold

388 SECTION FOUR

off and accounts receivable are collected, the firm’s investment in working capital is duced as it converts these assets into cash

Slick makes an initial (Year 0) investment of $10 million in inventories of plastic andsteel for its blade plant Then in Year 1 it accumulates an additional $20 million of rawmaterials The total level of inventories is now $10 million + $20 million = $30 million,but the cash expenditure in Year 1 is simply the $20 million addition to inventory The

$20 million investment in additional inventory results in a cash flow of –$20 million.Later on, say in Year 5, the company begins planning for the next-generation blade

At this point, it decides to reduce its inventory of raw material from $20 million to $15million This reduction in inventory investment frees up $5 million of cash, which is apositive cash flow Therefore, the cash flows from inventory investment are –$10 mil-lion in Year 0, –$20 million in Year 1, and +$5 million in Year 5

In general,

CASH FLOW FROM OPERATIONSThe third component of project cash flow is cash flow from operations There are sev-eral ways to work out this component

Method 1. Take only the items from the income statement that represent cash flows

We start with cash revenues and subtract cash expenses and taxes paid We do not, ever, subtract a charge for depreciation because depreciation is just an accounting entry,not a cash expense Thus

how-Cash flow from operations = revenues – cash expenses – taxes paid

Method 2. Alternatively, you can start with accounting profits and add back any ductions that were made for noncash expenses such as depreciation (Remember fromour earlier discussion that you want to discount cash flows, not profits.) By this rea-soning,

de-Cash flow from operations = net profit + depreciation

Method 3. Although the depreciation deduction is not a cash expense, it does affect net profits and therefore taxes paid, which is a cash item For example, if the firm’s tax

bracket is 35 percent, each additional dollar of depreciation reduces taxable income by

$1 Tax payments therefore fall by $.35, and cash flow increases by the same amount

The total depreciation tax shield equals the product of depreciation and the tax rate:

Depreciation tax shield = depreciation ⴛ tax rate

An increase in working capital implies a negative cash flow; a decrease

implies a positive cash flow.

The cash flow is measured by the change in working capital, not the level of

This suggests a third way to calculate cash flow from operations First calculate net

profit assuming zero depreciation This item would be (revenues – cash expenses) × (1– tax rate) Now add back the tax shield created by depreciation We then calculate op-erating cash flow as follows:

Cash flow from operations = (revenues – cash expenses) × (1 – tax rate)

+ (depreciation × tax rate)The following example confirms that the three methods for estimating cash flow fromoperations all give the same answer

A project generates revenues of $1,000, cash expenses of $600, and depreciationcharges of $200 in a particular year The firm’s tax bracket is 35 percent Net income iscalculated as follows:

Methods 1, 2, and 3 all show that cash flow from operations is $330:

Method 1: Cash flow from operations = revenues – cash expenses – taxes

= 1,000 – 600 – 70 = 330

Method 2: Cash flow from operations = net profit + depreciation

= 130 + 200 = 330

Method 3: Cash flow from operations = (revenues – cash expenses) × (1 – tax rate)

+ (depreciation × tax rate)

= (1,000 – 600) × (1 – 35) + (200 × 35) = 330

䉴 Self-Test 4 A project generates revenues of $600, expenses of $300, and depreciation charges of

$200 in a particular year The firm’s tax bracket is 35 percent Find the operating cashflow of the project using all three approaches

In many cases, a project will seek to improve efficiency or cut costs A new puter system may provide labor savings A new heating system may be more energy-efficient than the one it replaces These projects also contribute to the operating cashflow of the firm—not by increasing revenue, but by reducing costs As the next exam-ple illustrates, we calculate the addition to operating cash flow on cost-cutting projectsjust as we would for projects that increase revenues

Suppose the new heating system costs $100,000 but reduces heating expenditures by

$30,000 a year The system will be depreciated straight-line over a 5-year period, so the

390 SECTION FOUR

annual depreciation charge will be $20,000 The firm’s tax rate is 35 percent We

cal-culate the incremental effects on revenues, expenses, and depreciation charges as

fol-lows Notice that the reduction in expenses increases revenues minus cash expenses

Increase in (revenues minus expenses) 30,000– Additional depreciation expense – 20,000

= Incremental profit before tax = 10,000– Incremental tax at 35% – 3,500

Therefore, the increment to operating cash flow can be calculated by method 1 as

Increase in (revenues – cash expenses) – additional taxes =

Example: Blooper Industries

Now that we have examined many of the pieces of a cash-flow analysis, let’s try to putthem together into a coherent whole As the newly appointed financial manager ofBlooper Industries, you are about to analyze a proposal for mining and selling a smalldeposit of high-grade magnoosium ore.4You are given the forecasts shown in Table 4.3

We will walk through the lines in the table

TABLE 4.3

Profit projections for

Blooper’s magnoosium mine

4 Readers have inquired whether magnoosium is a real substance Here, now, are the facts Magnoosium was created in the early days of TV, when a splendid-sounding announcer closed a variety show by saying, “This program has been brought to you by Blooper Industries, proud producer of aleemium, magnoosium, and stool.” We forget the company, but the blooper really happened.

Note: Some entries subject to rounding error.

Capital Investment (line 1). The project requires an investment of $10 million inmining machinery At the end of 5 years the machinery has no further value.

Working Capital (lines 2 and 3). Line 2 shows the level of working capital As theproject gears up in the early years, working capital increases, but later in the project’slife, the investment in working capital is recovered

Line 3 shows the change in working capital from year to year Notice that in Years

1–4 the change is positive; in these years the project requires a continuing investment

in working capital Starting in Year 5 the change is negative; there is a disinvestment asworking capital is recovered

Revenues (line 4). The company expects to be able to sell 750,000 pounds of noosium a year at a price of $20 a pound in Year 1 That points to initial revenues of750,000× 20 = $15,000,000 But be careful; inflation is running at about 5 percent ayear If magnoosium prices keep pace with inflation, you should up your forecast of thesecond-year revenues by 5 percent Third-year revenues should increase by a further 5percent, and so on Line 4 in Table 4.3 shows revenues rising in line with inflation.The sales forecasts in Table 4.3 are cut off after 5 years That makes sense if the oredeposit will run out at that time But if Blooper could make sales for Year 6, you shouldinclude them in your forecasts We have sometimes encountered financial managerswho assume a project life of (say) 5 years, even when they confidently expect revenuesfor 10 years or more When asked the reason, they explain that forecasting beyond 5years is too hazardous We sympathize, but you just have to do your best Do not arbi-trarily truncate a project’s life

mag-Expenses (line 5). We assume that the expenses of mining and refining also increase

in line with inflation at 5 percent a year

Depreciation (line 6) The company applies straight-line depreciation to the

min-ing equipment over 5 years This means that it deducts one-fifth of the initial $10 lion investment from profits Thus line 6 shows that the annual depreciation deduction

mil-is $2 million

Pretax Profit (line 7). Pretax profit equals (revenues – expenses – depreciation)

Tax (line 8). Company taxes are 35 percent of pretax profits For example, in Year 1,

Tax = 35 × 3,000 = 1,050, or $1,050,000

Profit after Tax (line 9). Profit after tax equals pretax profit less taxes

CALCULATING BLOOPER’S PROJECT CASH FLOWSTable 4.3 provides all the information you need to figure out the cash flows on the mag-noosium project In Table 4.4 we use this information to set out the project cash flows

Capital Investment. Investment in plant and equipment is taken from line 1 of Table4.3 Blooper’s initial investment is a negative cash flow of –$10 million

STRAIGHT-LINE

DEPRECIATION

Constant depreciation for

each year of the asset’s

accounting life.

cal-Cash Flow from Operations. The necessary data for these calculations come fromlines 4–9 of Table 4.3 We’ve seen that there are at least three ways to compute thesecash flows (using any of methods 1, 2, or 3) For example, using the net profit + de-preciation approach, the first-year cash flow from operations (in thousands) is

profit after tax + depreciation expense = 1,950 + 2,000 = 3,950

or $3,950,000 You can apply the same calculation to the other years to obtain line 3 ofTable 4.3

CALCULATING THE NPV OF BLOOPER’S PROJECTYou have now derived (in the last line of Table 4.4) the forecast cash flows fromBlooper’s magnoosium mine Assume that investors expect a return of 12 percent frominvestments in the capital market with the same risk as the magnoosium project This isthe opportunity cost of the shareholders’ money that Blooper is proposing to invest in theproject Therefore, to calculate NPV you need to discount the cash flows at 12 percent.Table 4.5 sets out the calculations Remember that to calculate the present value of

a cash flow in Year t you can divide the cash flow by (1 + r) tor you can multiply by a

discount factor which is equal to 1/(1 + r) t When all cash flows are discounted andadded up, the magnoosium project is seen to offer a positive net present value of almost

$3.6 million

Now here is a small point that often causes confusion To calculate the present value

of the first year’s cash flow, we divide by (1 + r) = 1.12 Strictly speaking, this makes

sense only if all the sales and all the costs occur exactly 365 days, zero hours, and zerominutes from now But of course the year’s sales don’t all take place on the stroke of

TABLE 4.4

Cash flows for Blooper’s

magnoosium project (figures

Cash flows and net present

value of Blooper’s project

midnight on December 31 However, when making capital budgeting decisions, panies are usually happy to pretend that all cash flows occur at 1-year intervals Theypretend this for one reason only—simplicity When sales forecasts are sometimes littlemore than intelligent guesses, it may be pointless to inquire how the sales are likely to

com-be spread out during the year.5

FURTHER NOTES AND WRINKLES ARISING FROM BLOOPER’S PROJECTBefore we leave Blooper and its magnoosium project, we should cover a few extra wrinkles

A Further Note on Depreciation. We warned you earlier not to assume that all cashflows are likely to increase with inflation The depreciation tax shield is a case in point,because the Internal Revenue Service lets companies depreciate only the amount of theoriginal investment For example, if you go back to the IRS to explain that inflation mush-roomed since you made the investment and you should be allowed to depreciate more, the

IRS won’t listen The nominal amount of depreciation is fixed, and therefore the higher the rate of inflation, the lower the real value of the depreciation that you can claim.

We assumed in our calculations that Blooper could depreciate its investment in ing equipment by $2 million a year That produced an annual tax shield of $2 million ×.35 = $.70 million per year for 5 years These tax shields increase cash flows from op-erations and therefore increase present value So if Blooper could get those tax shieldssooner, they would be worth more, right? Fortunately for corporations, tax law allows

min-them to do just that It allows accelerated depreciation.

The rate at which firms are permitted to depreciate equipment is known as the

Mod-ified Accelerated Cost Recovery System, or MACRS MACRS places assets into one

of six classes, each of which has an assumed life Table 4.6 shows the rate of tion that the company can use for each of these classes Most industrial equipment fallsinto the 5- and 7-year classes To keep life simple, we will assume that all of Blooper’smining equipment goes into 5-year assets Thus Blooper can depreciate 20 percent ofits $10 million investment in Year 1 In the second year it could deduct depreciation of.32× 10 = $3.2 million, and so on.6

deprecia-How does use of MACRS depreciation affect the value of the depreciation tax shieldfor the magnoosium project? Table 4.7 gives the answer Notice that it does not affectthe total amount of depreciation that is claimed This remains at $10 million just as be-fore But MACRS allows companies to get the depreciation deduction earlier, which in-creases the present value of the depreciation tax shield from $2,523,000 to $2,583,000,

an increase of $60,000 Before we recognized MACRS depreciation, we calculatedproject NPV as $3,564,000 When we recognize MACRS, we should increase that figure by $60,000

5 Financial managers sometimes assume cash flows arrive in the middle of the calendar year, that is, at the

end of June This makes NPV also a midyear number If you are standing at the start of the year, the NPV must be discounted for a further half-year To do this, divide the midyear NPV by the square root of (1 + r).

This midyear convention is roughly equivalent to assuming cash flows are distributed evenly throughout the year This is a bad assumption for some industries In retailing, for example, most of the cash flow comes late in the year, as the holiday season approaches.

6 You might wonder why the 5-year asset class provides a depreciation deduction in Years 1 through 6 This is because the tax authorities assume that the assets are in service for only 6 months of the first year and 6 months of the last year The total project life is 5 years, but that 5-year life spans parts of 6 calendar years This assumption also explains why the depreciation is lower in the first year than it is in the second.

MODIFIED

ACCELERATED COST

RECOVERY SYSTEM

(MACRS) Depreciation

method that allows higher tax

deductions in early years and

lower deductions later.

EXCEL SPREADSHEET

A Spreadsheet Model for Blooper*

You might have guessed that discounted cash-flow analysis such as that of the Blooper case is tailor-made for spreadsheets The worksheet directly above shows the formu- las from the Excel spreadsheet that we used to generate the Blooper example The spreadsheet on the facing page shows the resulting values, which appear in the text in Tables 4.3 through 4.5.

The assumed values are the capital investment (cell B2), the initial level of revenues (cell C5), and expenses (cell C6) Rows 5 and 6 show that each entry for revenues and expenses equals the previous value times (1 + inflation rate), or 1.05 Row 3, which is the amount of working capital, is the sum of inventories and accounts receivable To capture the fact that inventories tend to rise with production, we set working capital equal to 15 times the following year’s expenses Similarly, accounts receivables rise with sales, so we assumed that accounts receivable would be 1/6 times the current year’s revenues Each entry in row 3 is the sum of these two quantities 1 Net investment

in working capital (row 4) is the increase in working capital from one year to the next Cash flow (row 12) is capital investment plus change in working capital plus profit after tax plus depreciation In row 13 we discount cash flow at a 12 percent discount rate and in cell B14 we add the present value of each cash flow to find project NPV Once the spreadsheet is up and running it is easy to do various sorts of “ what if” analysis Here are a few questions to try your hand.

Questions

1 What happens to cash flow in each year and the NPV of the project if the firm uses MACRS depreciation assuming a 3-year recovery period? Assume that Year 1 is the first year that de- preciation is taken.

2 Suppose the firm can economize on working capital by managing inventories more ciently If the firm can reduce inventories from 15 percent to 10 percent of next year’s cost

effi-of goods sold, what will be the effect on project NPV?

1 For convenience we assume that Blooper pays all its bills immediately and therefore accounts payable equals zero If it didn’t, working capital would be reduced by the amount of the payables.

3 What happens to NPV if the inflation rate falls from 5 percent to zero and the discount rate falls from 12 percent to 7 percent? Given that the real discount rate is almost unchanged, why does project NPV increase?

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All large corporations keep two sets of books, one for stockholders and one for theInternal Revenue Service It is common to use straight-line depreciation on the stock-holder books and MACRS depreciation on the tax books Only the tax books are rele-vant in capital budgeting

TABLE 4.6

Tax depreciation allowed

under the Modified

Accelerated Cost Recovery

System (figures in percent of

depreciable investment)

Recovery Period Class

1 Tax depreciation is lower in the first year because assets are assumed to be in service for 6 months.

2 Real property is depreciated straight-line over 27.5 years for residential property and 39 years for nonresidential property.

TABLE 4.7

The switch from straight-line to MACRS depreciation increases the value of Blooper’s depreciation tax shield from

$2,523,000 to $2,583,000 (figures in thousands of dollars)

䉴 Self-Test 5 Suppose that Blooper’s mining equipment could be put in the 3-year recovery period

class What is the present value of the depreciation tax shield? Confirm that the change

in the value of the depreciation tax shield equals the increase in project NPV from tion 1 of the spreadsheet exercises in the Excel box

ques-What to Do about Salvage Value. We assumed earlier that the mining equipmentwould be worthless when the magnoosium mine closed But suppose that it can be soldfor $2 million in Year 6 (The $2 million forecast salvage value recognizes inflation.)You recorded the initial $10 million investment as a negative cash flow Now in Year 6 you have a forecast return of $2 million of that investment That is a positive cash flow

When you sell the equipment, the IRS will check its books and see that you have already claimed depreciation of $10 million.7 So the value of your investment inBlooper’s tax books will be zero Any difference between the sale price ($2 million) andthe value in the tax books (zero) is treated as a taxable gain So your sale of the equip-ment will also land you with an additional tax bill in Year 6 of 35 × ($2 million – 0) =

$.70 million The extra cash flow in Year 6 is

Salvage value – tax on gain = $2 million – $.70 million

= $1.30 millionWhen discounted back to Year 0, this adds $.659 million, or $659,000, to the value ofthe project

Summary

How should the cash flows properly attributable to a proposed new project be culated?

cal-Here is a checklist to bear in mind when forecasting a project’s cash flows:

• Discount cash flows, not profits.

• Estimate the project’s incremental cash flows—that is, the difference between the cash flows with the project and those without the project.

• Include all indirect effects of the project, such as its impact on the sales of the firm’s other products.

• Forget sunk costs.

• Include opportunity costs, such as the value of land which you could otherwise sell.

• Beware of allocated overhead charges for heat, light, and so on These may not reflect the incremental effects of the project on these costs.

• Remember the investment in working capital As sales increase, the firm may need to make additional investments in working capital and, as the project finally comes to an end, it will recover these investments.

7 The MACRS tax depreciation schedules assume zero salvage value at the end of assets’ depreciable lives For reports to shareholders, however, positive expected salvage values are often recognized For example, Blooper’s financial statements might assume that its $10 million investment in mining equipment would be

worth $2 million in Year 6 In this case, the depreciation reported to shareholders would be based on the

dif-ference between investment and salvage value, that is, $8 million Straight-line depreciation would be $1.6

million per year.

398 SECTION FOUR

• Do not include debt interest or the cost of repaying a loan When calculating NPV, assume that the project is financed entirely by the shareholders and that they receive all the cash flows This isolates the investment decision from the financing decision.

How can the cash flows of a project be computed from standard financial ments?

state-Project cash flow does not equal profit You must allow for changes in working capital as well as noncash expenses such as depreciation Also, if you use a nominal cost of capital,

consistency requires that you forecast nominal cash flows—that is, cash flows that recognize

the effect of inflation.

How is the company’s tax bill affected by depreciation and how does this affect project value?

Depreciation is not a cash flow However, because depreciation reduces taxable income, it

reduces taxes This tax reduction is called the depreciation tax shield Modified Accelerated Cost Recovery System (MACRS) depreciation schedules allow more of the depreciation allowance to be taken in early years than under straight-line depreciation.

This increases the present value of the tax shield.

How do changes in working capital affect project cash flows?

Increases in net working capital such as accounts receivable or inventory are investments,

and therefore use cash—that is, they reduce the net cash flow provided by the project in that period When working capital is run down, cash is freed up, so cash flow increases.

www-ec.njit.edu/~mathis/interactive/FCCalcBase4.html A net present value calculator from

Professor Roswell Mathis

www.4pm.com/articles/palette.html Try the on-line demonstration here to see how good

busi-ness judgment is used to formulate cash-flow projections

www.irs.ustreas.gov/prod/bus_info/index.html Tax rules affecting project cash flows can be

found here

opportunity cost net working capital depreciation tax shield

Modified Accelerated Cost Recovery System (MACRS)

straight-line depreciation

1 Cash Flows A new project will generate sales of $74 million, costs of $42 million, and

de-preciation expense of $10 million in the coming year The firm’s tax rate is 35 percent culate cash flow for the year using all three methods discussed and confirm that they are equal.

Cal-2 Cash Flows Canyon Tours showed the following components of working capital last year:

Beginning End of Year

a What was the change in net working capital during the year?

b If sales were $36,000 and costs were $24,000, what was cash flow for the year? Ignore taxes.

3 Cash Flows Tubby Toys estimates that its new line of rubber ducks will generate sales of

$7 million, operating costs of $4 million, and a depreciation expense of $1 million If the tax rate is 40 percent, what is the firm’s operating cash flow? Show that you get the same an- swer using all three methods to calculate operating cash flow.

4 Cash Flows We’ve emphasized that the firm should pay attention only to cash flows when

assessing the net present value of proposed projects Depreciation is a noncash expense Why then does it matter whether we assume straight-line or MACRS depreciation when we assess project NPV?

5 Proper Cash Flows Quick Computing currently sells 10 million computer chips each

year at a price of $20 per chip It is about to introduce a new chip, and it forecasts annual sales of 12 million of these improved chips at a price of $25 each However, demand for the old chip will decrease, and sales of the old chip are expected to fall to 3 million per year The old chip costs $6 each to manufacture, and the new ones will cost $8 each What

is the proper cash flow to use to evaluate the present value of the introduction of the new chip?

6 Calculating Net Income The owner of a bicycle repair shop forecasts revenues of $160,000

a year Variable costs will be $45,000, and rental costs for the shop are $35,000 a year preciation on the repair tools will be $10,000 Prepare an income statement for the shop based on these estimates The tax rate is 35 percent.

De-7 Cash Flows Calculate the operating cash flow for the repair shop in the previous problem

using all three methods suggested in the material: (a) net income plus depreciation; (b) cash inflow/cash outflow analysis; and (c) the depreciation tax shield approach Confirm that all three approaches result in the same value for cash flow.

8 Cash Flows and Working Capital A house painting business had revenues of $16,000 and

expenses of $9,000 There were no depreciation expenses However, the business reported the following changes in various components of working capital:

Calculate net cash flow for the business for this period.

9 Incremental Cash Flows A corporation donates a valuable painting from its private

col-lection to an art museum Which of the following are incremental cash flows associated with the donation?

a The price the firm paid for the painting.

b The current market value of the painting.

c The deduction from income that it declares for its charitable gift.

d The reduction in taxes due to its declared tax deduction.

10 Operating Cash Flows Laurel’s Lawn Care, Ltd., has a new mower line that can generate

revenues of $120,000 per year Direct production costs are $40,000 and the fixed costs

of maintaining the lawn mower factory are $15,000 a year The factory originally cost

$1 million and is being depreciated for tax purposes over 25 years using straight-line preciation Calculate the operating cash flows of the project if the firm’s tax bracket is 35 percent.

de-400 SECTION FOUR

11 Operating Cash Flows Talia’s Tutus bought a new sewing machine for $40,000 that will be

depreciated using the MACRS depreciation schedule for a 5-year recovery period.

a Find the depreciation charge each year.

b If the sewing machine is sold after 3 years for $20,000, what will be the after-tax ceeds on the sale if the firm’s tax bracket is 35 percent?

pro-12 Proper Cash Flows Conference Services Inc has leased a large office building for $4

mil-lion per year The building is larger than the company needs: two of the building’s eight ries are almost empty A manager wants to expand one of her projects, but this will require using one of the empty floors In calculating the net present value of the proposed expan- sion, upper management allocates one-eighth of $4 million of building rental costs (i.e., $.5 million) to the project expansion, reasoning that the project will use one-eighth of the build- ing’s capacity.

sto-a Is this a reasonable procedure for purposes of calculating NPV?

b Can you suggest a better way to assess a cost of the office space used by the project?

13 Cash Flows and Working Capital A firm had net income last year of $1.2 million Its

de-preciation expenses were $.5 million, and its total cash flow was $1.2 million What pened to net working capital during the year?

hap-14 Cash Flows and Working Capital The only capital investment required for a small project

is investment in inventory Profits this year were $10,000, and inventory increased from

$4,000 to $5,000 What was the cash flow from the project?

15 Cash Flows and Working Capital A firm’s balance sheets for year-end 2000 and 2001

contain the following data What happened to investment in net working capital during 2001? All items are in millions of dollars.

Dec 31, 2000 Dec 31, 2001

16 Salvage Value Quick Computing (from problem 5) installed its previous generation of

com-puter chip manufacturing equipment 3 years ago Some of that older equipment will become unnecessary when the company goes into production of its new product The obsolete equip- ment, which originally cost $40 million, has been depreciated straight line over an assumed tax life of 5 years, but it can be sold now for $18 million The firm’s tax rate is 35 percent What is the after-tax cash flow from the sale of the equipment?

17 Salvage Value Your firm purchased machinery with a 7-year MACRS life for $10 million.

The project, however, will end after 5 years If the equipment can be sold for $4 million at the completion of the project, and your firm’s tax rate is 35 percent, what is the after-tax cash flow from the sale of the machinery?

18 Depreciation and Project Value Bottoms Up Diaper Service is considering the purchase

of a new industrial washer It can purchase the washer for $6,000 and sell its old washer for

$2,000 The new washer will last for 6 years and save $1,500 a year in expenses The portunity cost of capital is 15 percent, and the firm’s tax rate is 40 percent.

op-a If the firm uses straight-line depreciation to an assumed salvage value of zero over a year life, what are the cash flows of the project in Years 0–6? The new washer will in fact have zero salvage value after 6 years, and the old washer is fully depreciated.

6-b What is project NPV?

c What will NPV be if the firm uses MACRS depreciation with a 5-year tax life?Practice

Problems

19 Equivalent Annual Cost What is the equivalent annual cost of the washer in the previous

problem if the firm uses straight-line depreciation?

20 Cash Flows and NPV Johnny’s Lunches is considering purchasing a new, energy-efficient

grill The grill will cost $20,000 and will be depreciated according to the 3-year MACRS schedule It will be sold for scrap metal after 3 years for $5,000 The grill will have no effect

on revenues but will save Johnny’s $10,000 in energy expenses The tax rate is 35 percent.

a What are the operating cash flows in Years 1–3?

b What are total cash flows in Years 1–3?

c If the discount rate is 12 percent, should the grill be purchased?

21 Project Evaluation Revenues generated by a new fad product are forecast as follows:

a What is the initial investment in the product? Remember working capital.

b If the plant and equipment are depreciated over 4 years to a salvage value of zero using straight-line depreciation, and the firm’s tax rate is 40 percent, what are the project cash flows in each year?

c If the opportunity cost of capital is 10 percent, what is project NPV?

d What is project IRR?

22 Buy versus Lease You can buy a car for $25,000 and sell it in 5 years for $5,000 Or you

can lease the car for 5 years for $5,000 a year The discount rate is 10 percent per year.

a Which option do you prefer?

b What is the maximum amount you should be willing to pay to lease rather than buy the car?

23 Project Evaluation Kinky Copies may buy a high-volume copier The machine costs

$100,000 and will be depreciated straight-line over 5 years to a salvage value of $20,000 Kinky anticipates that the machine actually can be sold in 5 years for $30,000 The machine will save $20,000 a year in labor costs but will require an increase in working capital, mainly paper supplies, of $10,000 The firm’s marginal tax rate is 35 percent Should Kinky buy the machine?

24 Project Evaluation Blooper Industries must replace its magnoosium purification system.

Quick & Dirty Systems sells a relatively cheap purification system for $10 million The tem will last 5 years Do-It-Right sells a sturdier but more expensive system for $12 million;

sys-it will last for 8 years Both systems entail $1 million in operating costs; both will be preciated straight line to a final value of zero over their useful lives; neither will have any salvage value at the end of its life The firm’s tax rate is 35 percent, and the discount rate is

de-12 percent Which system should Blooper install?

25 Project Evaluation The following table presents sales forecasts for Golden Gelt Giftware.

The unit price is $40 The unit cost of the giftware is $25.

busi-26 Project Evaluation Ilana Industries, Inc., needs a new lathe It can buy a new high-speed

lathe for $1 million The lathe will cost $35,000 to run, will save the firm $125,000 in labor costs, and will be useful for 10 years Suppose that for tax purposes, the lathe will be de- preciated on a straight-line basis over its 10-year life to a salvage value of $100,000 The ac- tual market value of the lathe at that time also will be $100,000 The discount rate is 10 per- cent and the corporate tax rate is 35 percent What is the NPV of buying the new lathe?

27 Project Evaluation The efficiency gains resulting from a just-in-time inventory

manage-ment system will allow a firm to reduce its level of inventories permanently by $250,000 What is the most the firm should be willing to pay for installing the system?

28 Project Evaluation Better Mousetraps has developed a new trap It can go into production

for an initial investment in equipment of $6 million The equipment will be depreciated straight line over 5 years to a value of zero, but in fact it can be sold after 5 years for

$500,000 The firm believes that working capital at each date must be maintained at a level

of 10 percent of next year’s forecast sales The firm estimates production costs equal to

$1.50 per trap and believes that the traps can be sold for $4 each Sales forecasts are given

in the following table The project will come to an end in 5 years, when the trap becomes technologically obsolete The firm’s tax bracket is 35 percent, and the required rate of return

on the project is 12 percent What is project NPV?

29 Working Capital Management Return to the previous problem Suppose the firm can cut

its requirements for working capital in half by using better inventory control systems By how much will this increase project NPV?

30 Project Evaluation PC Shopping Network may upgrade its modem pool It last upgraded

2 years ago, when it spent $115 million on equipment with an assumed life of 5 years and

an assumed salvage value of $15 million for tax purposes The firm uses straight-line preciation The old equipment can be sold today for $80 million A new modem pool can be installed today for $150 million This will have a 3-year life, and will be depreciated to zero using straight-line depreciation The new equipment will enable the firm to increase sales by

de-$25 million per year and decrease operating costs by $10 million per year At the end of 3 years, the new equipment will be worthless Assume the firm’s tax rate is 35 percent and the discount rate for projects of this sort is 12 percent.

Challenge

Problems

a What is the net cash flow at time 0 if the old equipment is replaced?

b What are the incremental cash flows in Years 1, 2, and 3?

c What are the NPV and IRR of the replacement project?

op-c Demolition costs are incremental cash outflows.

d The cost of the access road is sunk and not incremental.

e Lost cash flows from other projects are incremental cash outflows.

f Depreciation is not a cash expense and should not be included, except as it affects taxes (Taxes are discussed later in this material.)

3 Actual health costs will be increasing at about 7 percent a year.

The present value at 10 percent is $9,214 if the first payment is made immediately If it is delayed a year, present value falls to $8,377.

4 The tax rate is T = 35 percent Taxes paid will be

T× (revenue – expenses – depreciation) = 35 × (600 – 300 – 200) = $35 Operating cash flow can be calculated as follows.

a Revenue – expenses – taxes = 600 – 300 – 35 = $265

b Net profit + depreciation = (600 – 300 – 200 – 35) + 200

= 65 + 200 = 265

c (Revenues – cash expenses) × (1 – tax rate) + (depreciation × tax rate)

= (600 – 300) × (1 – 35) + (200 × 35) = 265

Jack Tar, CFO of Sheetbend & Halyard, Inc., opened the

company-confidential envelope It contained a draft of a

com-petitive bid for a contract to supply duffel canvas to the U.S.

Navy The cover memo from Sheetbend’s CEO asked Mr Tar to

review the bid before it was submitted.

The bid and its supporting documents had been prepared by

Sheetbend’s sales staff It called for Sheetbend to supply 100,000

yards of duffel canvas per year for 5 years The proposed selling

price was fixed at $30 per yard.

Mr Tar was not usually involved in sales, but this bid was

un-usual in at least two respects First, if accepted by the navy, it

would commit Sheetbend to a fixed price, long-term contract.

Second, producing the duffel canvas would require an investment

of $1.5 million to purchase machinery and to refurbish

Sheet-bend’s plant in Pleasantboro, Maine.

Mr Tar set to work and by the end of the week had collected

the following facts and assumptions:

• The plant in Pleasantboro had been built in the early 1900s

and is now idle The plant was fully depreciated on

Sheet-bend’s books, except for the purchase cost of the land (in

1947) of $10,000.

• Now that the land was valuable shorefront property, Mr Tar

thought the land and the idle plant could be sold, immediately

or in the future, for $600,000.

• Refurbishing the plant would cost $500,000 This investment

would be depreciated for tax purposes on the 10-year MACRS

• Table 4.8 shows the sales staff ’s forecasts of income from the navy contract Mr Tar reviewed this forecast and decided that its assumptions were reasonable, except that the forecast used book, not tax, depreciation.

• But the forecast income statement contained no mention of working capital Mr Tar thought that working capital would average about 10 percent of sales.

Armed with this information, Mr Tar constructed a spreadsheet

to calculate the NPV of the duffel canvas project, assuming that Sheetbend’s bid would be accepted by the navy.

He had just finished debugging the spreadsheet when another confidential envelope arrived from Sheetbend’s CEO It con- tained a firm offer from a Maine real estate developer to purchase Sheetbend’s Pleasantboro land and plant for $1.5 million in cash Should Mr Tar recommend submitting the bid to the navy at the proposed price of $30 per yard? The discount rate for this project is 12 percent.

TABLE 4.8

Forecasted income statement

for the navy duffel canvas

project (dollar figures in

thousands, except price per

4 Cost of goods sold 2,100.00 2,184.00 2,271.36 2,362.21 2,456.70

5 Operating cash flow (3 – 4) 900.00 816.00 728.64 637.79 543.30

1 Yards sold and price per yard would be fixed by contract.

2 Cost of goods includes fixed cost of $300,000 per year plus variable costs of $18 per yard Costs are expected to increase at the inflation rate of 4 percent per year.

3 Depreciation: A $1 million investment in machinery is depreciated straight-line over 5 years ($200,000 per year) The $500,000 cost of refurbishing the Pleasantboro plant is depreciated straight-line over 10 years ($50,000 per year).

Risk and Return

Why the CAPM Works

The Security Market Line

How Well Does the CAPM Work?

Using the CAPM to Estimate Expected Returns

Capital Budgeting and Project Risk

Company versus Project Risk

Determinants of Project Risk

Don’t Add Fudge Factors to Discount Rates

Summary

Professor William F Sharpe receiving the Nobel Prize in Economics.

The prize was for Sharpe’s development of the capital asset pricing model This model shows

how risk should be measured and provides a formula relating risk to the opportunity cost of

capital.

Leif Jansson/Pica Pressfoto

arlier we began to come to grips with the topic of risk We made the

dis-tinction between unique risk and macro, or market, risk Unique risk

arises from events that affect only the individual firm or its immediatecompetitors; it can be eliminated by diversification But regardless of howmuch you diversify, you cannot avoid the macroeconomic events that create market risk.This is why investors do not require a higher rate of return to compensate for uniquerisk but do need a higher return to persuade them to take on market risk

How can you measure the market risk of a security or a project? We will see thatmarket risk is usually measured by the sensitivity of the investment’s returns to fluctu-ations in the market We will also see that the risk premium investors demand should beproportional to this sensitivity This relationship between risk and return is a useful way

to estimate the return that investors expect from investing in common stocks

Finally, we will distinguish between the risk of the company’s securities and the risk

of an individual project We will also consider what managers should do when the risk

of the project is different from that of the company’s existing business

After studying this material you should be able to

䉴 Measure and interpret the market risk, or beta, of a security

䉴 Relate the market risk of a security to the rate of return that investors demand

䉴 Calculate the opportunity cost of capital for a project

408

E

Measuring Market Risk

Changes in interest rates, government spending, monetary policy, oil prices, foreign change rates, and other macroeconomic events affect almost all companies and the re-turns on almost all stocks We can therefore assess the impact of “macro” news by

ex-tracking the rate of return on a market portfolio of all securities If the market is up on

a particular day, then the net impact of macroeconomic changes must be positive Weknow the performance of the market reflects only macro events, because firm-specificevents—that is, unique risks—average out when we look at the combined performance

of thousands of companies and securities

In principle the market portfolio should contain all assets in the world economy—not just stocks, but bonds, foreign securities, real estate, and so on In practice, however,financial analysts make do with indexes of the stock market, usually the Standard &Poor’s Composite Index (the S&P 500).1

Our task here is to define and measure the risk of individual common stocks You

can probably see where we are headed Risk depends on exposure to macroeconomicevents and can be measured as the sensitivity of a stock’s returns to fluctuations in re-

turns on the market portfolio This sensitivity is called the stock’s beta Beta is often

written as the Greek letter β

1 We discussed the most popular stock market indexes in Section 9.2.

MARKET PORTFOLIO

Portfolio of all assets in the

economy In practice a broad

stock market index, such as

the Standard & Poor’s

Composite, is used to

represent the market.

BETA Sensitivity of a

stock’s return to the return

on the market portfolio.

MEASURING BETAEarlier we looked at the variability of individual securities Compaq had the higheststandard deviation and Exxon the lowest If you had held Compaq on its own, your re-turns would have varied almost three times as much as if you had held Exxon But wiseinvestors don’t put all their eggs in just one basket: they reduce their risk by diversifi-cation An investor with a diversified portfolio will be interested in the effect each stockhas on the risk of the entire portfolio.

Diversification can eliminate the risk that is unique to individual stocks, but not therisk that the market as a whole may decline, carrying your stocks with it

Some stocks are less affected than others by market fluctuations Investment agers talk about “defensive” and “aggressive” stocks Defensive stocks are not very sen-sitive to market fluctuations In contrast, aggressive stocks amplify any market move-ments If the market goes up, it is good to be in aggressive stocks; if it goes down, it isbetter to be in defensive stocks (and better still to have your money in the bank)

man-Now we’ll show you how betas are measured

Suppose we look back at the trading history of Turbot-Charged Seafoods and pick out

6 months when the return on the market portfolio was plus or minus 1 percent

Month Market Return, % Turbot-Charged Seafood’s Return, %

Look at Figure 4.7, where these observations are plotted We’ve drawn a line through

the average performance of Turbot when the market is up or down by 1 percent The slope of this line is Turbot’s beta You can see right away that the beta is 8, because on

average Turbot stock gains or loses 8 percent when the market is up or down by 1 cent Notice that a 2-percentage-point difference in the market return (–1 to +1) gener-ates on average a 1.6-percentage-point difference for Turbot shareholders (–.8 to +.8).The ratio, 1.6/2 = 8, is beta

per-In 4 months, Turbot’s returns lie above or below the line in Figure 4.7 The distancefrom the line shows the response of Turbot’s stock returns to news or events that affected

Turbot but did not affect the overall market For example, in Month 2, investors in

Turbot stock benefited from good macroeconomic news (the market was up 1 percent)and also from some favorable news specific to Turbot The market rise gave a boost

of 8 percent to Turbot stock (beta of 8 times the 1 percent market return) Then

Aggressive stocks have high betas, betas greater than 1.0, meaning that their returns tend to respond more than one-for-one to changes in the return of the overall market The betas of defensive stocks are less than 1.0 The returns of these stocks vary less than one-for-one with market returns The average beta

of all stocks is—no surprises here—1.0 exactly.

䉴 Self-Test 1 Here are 6 months’ returns to stockholders in the Anchovy Queen restaurant chain:

Month Market Return, % Anchovy Queen Return, %

1 Observe rates of return, usually monthly, for the stock and the market

2 Plot the observations as in Figure 4.7

3 Fit a line showing the average return to the stock at different market returns.Beta is the slope of the fitted line

As this example illustrates, we can break down common stock returns into two parts: the part explained by market returns and the firm’s beta, and the part due to news that is specific to the firm Fluctuations in the first part reflect market risk; fluctuations in the second part reflect unique risk.

FIGURE 4.7

This figure is a plot of the

data presented in the table

from Example 1 Each point

shows the performance of

Turbot-Charged Seafoods

stock when the overall market

is either up or down by 1

percent On average,

Turbot-Charged moves in the same

direction as the market, but

not as far Therefore,

Turbot-Charged’s beta is less than

1.0 We can measure beta by

the slope of a line fitted to

the points in the figure In

0.5

1.0

1.5

.8 6

.4 2

Market return, percent

This may sound like a lot of work but in practice computers do it for you Here aretwo real examples.

BETAS FOR MCI WORLDCOM AND EXXON

Each point in Figure 4.8a shows the return on MCI WorldCom stock and the return on

the market index in a different month For example, the circled point shows that in themonth of May 1997 MCI stock price rose by 23 percent, whereas the market index rose

by 5.9 percent Notice that more often than not MCI outperformed the market when theindex rose and underperformed the market when the index fell Thus MCI was a rela-tively aggressive, high-beta stock

We have drawn a line of best fit through the points in the figure.2The slope of this

FIGURE 4.8

(a) Each point in this figure

shows the returns on MCI

common stock and the

overall market in a particular

month Sixty months are

plotted in all MCI’s beta is

the slope of the line fitted to

these points MCI has a

relatively high beta of 1.3.

(b) In this plot of 60 months’

returns for Exxon and the

overall market the slope of

the fitted line is much less

than MCI’s beta in (a) Exxon

has a relatively low beta of

5

2The line of best fit is usually known as a regression line The slope of the line can be calculated using

ordi-nary least squares regression The dependent variable is the return on the stock (MCI) The independent

vari-able is the return on the market index, in this case the S&P 500.

5

412 SECTION FOUR

line is 1.3 For each extra 1 percent rise in the market MCI stock price moved on age an extra 1.3 percent For each extra 1 percent fall in the market, MCI stock pricefell an extra 1.3 percent Thus MCI’s beta was 1.3

aver-Of course, MCI’s stock returns are not perfectly related to market returns The pany was also subject to unique risk, which shows up in the scatter of points around theline Sometimes MCI flew south while the market went north, or vice versa

com-Figure 4.8b shows a similar plot of the monthly returns for Exxon In contrast to

MCI, Exxon was a defensive, low-beta stock It was not highly sensitive to marketmovements, usually lagging when the market rose and yet doing better (or less badly)when the market fell The slope of the line of best fit shows that on average an extra 1percent change in the index resulted in an extra 61 percent change in the price of Exxonstock Thus Exxon’s beta was 61

You may find it interesting to look at Table 4.9, which shows how past marketmovements have affected several well-known stocks Exxon had the lowest beta: itsstock return was 61 times as sensitive as the average stock to market movements Mi-crosoft was at the other extreme: its return was 1.33 times as sensitive as the averagestock to market movements

PORTFOLIO BETASDiversification decreases variability from unique risk but not from market risk Thebeta of a portfolio is just an average of the betas of the securities in the portfolio,weighted by the investment in each security For example, a portfolio comprised of onlytwo stocks would have a beta as follows:

Thus a portfolio invested 50-50 in MCI and Exxon would have a beta of (.5 × 1.3) + (.5

× 61) = 95

A well-diversified portfolio of stocks all with betas of 1.3, like MCI, would still have

a portfolio beta of 1.3 However, most of the individual stocks’ unique risk would be versified away The market risk would remain, and such a portfolio would end up 1.3

di-Beta of portfolio = (fraction of portfolio in first stock × beta of first stock)

+ (fraction of portfolio in second stock × beta of

second stock)

TABLE 4.9

Betas for selected common

stocks, July 1994–June 1999

times as variable as the market For example, if the market has an annual standard viation of 20 percent (about the historical average reported earlier), a fully diversifiedportfolio with beta of 1.3 has a standard deviation of 1.3 × 20 = 26 percent.

de-Portfolios with betas between 0 and 1.0 tend to move in the same direction as themarket but not as far A well-diversified portfolio of low-beta stocks like Exxon, allwith betas of 61, has almost no unique risk and is relatively unaffected by marketmovements Such a portfolio is 61 times as variable as the market

Of course, on average stocks have a beta of 1.0 A well-diversified portfolio ing all kinds of stocks, with an average beta of 1, has the same variability as the mar-ket index

includ-䉴 Self-Test 2 Say you invested an equal amount in each of the stocks shown in Table 4.9 Calculate

the beta of your portfolio

You don’t have to be wealthy to own a diversified portfolio You can buy shares in one

of the more than 6,000 mutual funds in the United States

Investors buy shares of the funds, and the funds use the money to buy portfolios ofsecurities The returns on the portfolios are passed back to the funds’ owners in pro-portion to their shareholdings Therefore, the funds act like investment cooperatives, offering even the smallest investors diversification and professional management at low cost

Let’s look at the betas of two mutual funds that invest in stocks Figure 4.9a plots the

monthly returns of Vanguard’s Windsor II mutual fund and of the S&P index from July

1994 to June 1999 You can see that the stocks in the Windsor II fund had nearly age sensitivity to market changes: they had on average a beta of 87

aver-If the Windsor II fund had no unique risk, its portfolio would have been 87 times asvariable as the market portfolio But the fund had not diversified away quite all the

unique risk; there is still some scatter about the line in Figure 4.9a As a result, the

vari-ability of the fund was somewhat more than 87 times that of the market

Figure 4.9b shows the same sort of plot for Vanguard’s Index Trust 500 Portfolio

mu-tual fund Notice that this fund has a beta of 1.0 and only a tiny residual of unique risk—

the fitted line fits almost exactly because an index fund is designed to track the market

as closely as possible The managers of the fund do not attempt to pick good stocks butjust work to achieve full diversification at very low cost (The Vanguard index fundtakes investments of as little as $3,000 and manages the fund for an annual fee of less

than 20 percent of the fund’s assets.) The index fund is fully diversified Investors in

this fund buy the market as a whole and don’t have to worry at all about unique risk

䉴 Self-Test 3 Suppose you could achieve full diversification in a portfolio constructed from stocks

with an average beta of 5 If the standard deviation of the market is 20 percent per year,what is the standard deviation of the portfolio return?

414 SECTION FOUR

Risk and Return

Earlier we looked at past returns on selected investments The least risky investmentwas U.S Treasury bills Since the return on Treasury bills is fixed, it is unaffected by

what happens to the market Thus the beta of Treasury bills is zero The most risky

in-vestment that we considered was the market portfolio of common stocks This has erage market risk: its beta is 1.0

av-Wise investors don’t run risks just for fun They are playing with real money andtherefore require a higher return from the market portfolio than from Treasury bills Thedifference between the return on the market and the interest rate on bills is termed the

market risk premium Over the past 73 years the average market risk premium has

been just over 9 percent a year Of course, there is plenty of scope for argument as towhether the past 73 years constitute a typical period, but we will just assume here that

9 percent is the normal risk premium, that is, the additional return that an investor couldreasonably expect from investing in the stock market rather than Treasury bills

FIGURE 4.9

(a) The slope of the fitted line

shows that investors in the

Windsor II mutual fund bore

market risk slightly below

that of the S&P 500

portfolio Windsor II’s beta

was 87 This was the average

beta of the individual

common stocks held by the

fund They also bore some

unique risk, however; note

the scatter of Windsor II’s

returns above and below the

fitted line.

(b) The Vanguard 500

Portfolio is a fully diversified

index fund designed to track

the performance of the

market Note the fund’s beta

(1.0) and the absence of

unique risk The fund’s

returns lie almost precisely

on the fitted line relating its

returns to those of the S&P

500 portfolio.

20

10 0 10

20 Windsor II return,

percent

Market return, percent

20 Index 500 return,

percent

Market return, percent

Difference between market

return and return on risk-free

Treasury bills.

In Figure 4.10a we plotted the risk and expected return from Treasury bills and the

market portfolio You can see that Treasury bills have a beta of zero and a risk-free turn; we’ll assume that return is 5 percent The market portfolio has a beta of 1.0 and

re-an assumed expected return of 14 percent.3Now, given these two benchmarks, what expected rate of return should an investorrequire from a stock or portfolio with a beta of 5? Halfway between, of course Thus

in Figure 4.10b we drew a straight line through the Treasury bill return and the expected market return and marked with an X the expected return for a beta of 5, that is, 9.5

percent This includes a risk premium of 4.5 percent above the Treasury bill return of 5percent

You can calculate this return as follows: start with the difference between the

ex-pected market return r m and the Treasury bill rate r f This is the expected market riskpremium

FIGURE 4.10

(a) Here we begin the plot of

expected rate of return

against beta The first

benchmarks are Treasury

bills (beta = 0) and the

market portfolio (beta = 1.0).

We assume a Treasury bill

rate of 5 percent and a

market return of 14 percent.

The market risk premium is

14 – 5 = 9 percent.

(b) A portfolio split evenly

between Treasury bills and

the market will have beta =

.5 and an expected return of

9.5 percent (point X) A

portfolio invested 80 percent

in the market and 20 percent

in Treasury bills has beta =

.8 and an expected rate of

return of 12.2 percent (point

Y) Note that the expected

rate of return on any

portfolio mixing Treasury

bills and the market lies on a

straight line The risk

0

9%
market risk premium

Treasury bills

Beta (a)

0

Beta (b)

14 12.2

3 On past evidence the risk premium on the market is 9 percent With a 5 percent Treasury bill rate, the pected market return would be 5 + 9 = 14 percent.

ex-416 SECTION FOUR

Market risk premium = r m – r f= 14% – 5% = 9%

Beta measures risk relative to the market Therefore, the expected risk premium onany asset equals beta times the market risk premium:

Risk premium on any asset = r – r f=β(r m – r f)

With a beta of 5 and a market risk premium of 9 percent,

Risk premium = β(r m – r f) = 5 × 9 = 4.5%

The total expected rate of return is the sum of the risk-free rate and the risk premium:

Expected return = risk-free rate + risk premium

This formula states the basic risk–return relationship called the capital asset pricing

model, or CAPM The CAPM has a simple interpretation:

Note that the expected rate of return on an asset with β = 1 is just the market return.With a risk-free rate of 5 percent and market risk premium of 9 percent,

r = r f+β(r m – r f)

= 5% + (1 × 9%) = 14%

䉴 Self-Test 4 What are the risk premium and expected rate of return on a stock with β = 1.5? Assume

a Treasury bill rate of 6 percent and a market risk premium of 9 percent

WHY THE CAPM WORKSThe CAPM assumes that the stock market is dominated by well-diversified investorswho are concerned only with market risk That makes sense in a stock market wheretrading is dominated by large institutions and even small fry can diversify at very lowcost

Have you ever daydreamed about receiving a $1 million check, no strings attached, from

an unknown benefactor? Let’s daydream about how you would invest it

We have two good candidates: Treasury bills, which offer an absolutely safe return,and the market portfolio (possibly via the Vanguard index fund discussed earlier in this

The expected rates of return demanded by investors depend on two things:

(1) compensation for the time value of money (the risk-free rate r f), and (2) a risk premium, which depends on beta and the market risk premium.

CAPITAL ASSET

PRICING MODEL

(CAPM) Theory of the

relationship between risk and

return which states that the

expected risk premium on

any security equals its beta

times the market risk

premium.

material) The market has generated superior returns on average, but those returns havefluctuated a lot (Look back to Figure 3.15.) So your investment policy is going to de-pend on your tolerance for risk.

If you’re a cautious soul, you may invest only part of your money in the market folio and lend the remainder to the government by buying Treasury bills Suppose thatyou invest 80 percent of your money in the market portfolio and lend out the other 20percent to the government by buying U.S Treasury bills Then the beta of your portfo-lio will be a mixture of the beta of the market (βmarket= 1.0) and the beta of the T-bills(βT-bills= 0):

port-Beta of portfolio =(proportion × beta ofin market market)+(proportion × beta ofin T-bills T-bills)

β = (.8 × βmarket) + (.2 × βT-bills)

= (.8 × 1.0) + (.2 × 0) = 80The fraction of funds that you invest in the market also affects your return If you in-vest your entire million in the market portfolio, you earn the full market risk premium.But if you invest only 80 percent of your money in the market, you earn only 80 per-cent of the risk premium

Expectedrisk premium =on portfolio (proportion in × risk premiumT-bills on T-bills )+(proportion in × market riskmarket premium )

= (.2 × 0) + (.8 × expected market risk premium)

= 8 × expected market risk premium

= 8 × 9 = 7.2%

The expected return on your portfolio is equal to the risk-free interest rate plus theexpected risk premium:

Expected portfolio return = rportfolio= 5 + 7.2 = 12.2%

In Figure 4.10b we show the beta and expected return on this portfolio by the letter Y.

THE SECURITY MARKET LINEExample 3 illustrates a general point: by investing some proportion of your money inthe market portfolio and lending (or borrowing)4the balance, you can obtain any com-bination of risk and expected return along the sloping line in Figure 4.11 This line is

generally known as the security market line.

4 Notice that the security market line extends above the market return at β = 1 How would you generate a portfolio with, say, β = 2? It’s easy, but it’s risky Suppose you borrow $1 million and invest the loan plus $1 million in the market portfolio That gives you $2 million invested and a $1 million liability Your portfolio now has a beta of 2.0:

Beta of portfolio = (proportion in market × beta of market) + (proportion in loan × beta of loan)

β = (2 × β market ) + (–1 × β loan )

= (2 × 1.0) + (–1 × 0) = 2 Notice that the proportion in the loan is negative because you are borrowing, not lending money.

By the way, borrowing from a bank or stockbroker would not be difficult or unduly expensive as long as you put up your $2 million stock portfolio as security for the loan.

Can you calculate the risk premium and the expected rate of return on this borrow-and-invest strategy?

SECURITY MARKET

between expected return and

beta.

418 SECTION FOUR

䉴 Self-Test 5 How would you construct a portfolio with a beta of 25? What is the expected return to

this strategy? Assume Treasury bills yield 6 percent and the market risk premium is 9percent

Look back to Figure 4.10b, which asserts that an individual common stock with β =.5 must offer a 9.5 percent expected rate of return when Treasury bills yield 5 percentand the market risk premium is 9 percent You can now see why this has to be so If thatstock offered a lower rate of return, nobody would buy even a little of it—they could get9.5 percent just by investing 50-50 in Treasury bills and the market And if nobody wants

to hold the stock, its price has to drop A lower price means a better buy for investors,that is, a higher rate of return The price will fall until the stock’s expected rate of return

is pushed up to 9.5 percent At that price and expected return the CAPM holds

If, on the other hand, our stock offered more than 9.5 percent, diversified investorswould want to buy more of it That would push the price up and the expected returndown to the levels predicted by the CAPM

This reasoning holds for stocks with any beta That’s why the CAPM makes sense,and why the expected risk premium on an investment should be proportional to its beta

䉴 Self-Test 6 Suppose you invest $400,000 in Treasury bills and $600,000 in the market portfolio

What is the return on your portfolio if bills yield 6 percent and the expected return onthe market is 15 percent? What does the return on this portfolio imply for the expectedreturn on individual stocks with betas of 6?

The security market line describes the expected returns and risks from investing different fractions of your funds in the market It also sets a standard for other investments Investors will be willing to hold other investments only if they offer equally good prospects Thus the required risk

premium for any investment is given by the security market line:

Risk premium on investment = beta × expected market risk premium

FIGURE 4.11

The security market line

shows how expected rate of

return depends on beta.

According to the capital

asset pricing model, expected

rates of return for all

securities and all portfolios

lie on this line.