Revised SM 9th Ed (2)

Solutions Manual

Fundamentals of Corporate Finance 9th editionRoss, Westerfield, and Jordan

Updated 12-20-2008

CHAPTER 1INTRODUCTION TO CORPORATE FINANCEAnswers to Concepts Review and Critical Thinking Questions

1. Capital budgeting (deciding whether to expand a manufacturingplant), capital structure (deciding whether to issue new equity anduse the proceeds to retire outstanding debt), and working capitalmanagement (modifying the firm’s credit collection policy with itscustomers).

2. Disadvantages: unlimited liability, limited life, difficulty intransferring ownership, hard to raise capital funds. Someadvantages: simpler, less regulation, the owners are also themanagers, sometimes personal tax rates are better than corporatetax rates.

3. The primary disadvantage of the corporate form is the doubletaxation to shareholders of distributed earnings and dividends.Some advantages include: limited liability, ease oftransferability, ability to raise capital, and unlimited life.

4. In response to Sarbanes-Oxley, small firms have elected to go darkbecause of the costs of compliance. The costs to comply with Sarboxcan be several million dollars, which can be a large percentage ofa small firms profits. A major cost of going dark is less access tocapital. Since the firm is no longer publicly traded, it can nolonger raise money in the public market. Although the company willstill have access to bank loans and the private equity market, thecosts associated with raising funds in these markets are usuallyhigher than the costs of raising funds in the public market.

5. The treasurer’s office and the controller’s office are the twoprimary organizational groups that report directly to the chieffinancial officer. The controller’s office handles cost andfinancial accounting, tax management, and management informationsystems, while the treasurer’s office is responsible for cash and

CHAPTER 1 B-2

credit management, capital budgeting, and financial planning.Therefore, the study of corporate finance is concentrated withinthe treasury group’s functions.

6. To maximize the current market value (share price) of the equity ofthe firm (whether it’s publicly-traded or not).

7. In the corporate form of ownership, the shareholders are the ownersof the firm. The shareholders elect the directors of thecorporation, who in turn appoint the firm’s management. Thisseparation of ownership from control in the corporate form oforganization is what causes agency problems to exist. Managementmay act in its own or someone else’s best interests, rather thanthose of the shareholders. If such events occur, they maycontradict the goal of maximizing the share price of the equity ofthe firm.

8. A primary market transaction.

B-3 SOLUTIONS

9. In auction markets like the NYSE, brokers and agents meet at aphysical location (the exchange) to match buyers and sellers ofassets. Dealer markets like NASDAQ consist of dealers operating atdispersed locales who buy and sell assets themselves, communicatingwith other dealers either electronically or literally over-the-counter.

10. Such organizations frequently pursue social or political missions,so many different goals are conceivable. One goal that is oftencited is revenue minimization; i.e., provide whatever goods andservices are offered at the lowest possible cost to society. Abetter approach might be to observe that even a not-for-profitbusiness has equity. Thus, one answer is that the appropriate goalis to maximize the value of the equity.

11. Presumably, the current stock value reflects the risk, timing, andmagnitude of all future cash flows, both short-term and long-term.If this is correct, then the statement is false.

12. An argument can be made either way. At the one extreme, we couldargue that in a market economy, all of these things are priced.There is thus an optimal level of, for example, ethical and/orillegal behavior, and the framework of stock valuation explicitlyincludes these. At the other extreme, we could argue that these arenon-economic phenomena and are best handled through the politicalprocess. A classic (and highly relevant) thought question thatillustrates this debate goes something like this: “A firm hasestimated that the cost of improving the safety of one of itsproducts is $30 million. However, the firm believes that improvingthe safety of the product will only save $20 million in productliability claims. What should the firm do?”

13. The goal will be the same, but the best course of action towardthat goal may be different because of differing social, political,and economic institutions.

14. The goal of management should be to maximize the share price forthe current shareholders. If management believes that it canimprove the profitability of the firm so that the share price willexceed $35, then they should fight the offer from the outsidecompany. If management believes that this bidder or otherunidentified bidders will actually pay more than $35 per share toacquire the company, then they should still fight the offer.

CHAPTER 1 B-4

However, if the current management cannot increase the value of thefirm beyond the bid price, and no other higher bids come in, thenmanagement is not acting in the interests of the shareholders byfighting the offer. Since current managers often lose their jobswhen the corporation is acquired, poorly monitored managers have anincentive to fight corporate takeovers in situations such as this.

15. We would expect agency problems to be less severe in countries witha relatively small percentage of individual ownership. Fewerindividual owners should reduce the number of diverse opinionsconcerning corporate goals. The high percentage of institutionalownership might lead to a higher degree of agreement between ownersand managers on decisions concerning risky projects. In addition,institutions may be better able to implement effective monitoringmechanisms on managers than can individual owners, based on theinstitutions’ deeper resources and experiences with their ownmanagement. The increase in institutional ownership of stock in theUnited States and the growing activism of these large shareholdergroups may lead to a reduction in agency problems for U.S.corporations and a more efficient market for corporate control.

B-5 SOLUTIONS

16. How much is too much? Who is worth more, Ray Irani or Tiger Woods?The simplest answer is that there is a market for executives justas there is for all types of labor. Executive compensation is theprice that clears the market. The same is true for athletes andperformers. Having said that, one aspect of executive compensationdeserves comment. A primary reason executive compensation has grownso dramatically is that companies have increasingly moved to stock-based compensation. Such movement is obviously consistent with theattempt to better align stockholder and management interests. Inrecent years, stock prices have soared, so management has cleanedup. It is sometimes argued that much of this reward is simply dueto rising stock prices in general, not managerial performance.Perhaps in the future, executive compensation will be designed toreward only differential performance, i.e., stock price increasesin excess of general market increases.

CHAPTER 2FINANCIAL STATEMENTS, TAXES AND CASH FLOW

Answers to Concepts Review and Critical Thinking Questions

1. Liquidity measures how quickly and easily an asset can be convertedto cash without significant loss in value. It’s desirable for firmsto have high liquidity so that they have a large factor of safetyin meeting short-term creditor demands. However, since liquidityalso has an opportunity cost associated with it—namely that higherreturns can generally be found by investing the cash intoproductive assets—low liquidity levels are also desirable to thefirm. It’s up to the firm’s financial management staff to find areasonable compromise between these opposing needs.

2. The recognition and matching principles in financial accountingcall for revenues, and the costs associated with producing thoserevenues, to be “booked” when the revenue process is essentiallycomplete, not necessarily when the cash is collected or bills arepaid. Note that this way is not necessarily correct; it’s the wayaccountants have chosen to do it.

3. Historical costs can be objectively and precisely measured whereasmarket values can be difficult to estimate, and different analystswould come up with different numbers. Thus, there is a tradeoffbetween relevance (market values) and objectivity (book values).

4. Depreciation is a non-cash deduction that reflects adjustments madein asset book values in accordance with the matching principle infinancial accounting. Interest expense is a cash outlay, but it’s afinancing cost, not an operating cost.

5. Market values can never be negative. Imagine a share of stockselling for –$20. This would mean that if you placed an order for100 shares, you would get the stock along with a check for $2,000.How many shares do you want to buy? More generally, because of

B-7 SOLUTIONS

corporate and individual bankruptcy laws, net worth for a person ora corporation cannot be negative, implying that liabilities cannotexceed assets in market value.

6. For a successful company that is rapidly expanding, for example,capital outlays will be large, possibly leading to negative cashflow from assets. In general, what matters is whether the money isspent wisely, not whether cash flow from assets is positive ornegative.

7. It’s probably not a good sign for an established company, but itwould be fairly ordinary for a start-up, so it depends.

8. For example, if a company were to become more efficient ininventory management, the amount of inventory needed would decline.The same might be true if it becomes better at collecting itsreceivables. In general, anything that leads to a decline in endingNWC relative to beginning would have this effect. Negative netcapital spending would mean more long-lived assets were liquidatedthan purchased.

CHAPTER 2 B-8

9. If a company raises more money from selling stock than it pays individends in a particular period, its cash flow to stockholderswill be negative. If a company borrows more than it pays ininterest, its cash flow to creditors will be negative.

10. The adjustments discussed were purely accounting changes; they hadno cash flow or market value consequences unless the new accountinginformation caused stockholders to revalue the derivatives.

11. Enterprise value is the theoretical takeover price. In the event ofa takeover, an acquirer would have to take on the company's debt,but would pocket its cash. Enterprise value differs significantlyfrom simple market capitalization in several ways, and it may be amore accurate representation of a firm's value. In a takeover, thevalue of a firm's debt would need to be paid by the buyer whentaking over a company. This enterprise value provides a much moreaccurate takeover valuation because it includes debt in its valuecalculation.

12. In general, it appears that investors prefer companies that have asteady earnings stream. If true, this encourages companies tomanage earnings. Under GAAP, there are numerous choices for the waya company reports its financial statements. Although not the reasonfor the choices under GAAP, one outcome is the ability of a companyto manage earnings, which is not an ethical decision. Even thoughearnings and cash flow are often related, earnings managementshould have little effect on cash flow (except for taximplications). If the market is “fooled” and prefers steadyearnings, shareholder wealth can be increased, at leasttemporarily. However, given the questionable ethics of thispractice, the company (and shareholders) will lose value if thepractice is discovered.

Solutions to Questions and Problems

NOTE: All end of chapter problems were solved using a spreadsheet. Many problems requiremultiple steps. Due to space and readability constraints, when these intermediate steps areincluded in this solutions manual, rounding may appear to have occurred. However, the finalanswer for each problem is found without rounding during any step in the problem.

Basic

1. To find owner’s equity, we must construct a balance sheet asfollows:

B-9 SOLUTIONS

Balance SheetCA $5,100 CL $4,300NFA 23,800 LTD 7,400

OE ??TA $28,900 TL & OE$28,900

We know that total liabilities and owner’s equity (TL & OE) mustequal total assets of $28,900. We also know that TL & OE is equalto current liabilities plus long-term debt plus owner’s equity,so owner’s equity is:

OE = $28,900 – 7,400 – 4,300 = $17,200

NWC = CA – CL = $5,100 – 4,300 = $800

CHAPTER 2 B-10

2. The income statement for the company is:

Income StatementSales $586,000Costs 247,000Depreciation 43,000EBIT $296,000Interest 32,000EBT $264,000Taxes(35%) 92,400Net income $171,600

3. One equation for net income is:

Net income = Dividends + Addition to retained earnings

Rearranging, we get:

Addition to retained earnings = Net income – Dividends = $171,600– 73,000 = $98,600

4. EPS = Net income / Shares = $171,600 / 85,000 =$2.02 per share

DPS = Dividends / Shares = $73,000 / 85,000 = $0.86per share

5. To find the book value of current assets, we use: NWC = CA – CL.Rearranging to solve for current assets, we get:

CA = NWC + CL = $380,000 + 1,400,000 = $1,480,000

The market value of current assets and fixed assets is given, so:

Book value CA = $1,480,000 Market value CA = $1,600,000Book value NFA = $3,700,000 Market value NFA = $4,900,000Book value assets = $5,180,000 Market value assets= $6,500,000

6. Taxes = 0.15($50K) + 0.25($25K) + 0.34($25K) + 0.39($236K – 100K)= $75,290

B-11 SOLUTIONS

7. The average tax rate is the total tax paid divided by net income,so:

Average tax rate = $75,290 / $236,000 = 31.90%

The marginal tax rate is the tax rate on the next $1 of earnings,so the marginal tax rate = 39%.

CHAPTER 2 B-12

8. To calculate OCF, we first need the income statement:

Income StatementSales $27,500Costs 13,280Depreciation 2,300EBIT $11,920Interest 1,105Taxable income $10,815Taxes (35%) 3,785Net income $ 7,030

OCF = EBIT + Depreciation – Taxes = $11,920 + 2,300 – 3,785 = $10,435

9. Net capital spending = NFAend – NFAbeg + Depreciation Net capital spending = $4,200,000 – 3,400,000 + 385,000 Net capital spending = $1,185,000

10. Change in NWC = NWCend – NWCbeg Change in NWC = (CAend – CLend) – (CAbeg – CLbeg)Change in NWC = ($2,250 – 1,710) – ($2,100 – 1,380) Change in NWC = $540 – 720 = –$180

11. Cash flow to creditors = Interest paid – Net new borrowing Cash flow to creditors = Interest paid – (LTDend – LTDbeg) Cash flow to creditors = $170,000 – ($2,900,000 – 2,600,000) Cash flow to creditors = –$130,000

12. Cash flow to stockholders = Dividends paid – Net new equity Cash flow to stockholders = Dividends paid – [(Commonend + APISend)

– (Commonbeg + APISbeg)] Cash flow to stockholders = $490,000 – [($815,000 + 5,500,000) –

($740,000 + 5,200,000)]Cash flow to stockholders = $115,000

Note, APIS is the additional paid-in surplus.

13. Cash flow from assets = Cash flow to creditors + Cash flowto stockholders = –$130,000 + 115,000 = –$15,000

B-13 SOLUTIONS

Cash flow from assets = –$15,000 = OCF – Change in NWC – Netcapital spending

= –$15,000 = OCF – (–$85,000) – 940,000

Operating cash flow = –$15,000 – 85,000 + 940,000 Operating cash flow = $840,000

CHAPTER 2 B-14

Intermediate

14. To find the OCF, we first calculate net income.

Income StatementSales $196,000 Costs 104,000 Other expenses 6,800Depreciation 9,100 EBIT $76,100Interest 14,800 Taxable income $61,300 Taxes 21,455Net income $39,845

Dividends $10,400 Additions to RE$29,445

a. OCF = EBIT + Depreciation – Taxes = $76,100 + 9,100 – 21,455 =$63,745

b. CFC = Interest – Net new LTD = $14,800 – (–7,300) = $22,100 Note that the net new long-term debt is negative because the

company repaid part of its long- term debt.

c. CFS = Dividends – Net new equity = $10,400 – 5,700 = $4,700

d. We know that CFA = CFC + CFS, so:

CFA = $22,100 + 4,700 = $26,800

CFA is also equal to OCF – Net capital spending – Change inNWC. We already know OCF. Net capital spending is equal to:

Net capital spending = Increase in NFA + Depreciation =

$27,000 + 9,100 = $36,100

Now we can use:

CFA = OCF – Net capital spending – Change in NWC $26,800 = $63,745 – 36,100 – Change in NWC

B-15 SOLUTIONS

Solving for the change in NWC gives $845, meaning the companyincreased its NWC by $845.

15. The solution to this question works the income statementbackwards. Starting at the bottom:

Net income = Dividends + Addition to ret. earnings = $1,500 +5,100 = $6,600

CHAPTER 2 B-16

Now, looking at the income statement: EBT – EBT × Tax rate = Net income

Recognize that EBT × Tax rate is simply the calculation fortaxes. Solving this for EBT yields:

EBT = NI / (1– tax rate) = $6,600 / (1 – 0.35) = $10,154

Now you can calculate:

EBIT = EBT + Interest = $10,154 + 4,500 = $14,654

The last step is to use:

EBIT = Sales – Costs – Depreciation $14,654 = $41,000 – 19,500 – Depreciation

Solving for depreciation, we find that depreciation = $6,846

16. The balance sheet for the company looks like this:

Balance SheetCash $195,000 Accounts payable $405,000Accounts receivable 137,000 Notes payable 160,000Inventory 264,000 Current liabilities $565,000Current assets $596,000 Long-term debt 1,195,300

Total liabilities $1,760,300Tangible net fixed assets2,800,000Intangible net fixed assets 780,000 Common stock??

Accumulated ret. earnings1,934,000

Total assets $4,176,000 Total liab. & owners’ equity$4,176,000

Total liabilities and owners’ equity is:

TL & OE = CL + LTD + Common stock + Retained earnings

Solving for this equation for equity gives us:

Common stock = $4,176,000 – 1,934,000 – 1,760,300 = $481,700

B-17 SOLUTIONS

17. The market value of shareholders’ equity cannot be negative. Anegative market value in this case would imply that the companywould pay you to own the stock. The market value of shareholders’equity can be stated as: Shareholders’ equity = Max [(TA – TL),0]. So, if TA is $8,400, equity is equal to $1,100, and if TA is$6,700, equity is equal to $0. We should note here that the bookvalue of shareholders’ equity can be negative.

CHAPTER 2 B-18

18. a. Taxes Growth = 0.15($50,000) + 0.25($25,000) + 0.34($13,000) =$18,170

Taxes Income = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) +0.39($235,000)

+ 0.34($8,465,000)= $2,992,000

b. Each firm has a marginal tax rate of 34% on the next $10,000 of taxable income, despite their

different average tax rates, so both firms will pay an additional $3,400 in taxes.

19. Income StatementSales $730,000COGS 580,000 A&S expenses 105,000Depreciation 135,000EBIT –$90,000Interest 75,000Taxable income–$165,000Taxes (35%) 0

a. Net income –$165,000

b. OCF = EBIT + Depreciation – Taxes = –$90,000 + 135,000 – 0 =$45,000

c. Net income was negative because of the tax deductibility ofdepreciation and interest expense. However, the actual cashflow from operations was positive because depreciation is anon-cash expense and interest is a financing expense, not anoperating expense.

20. A firm can still pay out dividends if net income is negative; itjust has to be sure there is sufficient cash flow to make thedividend payments.

Change in NWC = Net capital spending = Net new equity = 0.(Given)

Cash flow from assets = OCF – Change in NWC – Net capitalspending

Cash flow from assets = $45,000 – 0 – 0 = $45,000Cash flow to stockholders = Dividends – Net new equity = $25,000

– 0 = $25,000

B-19 SOLUTIONS

Cash flow to creditors = Cash flow from assets – Cash flow tostockholders

Cash flow to creditors = $45,000 – 25,000 = $20,000Cash flow to creditors = Interest – Net new LTDNet new LTD = Interest – Cash flow to creditors = $75,000 –

20,000 = $55,000

21. a.Income Statement

Sales $22,800Cost of goodssold

16,050

Depreciation 4,050EBIT $ 2,700Interest 1,830Taxable income $ 870Taxes (34%) 29

6Net income $ 574

b. OCF = EBIT + Depreciation – Taxes= $2,700 + 4,050 – 296 = $6,454

CHAPTER 2 B-20

c. Change in NWC = NWCend – NWCbeg

= (CAend – CLend) – (CAbeg – CLbeg) = ($5,930 – 3,150) – ($4,800 – 2,700) = $2,780 – 2,100 = $680

Net capital spending = NFAend – NFAbeg + Depreciation= $16,800 – 13,650 + 4,050 = $7,200

CFA = OCF – Change in NWC – Net capital spending= $6,454 – 680 – 7,200 = –$1,426

The cash flow from assets can be positive or negative, sinceit represents whether the firm raised funds or distributedfunds on a net basis. In this problem, even though net incomeand OCF are positive, the firm invested heavily in both fixedassets and net working capital; it had to raise a net $1,426in funds from its stockholders and creditors to make theseinvestments.

d. Cash flow to creditors = Interest – Net new LTD = $1,830 – 0= $1,830

Cash flow to stockholders = Cash flow from assets – Cashflow to creditors

= –$1,426 – 1,830 = –$3,256

We can also calculate the cash flow to stockholders as:

Cash flow to stockholders = Dividends – Net new equity

Solving for net new equity, we get:

Net new equity = $1,300 – (–3,256) = $4,556

The firm had positive earnings in an accounting sense (NI > 0)and had positive cash flow from operations. The firm invested$680 in new net working capital and $7,200 in new fixedassets. The firm had to raise $1,426 from its stakeholders tosupport this new investment. It accomplished this by raising$4,556 in the form of new equity. After paying out $1,300 ofthis in the form of dividends to shareholders and $1,830 inthe form of interest to creditors, $1,426 was left to meet thefirm’s cash flow needs for investment.

B-21 SOLUTIONS

22. a. Total assets 2008 = $653 + 2,691 = $3,344 Total liabilities 2008 = $261 + 1,422 = $1,683Owners’ equity 2008 = $3,344 – 1,683 = $1,661

Total assets 2009 = $707 + 3,240 = $3,947 Total liabilities 2009 = $293 + 1,512 = $1,805Owners’ equity 2009 = $3,947 – 1,805 = $2,142

b. NWC 2008 = CA08 – CL08 = $653 – 261 = $392NWC 2009 = CA09 – CL09 = $707 – 293 = $414Change in NWC = NWC09 – NWC08 = $414 – 392 = $22

CHAPTER 2 B-22

c. We can calculate net capital spending as:

Net capital spending = Net fixed assets 2009 – Net fixedassets 2008 + Depreciation

Net capital spending = $3,240 – 2,691 + 738 = $1,287

So, the company had a net capital spending cash flow of$1,287. We also know that net capital spending is:

Net capital spending = Fixed assets bought – Fixed assetssold

$1,287 = $1,350 – Fixed assets soldFixed assets sold = $1,350 – 1,287 = $63

To calculate the cash flow from assets, we must firstcalculate the operating cash flow. The income statement is:

Income Statement

Sales $

8,280.00

Costs 3,86

1.00

Depreciation expense

738 .00

EBIT$3,681.0

0

Interest expense 211

.00

EBT $3,470.

00 Taxes (35%) 1,215.50

Net income $2,256.

50

So, the operating cash flow is:

OCF = EBIT + Depreciation – Taxes = $3,681 + 738 – 1,214.50 =$3,204.50

And the cash flow from assets is:

B-23 SOLUTIONS

Cash flow from assets = OCF – Change in NWC – Net capitalspending.

= $3,204.50 – 22 – 1,287 = $1,895.50

d. Net new borrowing = LTD09 – LTD08 = $1,512 – 1,422 = $90Cash flow to creditors = Interest – Net new LTD = $211 – 90 =

$121Net new borrowing = $90 = Debt issued – Debt retired Debt retired = $270 – 90 = $180

Challenge

23. Net capital spending = NFAend – NFAbeg + Depreciation= (NFAend – NFAbeg) + (Depreciation + ADbeg) – ADbeg

= (NFAend – NFAbeg)+ ADend – ADbeg

= (NFAend + ADend) – (NFAbeg + ADbeg) = FAend – FAbeg

CHAPTER 2 B-24

24. a. The tax bubble causes average tax rates to catch up tomarginal tax rates, thus eliminating the tax advantage of lowmarginal rates for high income corporations.

b. Taxes = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) +0.39($235,000) = $113,900

Average tax rate = $113,900 / $335,000 = 34%

The marginal tax rate on the next dollar of income is 34percent.

For corporate taxable income levels of $335,000 to $10million, average tax rates are equal to marginal tax rates.

Taxes = 0.34($10,000,000) + 0.35($5,000,000) +0.38($3,333,333)= $6,416,667

Average tax rate = $6,416,667 / $18,333,334 = 35%

The marginal tax rate on the next dollar of income is 35percent. For corporate taxable income levels over $18,333,334,average tax rates are again equal to marginal tax rates.

c. Taxes = 0.34($200,000) = $68,000 $68,000 = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) +

X($100,000);X($100,000) = $68,000 – 22,250X = $45,750 / $100,000 X = 45.75%

25. Balance sheet as of Dec. 31, 2008

Cash $3,792 Accounts payable $3,984

Accounts receivable 5,021 Notes payable 732

Inventory 8,927 Current liabilities $4,716

Current assets $17,740

Long-term debt $12,700

B-25 SOLUTIONS

Net fixed assets $31,805

Owners' equity 32,129

Total assets $49,545 Total liab. &equity $49,545

Balance sheet as of Dec. 31, 2009

Cash $4,041 Accounts payable $4,025

Accounts receivable 5,892 Notes payable 717

Inventory 9,555 Current liabilities $4,742

Current assets $19,488

Long-term debt $15,435

Net fixed assets $33,921

Owners' equity 33,232

Total assets $53,409 Total liab. &equity $53,409

CHAPTER 2 B-26

2008 Income Statement 2009 Income StatementSales

$7,233.00Sales $8,085.0

0COGS 2,487.00 COGS 2,942.00Other expenses 591.00

Other expenses 515.00

Depreciation 1,038.00

Depreciation1,085.00

EBIT$3,117.00

EBIT $3,543.00

Interest 485.00 Interest 579.00EBT

$2,632.00EBT $2,964.0

0Taxes (34%) 894.88 Taxes (34%) 1,007.76Net income

$1,737.12Net income $1,956.2

4

Dividends$882.00

Dividends $1,011.00

Additions to RE 855.12

Additions to RE 945.24

26. OCF = EBIT + Depreciation – Taxes = $3,543 + 1,085 – 1,007.76 = $3,620.24

Change in NWC = NWCend – NWCbeg = (CA – CL) end – (CA – CL) beg

= ($19,488 – 4,742) – ($17,740 – 4,716) = $1,722

Net capital spending = NFAend – NFAbeg + Depreciation = $33,921 – 31,805 + 1,085 = $3,201

Cash flow from assets = OCF – Change in NWC – Net capital spending

= $3,620.24 – 1,722 – 3,201 = –$1,302.76

Cash flow to creditors = Interest – Net new LTD Net new LTD = LTDend – LTDbeg

Cash flow to creditors = $579 – ($15,435 – 12,700) = –$2,156

B-27 SOLUTIONS

Net new equity = Common stockend – Common stockbeg

Common stock + Retained earnings = Total owners’ equityNet new equity = (OE – RE) end – (OE – RE) beg

= OEend – OEbeg + REbeg – REend

REend = REbeg + Additions to RE08 Net new equity = OEend – OEbeg + REbeg – (REbeg +

Additions to RE08)= OEend – OEbeg – Additions to RE

Net new equity= $33,232 – 32,129 – 945.24 = $157.76

CFS = Dividends – Net new equity CFS = $1,011 – 157.76 = $853.24

As a check, cash flow from assets is –$1,302.76.

CFA = Cash flow from creditors + Cash flow to stockholders CFA = –$2,156 + 853.24 = –$1,302.76

CHAPTER 3WORKING WITH FINANCIAL STATEMENTS


1. a. If inventory is purchased with cash, then there is no change inthe current ratio. If inventory is purchased on credit, thenthere is a decrease in the current ratio if it was initiallygreater than 1.0.

b. Reducing accounts payable with cash increases the current ratioif it was initially greater than 1.0.

c. Reducing short-term debt with cash increases the current ratioif it was initially greater than 1.0.

d. As long-term debt approaches maturity, the principal repaymentand the remaining interest expense become current liabilities.Thus, if debt is paid off with cash, the current ratio increasesif it was initially greater than 1.0. If the debt has not yetbecome a current liability, then paying it off will reduce thecurrent ratio since current liabilities are not affected.

e. Reduction of accounts receivables and an increase in cash leavesthe current ratio unchanged.

f. Inventory sold at cost reduces inventory and raises cash, so thecurrent ratio is unchanged.

g. Inventory sold for a profit raises cash in excess of theinventory recorded at cost, so the current ratio increases.

2. The firm has increased inventory relative to other current assets;therefore, assuming current liability levels remain unchanged,liquidity has potentially decreased.

3. A current ratio of 0.50 means that the firm has twice as much incurrent liabilities as it does in current assets; the firmpotentially has poor liquidity. If pressed by its short-termcreditors and suppliers for immediate payment, the firm might havea difficult time meeting its obligations. A current ratio of 1.50

B-29 SOLUTIONS

means the firm has 50% more current assets than it does currentliabilities. This probably represents an improvement in liquidity;short-term obligations can generally be met com-pletely with asafety factor built in. A current ratio of 15.0, however, might beexcessive. Any excess funds sitting in current assets generallyearn little or no return. These excess funds might be put to betteruse by investing in productive long-term assets or distributing thefunds to shareholders.

4. a. Quick ratio provides a measure of the short-term liquidity ofthe firm, after removing the effects of inventory, generally theleast liquid of the firm’s current assets.

b. Cash ratio represents the ability of the firm to completely payoff its current liabilities with its most liquid asset (cash).

c. Total asset turnover measures how much in sales is generated byeach dollar of firm assets.

d. Equity multiplier represents the degree of leverage for anequity investor of the firm; it measures the dollar worth offirm assets each equity dollar has a claim to.

e. Long-term debt ratio measures the percentage of total firmcapitalization funded by long-term debt.

CHAPTER 3 B-30

f. Times interest earned ratio provides a relative measure of howwell the firm’s operating earnings can cover current interestobligations.

g. Profit margin is the accounting measure of bottom-line profitper dollar of sales.

h. Return on assets is a measure of bottom-line profit per dollarof total assets.

i. Return on equity is a measure of bottom-line profit per dollarof equity.

j. Price-earnings ratio reflects how much value per share themarket places on a dollar of accounting earnings for a firm.

5. Common size financial statements express all balance sheet accountsas a percentage of total assets and all income statement accountsas a percentage of total sales. Using these percentage valuesrather than nominal dollar values facilitates comparisons betweenfirms of different size or business type. Common-base yearfinancial statements express each account as a ratio between theircurrent year nominal dollar value and some reference year nominaldollar value. Using these ratios allows the total growth trend inthe accounts to be measured.

6. Peer group analysis involves comparing the financial ratios andoperating performance of a particular firm to a set of peer groupfirms in the same industry or line of business. Comparing a firm toits peers allows the financial manager to evaluate whether someaspects of the firm’s operations, finances, or investmentactivities are out of line with the norm, thereby providing someguidance on appropriate actions to take to adjust these ratios ifappropriate. An aspirant group would be a set of firms whoseperformance the company in question would like to emulate. Thefinancial manager often uses the financial ratios of aspirantgroups as the target ratios for his or her firm; some managers areevaluated by how well they match the performance of an identifiedaspirant group.

7. Return on equity is probably the most important accounting ratiothat measures the bottom-line performance of the firm with respectto the equity shareholders. The Du Pont identity emphasizes therole of a firm’s profitability, asset utilization efficiency, andfinancial leverage in achieving an ROE figure. For example, a firmwith ROE of 20% would seem to be doing well, but this figure may bemisleading if it were marginally profitable (low profit margin) and

B-31 SOLUTIONS

highly levered (high equity multiplier). If the firm’s margins wereto erode slightly, the ROE would be heavily impacted.

8. The book-to-bill ratio is intended to measure whether demand isgrowing or falling. It is closely followed because it is abarometer for the entire high-tech industry where levels ofrevenues and earnings have been relatively volatile.

9. If a company is growing by opening new stores, then presumablytotal revenues would be rising. Comparing total sales at twodifferent points in time might be misleading. Same-store salescontrol for this by only looking at revenues of stores open withina specific period.

10. a. For an electric utility such as Con Ed, expressing costs on aper kilowatt hour basis would be a way to compare costs withother utilities of different sizes.b. For a retailer such as Sears, expressing sales on a per squarefoot basis would be useful in comparing revenue productionagainst other retailers.c. For an airline such as Southwest, expressing costs on a perpassenger mile basis allows for comparisons with other airlinesby examining how much it costs to fly one passenger one mile.

CHAPTER 3 B-32

d. For an on-line service provider such as AOL, using a per callbasis for costs would allow for comparisons with smallerservices. A per subscriber basis would also make sense.e. For a hospital such as Holy Cross, revenues and costs expressedon a per bed basis would be useful.f. For a college textbook publisher such as McGraw-Hill/Irwin, theleading publisher of finance textbooks for the college market, theobvious standardization would be per book sold.

11. Reporting the sale of Treasury securities as cash flow fromoperations is an accounting “trick”, and as such, should constitutea possible red flag about the companies accounting practices. Formost companies, the gain from a sale of securities should be placedin the financing section. Including the sale of securities in thecash flow from operations would be acceptable for a financialcompany, such as an investment or commercial bank.

12. Increasing the payables period increases the cash flow fromoperations. This could be beneficial for the company as it may be acheap form of financing, but it is basically a one time change. Thepayables period cannot be increased indefinitely as it willnegatively affect the company’s credit rating if the payablesperiod becomes too long.



Basic

1. Using the formula for NWC, we get:

NWC = CA – CLCA = CL + NWC = $3,720 + 1,370 = $5,090

So, the current ratio is:Current ratio = CA / CL = $5,090/$3,720 = 1.37 times

And the quick ratio is:

B-33 SOLUTIONS

Quick ratio = (CA – Inventory) / CL = ($5,090 – 1,950) / $3,720 =0.84 times

2. We need to find net income first. So:

Profit margin = Net income / SalesNet income = Sales(Profit margin)Net income = ($29,000,000)(0.08) = $2,320,000

ROA = Net income / TA = $2,320,000 / $17,500,000 = .1326 or 13.26%

CHAPTER 3 B-34

To find ROE, we need to find total equity. TL & OE = TD + TETE = TL & OE – TDTE = $17,500,000 – 6,300,000 = $11,200,000

ROE = Net income / TE = 2,320,000 / $11,200,000 = .2071 or 20.71%

3. Receivables turnover = Sales / Receivables Receivables turnover = $3,943,709 / $431,287 = 9.14 times

Days’ sales in receivables = 365 days / Receivables turnover =365 / 9.14 = 39.92 days

The average collection period for an outstanding accountsreceivable balance was 39.92 days.

4. Inventory turnover = COGS / Inventory Inventory turnover = $4,105,612 / $407,534 = 10.07 times

Days’ sales in inventory = 365 days / Inventory turnover = 365 /10.07 = 36.23 days

On average, a unit of inventory sat on the shelf 36.23 days beforeit was sold.

5. Total debt ratio = 0.63 = TD / TA

Substituting total debt plus total equity for total assets, we get:

0.63 = TD / (TD + TE)

Solving this equation yields:

0.63(TE) = 0.37(TD)

Debt/equity ratio = TD / TE = 0.63 / 0.37 = 1.70

Equity multiplier = 1 + D/E = 2.70

6. Net income = Addition to RE + Dividends = $430,000 + 175,000 = $605,000

B-35 SOLUTIONS

Earnings per share = NI / Shares = $605,000/ 210,000 = $2.88 per share

Dividends per share = Dividends / Shares =$175,000 / 210,000 = $0.83 per share

Book value per share = TE / Shares =$5,300,000 / 210,000 = $25.24 per share

Market-to-book ratio = Share price / BVPS = $63/ $25.24 = 2.50 times

P/E ratio = Share price / EPS = $63/ $2.88 = 21.87 times

Sales per share = Sales / Shares = $4,500,000 / 210,000 =$21.43

P/S ratio = Share price / Sales pershare = $63 / $21.43 = 2.94 times

CHAPTER 3 B-36

7. ROE = (PM)(TAT)(EM) ROE = (.055)(1.15)(2.80) = .1771 or 17.71%

8. This question gives all of the necessary ratios for the DuPontIdentity except the equity multiplier, so, using the DuPontIdentity:

ROE = (PM)(TAT)(EM)ROE = .1827 = (.068)(1.95)(EM)

EM = .1827 / (.068)(1.95) = 1.38

D/E = EM – 1 = 1.38 – 1 = 0.38

9. Decrease in inventory is a source of cashDecrease in accounts payable is a use of cashIncrease in notes payable is a source of cashIncrease in accounts receivable is a use of cashChanges in cash = sources – uses = $375 – 190 + 210 – 105 = $290Cash increased by $290

10. Payables turnover = COGS / Accounts payable Payables turnover = $28,384 / $6,105 = 4.65 times

Days’ sales in payables = 365 days / Payables turnover Days’ sales in payables = 365 / 4.65 = 78.51 days

The company left its bills to suppliers outstanding for 78.51 dayson average. A large value for this ratio could imply that either(1) the company is having liquidity problems, making it difficultto pay off its short-term obligations, or (2) that the company hassuccessfully negotiated lenient credit terms from its suppliers.

11. New investment in fixed assets is found by:

Net investment in FA = (NFAend – NFAbeg) + Depreciation Net investment in FA = $835 + 148 = $983

The company bought $983 in new fixed assets; this is a use of cash.

12. The equity multiplier is:

EM = 1 + D/E

B-37 SOLUTIONS

EM = 1 + 0.65 = 1.65

One formula to calculate return on equity is:

ROE = (ROA)(EM) ROE = .085(1.65) = .1403 or 14.03%

CHAPTER 3 B-38

ROE can also be calculated as:

ROE = NI / TE

So, net income is:

NI = ROE(TE)NI = (.1403)($540,000) = $75,735

13. through 15:

2008 #13 2009 #13 #14 #15Assets

Current assets

Cash $8,436 2.86%$10,15

7 3.13%1.204

01.096

1

Accounts receivable 21,530 7.29% 23,406 7.21%1.087

10.989

7

Inventory 38,760 13.12% 42,650 13.14%1.100

41.001

7

Total$68,72

6 23.26%$76,21

3 23.48%1.108

91.009

5Fixed assets Net plant and equipment

226,706 76.74%

248,306 76.52%

1.0953

0.9971

Total assets$295,4

32 100%$324,5

19 100%1.098

51.000

0 Liabilities and Owners’

EquityCurrent liabilities

Accounts payable$43,05

0 14.57%$46,82

1 14.43%1.087

60.990

1

Notes payable 18,384 6.22% 17,382 5.36%0.945

50.860

8

Total$61,43

4 20.79%$64,20

3 19.78%1.045

10.951

4

Long-term debt 25,000 8.46% 32,000 9.86%1.280

01.165

3Owners' equity Common stock and paid-in surplus

$40,000 13.54%

$40,000 12.33%

1.0000

0.9104

B-39 SOLUTIONS

Accumulated retained earnings

168,998 57.20%

188,316 58.03%

1.1143

1.0144

Total$208,9

98 70.74%$228,3

16 70.36%1.092

40.994

5Total liabilities and owners' equity

$295,432 100%

$324,519 100%

1.0985

1.0000

The common-size balance sheet answers are found by dividing eachcategory by total assets. For example, the cash percentage for 2008is:

$8,436 / $295,432 = .0286 or 2.86%

This means that cash is 2.86% of total assets.

CHAPTER 3 B-40

The common-base year answers for Question 14 are found by dividingeach category value for 2009 by the same category value for 2008.For example, the cash common-base year number is found by:

$10,157 / $8,436 = 1.2040

This means the cash balance in 2009 is 1.2040 times as large as thecash balance in 2008.

The common-size, common-base year answers for Question 15 are foundby dividing the common-size percentage for 2009 by the common-sizepercentage for 2008. For example, the cash calculation is found by:

3.13% / 2.86% = 1.0961

This tells us that cash, as a percentage of assets, increased by9.61%.

16. 2008 Sources/Uses 2008

AssetsCurrent assets

Cash $8,436 $1,721 U$10,15

7 Accounts receivable 21,530 1,876 U 23,406 Inventory 38,760 3,890 U 42,650

Total $68,726 $7,487 U$76,21

3Fixed assets Net plant and equipment

$226,706 $21,600 U

$248,306

Total assets$295,43

2 $29,087 U$324,5

19 Liabilities and Owners’

EquityCurrent liabilities

Accounts payable $43,050 3,771 S$46,82

1 Notes payable 18,384 –1,002 U 17,382

Total $61,434 2,769 S$64,20

3

B-41 SOLUTIONS

Long-term debt 25,000 $7,000 S 32,000Owners' equity Common stock and paid-in surplus $40,000 $0

$40,000

Accumulated retained earnings 168,998 19,318 S

188,316

Total$208,99

8 $19,318 S$228,3

16Total liabilities and owners' equity

$295,432 $29,087 S

$324,519

The firm used $29,087 in cash to acquire new assets. It raised thisamount of cash by increasing liabilities and owners’ equity by$29,087. In particular, the needed funds were raised by internalfinancing (on a net basis), out of the additions to retainedearnings, an increase in current liabilities, and by an issue oflong-term debt.

CHAPTER 3 B-42

17. a. Current ratio = Current assets / CurrentliabilitiesCurrent ratio 2008 = $68,726 / $61,434 = 1.12times

Current ratio 2009 = $76,213 / $64,203 = 1.19times

b. Quick ratio = (Current assets – Inventory) / CurrentliabilitiesQuick ratio 2008 = ($67,726 – 38,760) / $61,434 =0.49 timesQuick ratio 2009 = ($76,213 – 42,650) / $64,203 =0.52 times

c. Cash ratio = Cash / Current liabilitiesCash ratio 2008 = $8,436 / $61,434 = 0.14 times Cash ratio 2009 = $10,157 / $64,203 = 0.16 times

d. NWC ratio = NWC / Total assetsNWC ratio 2008 = ($68,726 – 61,434) / $295,432= 2.47%NWC ratio 2009 = ($76,213 – 64,203) / $324,519= 3.70%

e. Debt-equity ratio = Total debt / Total equityDebt-equity ratio 2008 = ($61,434 + 25,000) / $208,998= 0.41 times Debt-equity ratio 2009 = ($64,206 + 32,000) / $228,316= 0.42 times

Equity multiplier = 1 + D/E Equity multiplier 2008 = 1 + 0.41 = 1.41Equity multiplier 2009 = 1 + 0.42 = 1.42

f. Total debt ratio = (Total assets – Total equity) / Totalassets Total debt ratio 2008 = ($295,432 – 208,998) /$295,432 = 0.29Total debt ratio 2009 = ($324,519 – 228,316) /$324,519 = 0.30

Long-term debt ratio = Long-term debt / (Long-termdebt + Total equity)

B-43 SOLUTIONS

Long-term debt ratio 2008 = $25,000 / ($25,000 + 208,998)= 0.11Long-term debt ratio 2009 = $32,000 / ($32,000 + 228,316)= 0.12

Intermediate

18. This is a multi-step problem involving several ratios. The ratiosgiven are all part of the DuPont Identity. The only DuPont Identityratio not given is the profit margin. If we know the profit margin,we can find the net income since sales are given. So, we beginwith the DuPont Identity:

ROE = 0.15 = (PM)(TAT)(EM) = (PM)(S / TA)(1 + D/E)

Solving the DuPont Identity for profit margin, we get:

PM = [(ROE)(TA)] / [(1 + D/E)(S)] PM = [(0.15)($3,105)] / [(1 + 1.4)( $5,726)] = .0339

Now that we have the profit margin, we can use this number and thegiven sales figure to solve for net income:

PM = .0339 = NI / SNI = .0339($5,726) = $194.06

CHAPTER 3 B-44

19. This is a multi-step problem involving several ratios. It is ofteneasier to look backward to determine where to start. We needreceivables turnover to find days’ sales in receivables. Tocalculate receivables turnover, we need credit sales, and to findcredit sales, we need total sales. Since we are given the profitmargin and net income, we can use these to calculate total salesas:

PM = 0.087 = NI / Sales = $218,000 / Sales; Sales = $2,505,747

Credit sales are 70 percent of total sales, so:

Credit sales = $2,515,747(0.70) = $1,754,023

Now we can find receivables turnover by:

Receivables turnover = Credit sales / Accounts receivable =$1,754,023 / $132,850 = 13.20 times

Days’ sales in receivables = 365 days / Receivables turnover =365 / 13.20 = 27.65 days

20. The solution to this problem requires a number of steps. First,remember that CA + NFA = TA. So, if we find the CA and the TA, wecan solve for NFA. Using the numbers given for the current ratioand the current liabilities, we solve for CA:

CR = CA / CLCA = CR(CL) = 1.25($875) = $1,093.75

To find the total assets, we must first find the total debt andequity from the information given. So, we find the sales using theprofit margin:

PM = NI / SalesNI = PM(Sales) = .095($5,870) = $549.10

We now use the net income figure as an input into ROE to find thetotal equity:

ROE = NI / TETE = NI / ROE = $549.10 / .185 = $2,968.11

B-45 SOLUTIONS

Next, we need to find the long-term debt. The long-term debt ratiois:

Long-term debt ratio = 0.45 = LTD / (LTD + TE)

Inverting both sides gives:

1 / 0.45 = (LTD + TE) / LTD = 1 + (TE / LTD)

Substituting the total equity into the equation and solving for long-term debt gives the following:

2.222 = 1 + ($2,968.11 / LTD) LTD = $2,968.11 / 1.222 = $2,428.45

CHAPTER 3 B-46

Now, we can find the total debt of the company:

TD = CL + LTD = $875 + 2,428.45 = $3,303.45

And, with the total debt, we can find the TD&E, which is equal toTA:

TA = TD + TE = $3,303.45 + 2,968.11 = $6,271.56

And finally, we are ready to solve the balance sheet identity as:

NFA = TA – CA = $6,271.56 – 1,093.75 = $5,177.81

21. Child: Profit margin = NI / S = $3.00 / $50 = .06 or 6%

Store: Profit margin = NI / S = $22,500,000 /$750,000,000 = .03 or 3%

The advertisement is referring to the store’s profit margin, but amore appropriate earnings measure for the firm’s owners is thereturn on equity.

ROE = NI / TE = NI / (TA – TD) ROE = $22,500,000 / ($420,000,000 – 280,000,000) = .1607 or 16.07%

22. The solution requires substituting two ratios into a third ratio.Rearranging D/TA:

Firm A Firm BD / TA = .35 D / TA = .30(TA – E) / TA = .35 (TA – E) / TA = .30(TA / TA) – (E / TA) = .35 (TA / TA) – (E / TA)

= .301 – (E / TA) = .35 1 – (E / TA) = .30E / TA = .65 E / TA = .30E = .65(TA) E = .70 (TA)

Rearranging ROA, we find:

NI / TA = .12 NI / TA = .11NI = .12(TA) NI = .11(TA)

B-47 SOLUTIONS

Since ROE = NI / E, we can substitute the above equations into theROE formula, which yields:

ROE = .12(TA) / .65(TA) = .12 / .65 = 18.46% ROE = .11(TA) / .70(TA) = .11 / .70 = 15.71%

23. This problem requires you to work backward through the incomestatement. First, recognize that Net income = (1 – t)EBT.Plugging in the numbers given and solving for EBT, we get:

EBT = $13,168 / (1 – 0.34) = $19,951.52

Now, we can add interest to EBT to get EBIT as follows:

EBIT = EBT + Interest paid = $19,951.52 + 3,605 = $23,556.52

CHAPTER 3 B-48

To get EBITD (earnings before interest, taxes, and depreciation),the numerator in the cash coverage

ratio, add depreciation to EBIT:

EBITD = EBIT + Depreciation = $23,556.52 + 2,382 = $25,938.52

Now, simply plug the numbers into the cash coverage ratio andcalculate:

Cash coverage ratio = EBITD / Interest = $25,938.52 / $3,605 = 7.20times

24. The only ratio given which includes cost of goods sold is theinventory turnover ratio, so it is the last ratio used. Sincecurrent liabilities is given, we start with the current ratio:

Current ratio = 1.40 = CA / CL = CA / $365,000CA = $511,000

Using the quick ratio, we solve for inventory:

Quick ratio = 0.85 = (CA – Inventory) / CL = ($511,000 – Inventory)/ $365,000

Inventory = CA – (Quick ratio × CL) Inventory = $511,000 – (0.85 × $365,000)Inventory = $200,750

Inventory turnover = 5.82 = COGS / Inventory = COGS / $200,750 COGS = $1,164,350

25. PM = NI / S = –£13,482,000 / £138,793 = –0.0971 or –9.71%

As long as both net income and sales are measured in the samecurrency, there is no problem; in fact, except for some marketvalue ratios like EPS and BVPS, none of the financial ratiosdiscussed in the text are measured in terms of currency. This isone reason why financial ratio analysis is widely used ininternational finance to compare the business operations of firmsand/or divisions across national economic borders. The net incomein dollars is:

NI = PM × SalesNI = –0.0971($274,213,000) = –$26,636,355

B-49 SOLUTIONS

26. Short-term solvency ratios:Current ratio = Current assets / Current liabilitiesCurrent ratio 2008 = $56,260 / $38,963 = 1.44 timesCurrent ratio 2009 = $60,550 / $43,235 = 1.40 times

Quick ratio = (Current assets – Inventory) / Currentliabilities

Quick ratio 2008 = ($56,260 – 23,084) / $38,963 = 0.85 timesQuick ratio 2009 = ($60,550 – 24,650) / $43,235 = 0.83 times

Cash ratio = Cash / Current liabilitiesCash ratio 2008 = $21,860 / $38,963 = 0.56 timesCash ratio 2009 = $22,050 / $43,235 = 0.51 times

CHAPTER 3 B-50

Asset utilization ratios:Total asset turnover = Sales / Total assetsTotal asset turnover = $305,830 / $321,075 = 0.95 times

Inventory turnover = Cost of goods sold / InventoryInventory turnover = $210,935 / $24,650 = 8.56 times

Receivables turnover = Sales / Accounts receivableReceivables turnover = $305,830 / $13,850 = 22.08 times

Long-term solvency ratios:Total debt ratio = (Total assets – Total equity) / Total

assetsTotal debt ratio 2008 = ($290,328 – 176,365) / $290,328 = 0.39Total debt ratio 2009 = ($321,075 – 192,840) / $321,075 = 0.40

Debt-equity ratio = Total debt / Total equityDebt-equity ratio 2008 = ($38,963 + 75,000) / $176,365 = 0.65Debt-equity ratio 2009 = ($43,235 + 85,000) / $192,840 = 0.66

Equity multiplier = 1 + D/EEquity multiplier 2008 = 1 + 0.65 = 1.65Equity multiplier 2009 = 1 + 0.66 = 1.66

Times interest earned = EBIT / InterestTimes interest earned = $68,045 / $11,930 = 5.70 times

Cash coverage ratio = (EBIT + Depreciation) / InterestCash coverage ratio = ($68,045 + 26,850) / $11,930 = 7.95 times

Profitability ratios:Profit margin = Net income / SalesProfit margin = $36,475 / $305,830 = 0.1193 or 11.93%

Return on assets = Net income / Total assetsReturn on assets = $36,475 / $321,075 = 0.1136 or 11.36%

Return on equity = Net income / Total equityReturn on equity = $36,475 / $192,840 = 0.1891 or 18.91%

27. The DuPont identity is:

ROE = (PM)(TAT)(EM)

B-51 SOLUTIONS

ROE = (0.1193)(0.95)(1.66) = 0.1891 or 18.91%

CHAPTER 3 B-52

28. SMOLIRA GOLF CORP. Statement of Cash Flows

For 2009 Cash, beginning of the year $ 21,860 Operating activities Net income $ 36,475 Plus: Depreciation $ 26,850

Increase in accounts payable

3,530

Increase in other currentliabilities

1,742

Less:

Increase in accounts receivable $ (2,534)

Increase in inventory (1,566) Net cash from operating activities $ 64,497 Investment activities Fixed asset acquisition $(53,307) Net cash from investment activities $(53,307) Financing activities

Increase in notes payable $

(1,000) Dividends paid (20,000)

Increase in long-term debt

10,000

Net cash from financing activities $(11,000)

Net increase in cash $

190 Cash, end of year $ 22,050

29. Earnings per share = Net income / SharesEarnings per share = $36,475 / 25,000 = $1.46 per share

P/E ratio = Shares price / Earnings per shareP/E ratio = $43 / $1.46 = 29.47 times

B-53 SOLUTIONS

Dividends per share = Dividends / SharesDividends per share = $20,000 / 25,000 = $0.80 per share

Book value per share = Total equity / SharesBook value per share = $192,840 / 25,000 shares = $7.71 per

share

CHAPTER 3 B-54

Market-to-book ratio = Share price / Book value per shareMarket-to-book ratio = $43 / $7.71 = 5.57 times

PEG ratio = P/E ratio / Growth ratePEG ratio = 29.47 / 9 = 3.27 times

30. First, we will find the market value of the company’s equity, whichis:

Market value of equity = Shares × Share priceMarket value of equity = 25,000($43) = $1,075,000

The total book value of the company’s debt is:

Total debt = Current liabilities + Long-term debtTotal debt = $43,235 + 85,000 = $128,235

Now we can calculate Tobin’s Q, which is:

Tobin’s Q = (Market value of equity + Book value of debt) / Bookvalue of assets

Tobin’s Q = ($1,075,000 + 128,235) / $321,075Tobin’s Q = 3.75

Using the book value of debt implicitly assumes that the book valueof debt is equal to the market value of debt. This will bediscussed in more detail in later chapters, but this assumption isgenerally true. Using the book value of assets assumes that theassets can be replaced at the current value on the balance sheet.There are several reasons this assumption could be flawed. First,inflation during the life of the assets can cause the book value ofthe assets to understate the market value of the assets. Sinceassets are recorded at cost when purchased, inflation means that itis more expensive to replace the assets. Second, improvements intechnology could mean that the assets could be replaced with moreproductive, and possibly cheaper, assets. If this is true, the bookvalue can overstate the market value of the assets. Finally, thebook value of assets may not accurately represent the market valueof the assets because of depreciation. Depreciation is doneaccording to some schedule, generally straight-line or MACRS. Thus,the book value and market value can often diverge.

CHAPTER 4LONG-TERM FINANCIAL PLANNING ANDGROWTHAnswers to Concepts Review and Critical Thinking Questions

1. The reason is that, ultimately, sales are the driving force behind abusiness. A firm’s assets, employees, and, in fact, just about everyaspect of its operations and financing exist to directly or indirectly support sales. Put differently, a firm’s future need for things like capital assets, employees, inventory, and financing are determined by its future sales level.

2. Two assumptions of the sustainable growth formula are that the company does not want to sell new equity, and that financial policy is fixed. If the company raises outside equity, or increases its debt-equity ratio it can grow at a higher rate than the sustainable growth rate. Of course the company could also grow faster than its profit margin increases, if it changes its dividend policy by increasing the retention ratio, or its total asset turnover increases.

3. The internal growth rate is greater than 15%, because at a 15% growth rate the negative EFN indicates that there is excess internalfinancing. If the internal growth rate is greater than 15%, then thesustainable growth rate is certainly greater than 15%, because thereis additional debt financing used in that case (assuming the firm isnot 100% equity-financed). As the retention ratio is increased, the firm has more internal sources of funding, so the EFN will decline. Conversely, as the retention ratio is decreased, the EFN will rise. If the firm pays out all its earnings in the form of dividends, thenthe firm has no internal sources of funding (ignoring the effects ofaccounts payable); the internal growth rate is zero in this case andthe EFN will rise to the change in total assets.

4. The sustainable growth rate is greater than 20%, because at a 20% growth rate the negative EFN indicates that there is excess

CHAPTER 4 B-56

financing still available. If the firm is 100% equity financed, thenthe sustainable and internal growth rates are equal and the internalgrowth rate would be greater than 20%. However, when the firm has some debt, the internal growth rate is always less than the sustainable growth rate, so it is ambiguous whether the internal growth rate would be greater than or less than 20%. If the retentionratio is increased, the firm will have more internal funding sourcesavailable, and it will have to take on more debt to keep the debt/equity ratio constant, so the EFN will decline. Conversely, if the retention ratio is decreased, the EFN will rise. If the retention rate is zero, both the internal and sustainable growth rates are zero, and the EFN will rise to the change in total assets.

5. Presumably not, but, of course, if the product had been much less popular, then a similar fate would have awaited due to lack of sales.

6. Since customers did not pay until shipment, receivables rose. The firm’s NWC, but not its cash, increased. At the same time, costs were rising faster than cash revenues, so operating cash flow declined. The firm’s capital spending was also rising. Thus, all three components of cash flow from assets were negatively impacted.

B-57 SOLUTIONS

7. Apparently not! In hindsight, the firm may have underestimated costsand also underestimated the extra demand from the lower price.

8. Financing possibly could have been arranged if the company had takenquick enough action. Sometimes it becomes apparent that help is needed only when it is too late, again emphasizing the need for planning.

9. All three were important, but the lack of cash or, more generally, financial resources ultimately spelled doom. An inadequate cash resource is usually cited as the most common cause of small businessfailure.

10. Demanding cash up front, increasing prices, subcontracting production, and improving financial resources via new owners or new sources of credit are some of the options. When orders exceed capacity, price increases may be especially beneficial.


NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.

Basic

1. It is important to remember that equity will not increase by the same percentage as the other assets. If every other item on the income statement and balance sheet increases by 15 percent, the proforma income statement and balance sheet will look like this:

Pro forma income statement Pro forma balance sheetSales $26,450 Assets $18,170 Debt $5,980Costs 19,205 Equity 12,190 Net income $ 7,245 Total $ 18,170 Total $ 18,170

In order for the balance sheet to balance, equity must be:

Equity = Total liabilities and equity – Debt

CHAPTER 4 B-58

Equity = $18,170 – 5,980Equity = $12,190

Equity increased by:

Equity increase = $12,190 – 10,600Equity increase = $1,590

B-59 SOLUTIONS

Net income is $7,245 but equity only increased by $1,590; therefore, a dividend of:

Dividend = $7,245 – 1,590Dividend = $5,655

must have been paid. Dividends paid is the plug variable.

2. Here we are given the dividend amount, so dividends paid is not a plug variable. If the company pays out one-half of its net income as dividends, the pro forma income statement and balance sheet willlook like this:

Pro forma income statement

Pro forma balance sheet

Sales $26,450.00

Assets

$18,170.00

Debt $5,980.00

Costs 19,205.00 Equity 14,222.50

Net income

$7,245.00

Total $18,170.00

Total $19,422.50

Dividends $ 3,622.50Add. to RE$3,622.50

Note that the balance sheet does not balance. This is due to EFN. The EFN for this company is:

EFN = Total assets – Total liabilities and equityEFN = $18,170 – 19,422.50 EFN = –$1,252.50

3. An increase of sales to $7,424 is an increase of:

Sales increase = ($7,424 – 6,300) / $6,300 Sales increase = .18 or 18%

Assuming costs and assets increase proportionally, the pro forma financial statements will look like this:

Pro forma income statement Pro forma balance sheet

CHAPTER 4 B-60

Sales $ 7,434 Assets $21,594 Debt $12,400Costs 4,590 Equity 8,744 Net income $ 2,844 Total $ 21,594 Total $ 21,144

If no dividends are paid, the equity account will increase by the net income, so:

Equity = $5,900 + 2,844 Equity = $8,744

So the EFN is:

EFN = Total assets – Total liabilities and equityEFN = $21,594 – 21,144 = $450

B-61 SOLUTIONS

4. An increase of sales to $21,840 is an increase of:

Sales increase = ($21,840 – 19,500) / $19,500 Sales increase = .12 or 12%

Assuming costs and assets increase proportionally, the pro forma financial statements will look like this:

Pro forma income statement Pro forma balance sheetSales $ 21,840 Assets $109,760 Debt $52,500Costs 16,800 Equity 79,208EBIT 5,040 Total $109,760 Total $99,456Taxes (40%) 2,016 Net income $ 3,024

The payout ratio is constant, so the dividends paid this year is the payout ratio from last year times net income, or:

Dividends = ($1,400 / $2,700)($3,024) Dividends = $1,568

The addition to retained earnings is:

Addition to retained earnings = $3,024 – 1,568 Addition to retained earnings = $1,456

And the new equity balance is:

Equity = $45,500 + 1,456Equity = $46,956

So the EFN is:

EFN = Total assets – Total liabilities and equityEFN = $109,760 – 99,456 EFN = $10,304

5. Assuming costs and assets increase proportionally, the pro forma financial statements will look like this:

Pro forma income Pro forma balance sheet

CHAPTER 4 B-62

statementSales $4,830.

00CA $4,140.

00CL $2,145.0

0Costs 3,795.0

0FA 9,085.0

0LTD 3,650.00

Taxable income

$1,035.00

Equity 6,159.86

Taxes (34%) 351.90 TA $13,225.00

Total D&E

$12,224.86

Net income $683.10

B-63 SOLUTIONS

The payout ratio is 40 percent, so dividends will be:

Dividends = 0.40($683.10) Dividends = $273.24

The addition to retained earnings is:

Addition to retained earnings = $683.10 – 273.24Addition to retained earnings = $409.86

So the EFN is:

EFN = Total assets – Total liabilities and equityEFN = $13,225 – 12,224.86 EFN = $1,000.14

6. To calculate the internal growth rate, we first need to calculate the ROA, which is:

ROA = NI / TA ROA = $2,262 / $39,150 ROA = .0578 or 5.78%

The plowback ratio, b, is one minus the payout ratio, so:

b = 1 – .30 b = .70

Now we can use the internal growth rate equation to get:

Internal growth rate = (ROA × b) / [1 – (ROA × b)]Internal growth rate = [0.0578(.70)] / [1 – 0.0578(.70)] Internal growth rate = .0421 or 4.21%

7. To calculate the sustainable growth rate, we first need to calculate the ROE, which is:

ROE = NI / TE ROE = $2,262 / $21,650 ROE = .1045 or 10.45%


CHAPTER 4 B-64

b = 1 – .30 b = .70

Now we can use the sustainable growth rate equation to get:

Sustainable growth rate = (ROE × b) / [1 – (ROE × b)]Sustainable growth rate = [0.1045(.70)] / [1 – 0.1045(.70)] Sustainable growth rate = .0789 or 7.89%

B-65 SOLUTIONS

8. The maximum percentage sales increase is the sustainable growth rate. To calculate the sustainable growth rate, we first need to calculate the ROE, which is:

ROE = NI / TE ROE = $8,910 / $56,000 ROE = .1591 or 15.91%


b = 1 – .30 b = .70


Sustainable growth rate = (ROE × b) / [1 – (ROE × b)]Sustainable growth rate = [.1591(.70)] / [1 – .1591(.70)] Sustainable growth rate = .1253 or 12.53%

So, the maximum dollar increase in sales is:

Maximum increase in sales = $42,000(.1253) Maximum increase in sales = $5,264.03

9. Assuming costs vary with sales and a 20 percent increase in sales, the pro forma income statement will look like this:

HEIR JORDAN CORPORATIONPro Forma Income StatementSales $45,600.00Costs 22,080.00Taxable income

$23,520.00

Taxes (34%) 7,996.80

Net income $15,523.20


Dividends = ($5,200/$12,936)($15,523.20)Dividends = $6,240.00

CHAPTER 4 B-66

And the addition to retained earnings will be:

Addition to retained earnings = $15,523.20 – 6,240Addition to retained earnings = $9,283.20

B-67 SOLUTIONS

10. Below is the balance sheet with the percentage of sales for each account on the balance sheet. Notes payable, total current liabilities, long-term debt, and all equity accounts do not vary directly with sales.

HEIR JORDAN CORPORATIONBalance Sheet

($) (%) ($) (%)

Assets Liabilities and Owners’ EquityCurrent assets Current liabilities

Cash $ 3,050 8.03 Accounts payable $ 1,300 3.42

Accounts receivable 6,900 18.16 Notes payable 6,800 n/a

Inventory 7,600 20.00 Total $ 8,100 n/aTotal $ 17,550 46.18 Long-term debt 25,000 n/a

Fixed assets Owners’ equityNet plant and Common stock andequipment 34,500 90.79 paid-in surplus $15,000

n/aRetained earnings 3,950

n/aTotal $ 18,950

n/aTotal liabilities and owners’

Total assets $ 52,050 136.97 equity $ 52,050 n/a


HEIR JORDAN CORPORATIONPro Forma Income StatementSales $43,700.00Costs 21,160.00Taxable income

$22,540.00

Taxes (34%) 7,663.60

Net income $14,876.40

CHAPTER 4 B-68


Dividends = ($5,200/$12,936)($14,876.40)Dividends = $5,980.00


Addition to retained earnings = $14,876.40 – 5,980Addition to retained earnings = $8,896.40

The new accumulated retained earnings on the pro forma balance sheet will be:

New accumulated retained earnings = $3,950 + 8,896.40New accumulated retained earnings = $12,846.40

B-69 SOLUTIONS

The pro forma balance sheet will look like this:

HEIR JORDAN CORPORATIONPro Forma Balance Sheet


Cash $ 3,507.50 Accounts payable $ 1,495.00

Accounts receivable 7,935.00 Notes payable 6,800.00

Inventory 8,740.00 Total $8,295.00Total $20,182.50 Long-term debt 25,000.00

Fixed assetsNet plant and Owners’ equityequipment 39.675.00 Common stock and

paid-in surplus $15,000.00

Retained earnings 12,846.40 Total $

27,846.40 Total liabilities and owners’

Total assets $ 59,857.50 equity $ 61,141.40

So the EFN is:

EFN = Total assets – Total liabilities and equityEFN = $59,857.50 – 61,141.40 EFN = –$1,283.90

12. We need to calculate the retention ratio to calculate the internal growth rate. The retention ratio is:

b = 1 – .20 b = .80

Now we can use the internal growth rate equation to get:

Internal growth rate = (ROA × b) / [1 – (ROA × b)]Internal growth rate = [.08(.80)] / [1 – .08(.80)] Internal growth rate = .0684 or 6.84%

CHAPTER 4 B-70

13. We need to calculate the retention ratio to calculate the sustainable growth rate. The retention ratio is:

b = 1 – .25 b = .75



B-71 SOLUTIONS

14. We first must calculate the ROE to calculate the sustainable growthrate. To do this we must realize two other relationships. The totalasset turnover is the inverse of the capital intensity ratio, and the equity multiplier is 1 + D/E. Using these relationships, we get:

ROE = (PM)(TAT)(EM) ROE = (.082)(1/.75)(1 + .40) ROE = .1531 or 15.31%

The plowback ratio is one minus the dividend payout ratio, so:

b = 1 – ($12,000 / $43,000) b = .7209



15. We must first calculate the ROE using the DuPont ratio to calculatethe sustainable growth rate. The ROE is:

ROE = (PM)(TAT)(EM) ROE = (.078)(2.50)(1.80) ROE = .3510 or 35.10%

The plowback ratio is one minus the dividend payout ratio, so:

b = 1 – .60 b = .40



Intermediate

16. To determine full capacity sales, we divide the current sales by the capacity the company is currently

CHAPTER 4 B-72

using, so:

Full capacity sales = $550,000 / .95 Full capacity sales = $578,947

The maximum sales growth is the full capacity sales divided by the current sales, so:

Maximum sales growth = ($578,947 / $550,000) – 1 Maximum sales growth = .0526 or 5.26%

B-73 SOLUTIONS

17. To find the new level of fixed assets, we need to find the current percentage of fixed assets to full capacity sales. Doing so, we find:

Fixed assets / Full capacity sales = $440,000 / $578,947 Fixed assets / Full capacity sales = .76

Next, we calculate the total dollar amount of fixed assets needed at the new sales figure.

Total fixed assets = .76($630,000) Total fixed assets = $478,800

The new fixed assets necessary is the total fixed assets at the newsales figure minus the current level of fixed assts.

New fixed assets = $478,800 – 440,000 New fixed assets = $38,800

18. We have all the variables to calculate ROE using the DuPont identity except the profit margin. If we find ROE, we can solve theDuPont identity for profit margin. We can calculate ROE from the sustainable growth rate equation. For this equation we need the retention ratio, so:

b = 1 – .30 b = .70

Using the sustainable growth rate equation and solving for ROE, we get:

Sustainable growth rate = (ROE × b) / [1 – (ROE × b)].12 = [ROE(.70)] / [1 – ROE(.70)] ROE = .1531 or 15.31%

Now we can use the DuPont identity to find the profit margin as:

ROE = PM(TAT)(EM).1531 = PM(1 / 0.75)(1 + 1.20) PM = (.1531) / [(1 / 0.75)(2.20)] PM = .0522 or 5.22%

CHAPTER 4 B-74

19. We have all the variables to calculate ROE using the DuPont identity except the equity multiplier. Remember that the equity multiplier is one plus the debt-equity ratio. If we find ROE, we can solve the DuPont identity for equity multiplier, then the debt-equity ratio. We can calculate ROE from the sustainable growth rateequation. For this equation we need the retention ratio, so:

b = 1 – .30 b = .70

Using the sustainable growth rate equation and solving for ROE, we get:

Sustainable growth rate = (ROE × b) / [1 – (ROE × b)].115 = [ROE(.70)] / [1 – ROE(.70)] ROE = .1473 or 14.73%

B-75 SOLUTIONS

Now we can use the DuPont identity to find the equity multiplier as:

ROE = PM(TAT)(EM).1473 = (.062)(1 / .60)EM EM = (.1473)(.60) / .062 EM = 1.43

So, the D/E ratio is:

D/E = EM – 1D/E = 1.43 – 1D/E = 0.43

20. We are given the profit margin. Remember that:

ROA = PM(TAT)

We can calculate the ROA from the internal growth rate formula, andthen use the ROA in this equation to find the total asset turnover.The retention ratio is:

b = 1 – .25b = .75

Using the internal growth rate equation to find the ROA, we get:

Internal growth rate = (ROA × b) / [1 – (ROA × b)].07 = [ROA(.75)] / [1 – ROA(.75)] ROA = .0872 or 8.72%

Plugging ROA and PM into the equation we began with and solving forTAT, we get:

ROA = (PM)(TAT).0872 = .05(PM)TAT = .0872 / .05 TAT = 1.74 times

21. We should begin by calculating the D/E ratio. We calculate the D/E ratio as follows:

Total debt ratio = .65 = TD / TA

CHAPTER 4 B-76

Inverting both sides we get:

1 / .65 = TA / TD

Next, we need to recognize that

TA / TD = 1 + TE / TD

Substituting this into the previous equation, we get:

1 / .65 = 1 + TE /TD

B-77 SOLUTIONS

Subtract 1 (one) from both sides and inverting again, we get:

D/E = 1 / [(1 / .65) – 1] D/E = 1.86

With the D/E ratio, we can calculate the EM and solve for ROE usingthe DuPont identity:

ROE = (PM)(TAT)(EM) ROE = (.048)(1.25)(1 + 1.86) ROE = .1714 or 17.14%

Now we can calculate the retention ratio as:

b = 1 – .30 b = .70

Finally, putting all the numbers we have calculated into the sustainable growth rate equation, we get:


22. To calculate the sustainable growth rate, we first must calculate the retention ratio and ROE. The retention ratio is:

b = 1 – $9,300 / $17,500 b = .4686

And the ROE is:

ROE = $17,500 / $58,000 ROE = .3017 or 30.17%

So, the sustainable growth rate is:


If the company grows at the sustainable growth rate, the new level of total assets is:

CHAPTER 4 B-78

New TA = 1.1647($86,000 + 58,000) = $167,710.84

To find the new level of debt in the company’s balance sheet, we take the percentage of debt in the capital structure times the new level of total assets. The additional borrowing will be the new level of debt minus the current level of debt. So:

New TD = [D / (D + E)](TA) New TD = [$86,000 / ($86,000 + 58,000)]($167,710.84) New TD = $100,160.64

B-79 SOLUTIONS

And the additional borrowing will be:

Additional borrowing = $100,160.04 – 86,000 Additional borrowing = $14,160.64

The growth rate that can be supported with no outside financing is the internal growth rate. To calculate the internal growth rate, wefirst need the ROA, which is:

ROA = $17,500 / ($86,000 + 58,000) ROA = .1215 or 12.15%

This means the internal growth rate is:

Internal growth rate = (ROA × b) / [1 – (ROA × b)]Internal growth rate = [.1215(.4686)] / [1 – .1215(.4686)] Internal growth rate = .0604 or 6.04%

23. Since the company issued no new equity, shareholders’ equity increased by retained earnings. Retained earnings for the year were:

Retained earnings = NI – DividendsRetained earnings = $19,000 – 2,500 Retained earnings = $16,500

So, the equity at the end of the year was:

Ending equity = $135,000 + 16,500 Ending equity = $151,500

The ROE based on the end of period equity is:

ROE = $19,000 / $151,500 ROE = .1254 or 12.54%

The plowback ratio is:

Plowback ratio = Addition to retained earnings/NIPlowback ratio = $16,500 / $19,000 Plowback ratio = .8684 or 86.84%

CHAPTER 4 B-80

Using the equation presented in the text for the sustainable growthrate, we get:


The ROE based on the beginning of period equity is

ROE = $16,500 / $135,000 ROE = .1407 or 14.07%

B-81 SOLUTIONS

Using the shortened equation for the sustainable growth rate and the beginning of period ROE, we get:

Sustainable growth rate = ROE × bSustainable growth rate = .1407 × .8684 Sustainable growth rate = .1222 or 12.22%

Using the shortened equation for the sustainable growth rate and the end of period ROE, we get:

Sustainable growth rate = ROE × b Sustainable growth rate = .1254 × .8684 Sustainable growth rate = .1089 or 10.89%

Using the end of period ROE in the shortened sustainable growth rate results in a growth rate that is too low. This will always occur whenever the equity increases. If equity increases, the ROE based on end of period equity is lower than the ROE based on the beginning of period equity. The ROE (and sustainable growth rate) in the abbreviated equation is based on equity that did not exist when the net income was earned.

24. The ROA using end of period assets is:

ROA = $19,000 / $250,000 ROA = .0760 or 7.60%

The beginning of period assets had to have been the ending assets minus the addition to retained earnings, so:

Beginning assets = Ending assets – Addition to retained earningsBeginning assets = $250,000 – 16,500Beginning assets = $233,500

And the ROA using beginning of period assets is:

ROA = $19,000 / $233,500 ROA = .0814 or 8.14%

Using the internal growth rate equation presented in the text, we get:

Internal growth rate = (ROA × b) / [1 – (ROA × b)]

CHAPTER 4 B-82

Internal growth rate = [.0814(.8684)] / [1 – .0814(.8684)] Internal growth rate = .0707 or 7.07%

Using the formula ROA × b, and end of period assets:

Internal growth rate = .0760 × .8684 Internal growth rate = .0660 or 6.60%

Using the formula ROA × b, and beginning of period assets:

Internal growth rate = .0814 × .8684 Internal growth rate = .0707 or 7.07%

B-83 SOLUTIONS


MOOSE TOURS INC. Pro Forma Income Statement

Sales $1,114,800Costs 867,600Other expenses 22,800 EBIT $224,400Interest 14,000 Taxable income $ 210,400Taxes(35%) 73,640 Net income $ 136,760


Dividends = ($33,735/$112,450)($136,760)Dividends = $41,028


Addition to retained earnings = $136,760 – 41,028Addition to retained earnings = $95,732

The new retained earnings on the pro forma balance sheet will be:

New retained earnings = $182,900 + 95,732New retained earnings = $278,632


MOOSE TOURS INC.Pro Forma Balance Sheet

Assets Liabilities and Owners’ Equity

Current assets Current liabilitiesCash $ 30,360 Accounts

payable $ 81,600Accounts receivable 48,840 Notes

payable 17,000 Inventory 104,280 Total $ 98,600

CHAPTER 4 B-84

Total $ 183,480 Long-term debt 158,000 Fixed assets

Net plant and Owners’ equityequipment 495,600 Common stock and

paid-in surplus $ 140,000Retained earnings 278,632

Total $ 418,632 Total liabilities and owners’

Total assets $ 679,080 equity $ 675,232

B-85 SOLUTIONS

So the EFN is:


26. First, we need to calculate full capacity sales, which is:

Full capacity sales = $929,000 / .80 Full capacity sales = $1,161,250

The capital intensity ratio at full capacity sales is:

Capital intensity ratio = Fixed assets / Full capacity sales Capital intensity ratio = $413,000 / $1,161,250 Capital intensity ratio = .35565

The fixed assets required at full capacity sales is the capital intensity ratio times the projected sales level:

Total fixed assets = .35565($1,161,250) = $396,480

So, EFN is:

EFN = ($183,480 + 396,480) – $613,806 = –$95,272

Note that this solution assumes that fixed assets are decreased (sold) so the company has a 100 percent fixed asset utilization. Ifwe assume fixed assets are not sold, the answer becomes:

EFN = ($183,480 + 413,000) – $613,806 = –$166,154

27. The D/E ratio of the company is:

D/E = ($85,000 + 158,000) / $322,900 D/E = .7526

So the new total debt amount will be:

New total debt = .7526($418,632) New total debt = $315,044

CHAPTER 4 B-86

This is the new total debt for the company. Given that our calculation for EFN is the amount that must be raised externally and does not increase spontaneously with sales, we need to subtractthe spontaneous increase in accounts payable. The new level of accounts payable will be, which is the current accounts payable times the sales growth, or:

Spontaneous increase in accounts payable = $68,000(.20)Spontaneous increase in accounts payable = $13,600

B-87 SOLUTIONS

This means that $13,600 of the new total debt is not raised externally. So, the debt raised externally, which will be the EFN is:

EFN = New total debt – (Beginning LTD + Beginning CL + Spontaneous increase in AP) EFN = $315,044 – ($158,000 + 68,000 + 17,000 + 13,600) = $58,444

The pro forma balance sheet with the new long-term debt will be:











The funds raised by the debt issue can be put into an excess cashaccount to make the balance sheet balance. The excess debt will be:

Excess debt = $733,676 – 697,080 = $54,596

To make the balance sheet balance, the company will have toincrease its assets. We will put this amount in an account calledexcess cash, which will give us the following balance sheet:

CHAPTER 4 B-88



Cash $ 30,360 Accounts payable $ 81,600

Excess cash 54,596Accounts receivable 44,400 Notes







The excess cash has an opportunity cost that we discussed earlier.Increasing fixed assets would also not be a good idea since thecompany already has enough fixed assets. A likely scenario would bethe repurchase of debt and equity in its current capital structureweights. The company’s debt-assets and equity assets are:

Debt-assets = .7526 / (1 + .7526) = .43Equity-assets = 1 / (1 + .7526) = .57

So, the amount of debt and equity needed will be:

Total debt needed = .43($697,080) = $291,600Equity needed = .57($697,080) = $387,480

So, the repurchases of debt and equity will be:

Debt repurchase = ($98,600 + 216,444) – 291,600 = $23,444Equity repurchase = $418,632 – 387,480 = $31,152

B-89 SOLUTIONS

Assuming all of the debt repurchase is from long-term debt, and theequity repurchase is entirely from the retained earnings, the finalpro forma balance sheet will be:

CHAPTER 4 B-90











Challenge

28. The pro forma income statements for all three growth rates will be:

MOOSE TOURS INC.Pro Forma Income Statement

15 % SalesGrowth

20% SalesGrowth

25% SalesGrowth

Sales $1,068,350 $1,114,800 $1,161,250Costs 831,450 867,600 903,750Other expenses 21,850 22,800 23,750EBIT $215,050 $224,400 $233,750Interest 14,000 14,000 14,000Taxable income $201,050 $210,400 $219,750Taxes (35%) 70,368 73,640 76,913Net income $130,683 $136,760 $142,838 Dividends $39,205 $41,028 $42,851 Add to RE 91,478 95,732 99,986

B-91 SOLUTIONS

We will calculate the EFN for the 15 percent growth rate first. Assuming the payout ratio is constant, the dividends paid will be:







15% Sales Growth:MOOSE TOURS INC.

Pro Forma Balance Sheet





Total $ 175,835 Long-term debt $ 158,000Fixed assets





So the EFN is:

CHAPTER 4 B-92

EFN = Total assets – Total liabilities and equityEFN = $650,785 – 667,578 EFN = –$16,793

At a 20 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be:






B-93 SOLUTIONS


20% Sales Growth:











So the EFN is:







CHAPTER 4 B-94



B-95 SOLUTIONS

25% Sales Growth:MOOSE TOURS INC.

Pro Forma Balance Sheet










So the EFN is:


29. The pro forma income statements for all three growth rates will be:

MOOSE TOURS INC.Pro Forma Income Statement

20% SalesGrowth

30% SalesGrowth

35% SalesGrowth

Sales $1,114,800 $1,207,700 $1,254,150Costs 867,600 939,900 976,050Other expenses 22,800 24,700 25,650EBIT $224,400 $243,100 $252,450Interest 14,000 14,000 14,000Taxable income $210,400 $229,100 $238,450Taxes (35%) 73,640 80,185 83,458Net income $136,760 $148,915 $154,993

CHAPTER 4 B-96

Dividends $41,028 $44,675 $46,498 Add to RE 95,732 104,241 108,495





The new addition to retained earnings on the pro forma balance sheet will be:

New addition to retained earnings = $182,900 + 104,241New addition to retained earnings = $287,141The new total debt will be:

New total debt = .7556($427,141)New total debt = $321,447

So, the new long-term debt will be the new total debt minus the newshort-term debt, or:

New long-term debt = $321,447 – 105,400New long-term debt = $58,047


Sales growth rate = 30% and debt/equity ratio = .7526:



Current assets Current liabilities

B-97 SOLUTIONS

Cash $ 32,890 Accounts payable $ 88,400

Accounts receivable 52,910 Notes payable 17,000

Inventory 112,970 Total $ 105,400Total $ 198,770 Long-term debt 216,047

Fixed assetsNet plant and Owners’ equityequipment 536,900 Common stock and




So the excess debt raised is:

Excess debt = $748,587 – 735,670Excess debt = $12,917







The new total debt will be:

New total debt = .75255($431,395)New total debt = $324,648

So, the new long-term debt will be the new total debt minus the newshort-term debt, or:

CHAPTER 4 B-98

New long-term debt = $324,648 – 108,800New long-term debt = $215,848

B-99 SOLUTIONS

Sales growth rate = 35% and debt/equity ratio = .75255:











So the excess debt raised is:

Excess debt = $756,043 – 763,965Excess debt = –$7,922

At a 35 percent growth rate, the firm will need funds in the amountof $7,922 in addition to the external debt already raised. So, the EFN will be:

EFN = $57,848 + 7,922EFN = $65,770

30. We must need the ROE to calculate the sustainable growth rate. The ROE is:

ROE = (PM)(TAT)(EM) ROE = (.067)(1 / 1.35)(1 + 0.30) ROE = .0645 or 6.45%

CHAPTER 4 B-100

Now we can use the sustainable growth rate equation to find the retention ratio as:

Sustainable growth rate = (ROE × b) / [1 – (ROE × b)]Sustainable growth rate = .12 = [.0645(b)] / [1 – .0645(b) b = 1.66

This implies the payout ratio is:

Payout ratio = 1 – b Payout ratio = 1 – 1.66Payout ratio = –0.66

This is a negative dividend payout ratio of 66 percent, which is impossible. The growth rate is not consistent with the other constraints. The lowest possible payout rate is 0, which corresponds to retention ratio of 1, or total earnings retention.

The maximum sustainable growth rate for this company is:

Maximum sustainable growth rate = (ROE × b) / [1 – (ROE × b)]Maximum sustainable growth rate = [.0645(1)] / [1 – .0645(1)] Maximum sustainable growth rate = .0690 or 6.90%

31. We know that EFN is:

EFN = Increase in assets – Addition to retained earnings

The increase in assets is the beginning assets times the growth rate, so:

Increase in assets = A g

The addition to retained earnings next year is the current net income times the retention ratio, times one plus the growth rate, so:

Addition to retained earnings = (NI b)(1 + g)

And rearranging the profit margin to solve for net income, we get:

NI = PM(S)

B-101 SOLUTIONS

Substituting the last three equations into the EFN equation we started with and rearranging, we get:

EFN = A(g) – PM(S)b(1 + g)EFN = A(g) – PM(S)b – [PM(S)b]gEFN = – PM(S)b + [A – PM(S)b]g

32. We start with the EFN equation we derived in Problem 31 and set it equal to zero:

EFN = 0 = – PM(S)b + [A – PM(S)b]g

Substituting the rearranged profit margin equation into the internal growth rate equation, we have:

Internal growth rate = [PM(S)b ] / [A – PM(S)b]

Since:

ROA = NI / A ROA = PM(S) / A

We can substitute this into the internal growth rate equation and divide both the numerator and denominator by A. This gives:

Internal growth rate = {[PM(S)b] / A} / {[A – PM(S)b] / A}Internal growth rate = b(ROA) / [1 – b(ROA)]

CHAPTER 4 B-102

To derive the sustainable growth rate, we must realize that to maintain a constant D/E ratio with no external equity financing, EFN must equal the addition to retained earnings times the D/E ratio:

EFN = (D/E)[PM(S)b(1 + g)] EFN = A(g) – PM(S)b(1 + g)

Solving for g and then dividing numerator and denominator by A:

Sustainable growth rate = PM(S)b(1 + D/E) / [A – PM(S)b(1 + D/E )]Sustainable growth rate = [ROA(1 + D/E )b] / [1 – ROA(1 + D/E )b]Sustainable growth rate = b(ROE) / [1 – b(ROE)]

33. In the following derivations, the subscript “E” refers to end of period numbers, and the subscript “B” refers to beginning of periodnumbers. TE is total equity and TA is total assets.

For the sustainable growth rate:

Sustainable growth rate = (ROEE × b) / (1 – ROEE × b)Sustainable growth rate = (NI/TEE × b) / (1 – NI/TEE × b)

We multiply this equation by:

(TEE / TEE)

Sustainable growth rate = (NI / TEE × b) / (1 – NI / TEE × b) × (TEE

/ TEE)Sustainable growth rate = (NI × b) / (TEE – NI × b)

Recognize that the numerator is equal to beginning of period equity, that is:

(TEE – NI × b) = TEB


Sustainable rate = (NI × b) / TEB

Which is equivalent to:

Sustainable rate = (NI / TEB) × b

B-103 SOLUTIONS

Since ROEB = NI / TEB

The sustainable growth rate equation is:

Sustainable growth rate = ROEB × b

For the internal growth rate:

Internal growth rate = (ROAE × b) / (1 – ROAE × b)Internal growth rate = (NI / TAE × b) / (1 – NI / TAE × b)

CHAPTER 4 B-104

We multiply this equation by:

(TAE / TAE)

Internal growth rate = (NI / TAE × b) / (1 – NI / TAE × b) × (TAE / TAE)Internal growth rate = (NI × b) / (TAE – NI × b)

Recognize that the numerator is equal to beginning of period assets, that is:

(TAE – NI × b) = TAB


Internal growth rate = (NI × b) / TAB

Which is equivalent to:

Internal growth rate = (NI / TAB) × b

Since ROAB = NI / TAB

The internal growth rate equation is:

Internal growth rate = ROAB × b

CHAPTER 5INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEYAnswers to Concepts Review and Critical Thinking Questions

1. The four parts are the present value (PV), the future value (FV),the discount rate (r), and the life of the investment (t).

2. Compounding refers to the growth of a dollar amount through timevia reinvestment of interest earned. It is also the process ofdetermining the future value of an investment. Discounting is theprocess of determining the value today of an amount to be receivedin the future.

3. Future values grow (assuming a positive rate of return); presentvalues shrink.

4. The future value rises (assuming it’s positive); the present valuefalls.

5. It would appear to be both deceptive and unethical to run such anad without a disclaimer or explanation.

6. It’s a reflection of the time value of money. TMCC gets to use the$24,099. If TMCC uses it wisely, it will be worth more than$100,000 in thirty years.

7. This will probably make the security less desirable. TMCC will onlyrepurchase the security prior to maturity if it is to itsadvantage, i.e. interest rates decline. Given the drop in interestrates needed to make this viable for TMCC, it is unlikely thecompany will repurchase the security. This is an example of a“call” feature. Such features are discussed at length in a laterchapter.

8. The key considerations would be: (1) Is the rate of return implicitin the offer attractive relative to other, similar risk

CHAPTER 5 B-106

investments? and (2) How risky is the investment; i.e., how certainare we that we will actually get the $100,000? Thus, our answerdoes depend on who is making the promise to repay.

9. The Treasury security would have a somewhat higher price becausethe Treasury is the strongest of all borrowers.

10. The price would be higher because, as time passes, the price of thesecurity will tend to rise toward $100,000. This rise is just areflection of the time value of money. As time passes, the timeuntil receipt of the $100,000 grows shorter, and the present valuerises. In 2019, the price will probably be higher for the samereason. We cannot be sure, however, because interest rates could bemuch higher, or TMCC’s financial position could deteriorate. Eitherevent would tend to depress the security’s price.

B-107 SOLUTIONS



Basic

1. The simple interest per year is:

$5,000 × .08 = $400

So after 10 years you will have:

$400 × 10 = $4,000 in interest.

The total balance will be $5,000 + 4,000 = $9,000

With compound interest we use the future value formula:

FV = PV(1 +r)t FV = $5,000(1.08)10 = $10,794.62

The difference is:

$10,794.62 – 9,000 = $1,794.62

2. To find the FV of a lump sum, we use:

FV = PV(1 + r)t

FV = $2,250(1.10)11 = $ 6,419.51FV = $8,752(1.08)7 = $ 14,999.39FV = $76,355(1.17)14 = $687,764.17FV = $183,796(1.07)8 = $315,795.75

3. To find the PV of a lump sum, we use:

PV = FV / (1 + r)t

PV = $15,451 / (1.07)6 = $ 10,295.65PV = $51,557 / (1.13)7 = $ 21,914.85

CHAPTER 5 B-108

PV = $886,073 / (1.14)23 = $ 43,516.90PV = $550,164 / (1.09)18 = $116,631.32

B-109 SOLUTIONS

4. To answer this question, we can use either the FV or the PVformula. Both will give the same answer since they are the inverseof each other. We will use the FV formula, that is:

FV = PV(1 + r)t

Solving for r, we get:

r = (FV / PV)1 / t – 1

FV = $297 = $240(1 + r)2; r = ($297 / $240)1/2 – 1 = 11.24%FV = $1,080 = $360(1 + r)10; r = ($1,080 / $360)1/10 – 1 = 11.61%FV = $185,382 = $39,000(1 + r)15; r = ($185,382 / $39,000)1/15 – 1 = 10.95%FV = $531,618 = $38,261(1 + r)30;r = ($531,618 / $38,261)1/30 – 1=

9.17%


FV = PV(1 + r)t

Solving for t, we get:

t = ln(FV / PV) / ln(1 + r)

FV = $1,284 = $560(1.09)t; t = ln($1,284/ $560) / ln 1.09 = 9.63 yearsFV = $4,341 = $810(1.10)t; t = ln($4,341/ $810) / ln 1.10 = 17.61 yearsFV = $364,518 = $18,400(1.17)t; t = ln($364,518 /

$18,400) / ln 1.17 = 19.02 yearsFV = $173,439 = $21,500(1.15)t; t = ln($173,439 /

$21,500) / ln 1.15 = 14.94 years


FV = PV(1 + r)t

CHAPTER 5 B-110


r = (FV / PV)1 / t – 1r = ($290,000 / $55,000)1/18 – 1 = .0968 or 9.68%

B-111 SOLUTIONS

7. To find the length of time for money to double, triple, etc., thepresent value and future value are irrelevant as long as the futurevalue is twice the present value for doubling, three times as largefor tripling, etc. To answer this question, we can use either theFV or the PV formula. Both will give the same answer since they arethe inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t


t = ln(FV / PV) / ln(1 + r)

The length of time to double your money is:

FV = $2 = $1(1.07)t

t = ln 2 / ln 1.07 = 10.24 years

The length of time to quadruple your money is:

FV = $4 = $1(1.07)t t = ln 4 / ln 1.07 = 20.49 years

Notice that the length of time to quadruple your money is twice aslong as the time needed to double your money (the difference inthese answers is due to rounding). This is an important concept oftime value of money.


FV = PV(1 + r)t


r = (FV / PV)1 / t – 1r = ($314,600 / $200,300)1/7 – 1 = .0666 or 6.66%


CHAPTER 5 B-112

FV = PV(1 + r)t


t = ln(FV / PV) / ln(1 + r)t = ln ($170,000 / $40,000) / ln 1.053 = 28.02 years


PV = FV / (1 + r)t

PV = $650,000,000 / (1.074)20 = $155,893,400.13

B-113 SOLUTIONS


PV = FV / (1 + r)t

PV = $1,000,000 / (1.10)80 = $488.19


FV = PV(1 + r)t

FV = $50(1.045)105 = $5,083.71


FV = PV(1 + r)t


r = (FV / PV)1 / t – 1r = ($1,260,000 / $150)1/112 – 1 = .0840 or 8.40%

To find the FV of the first prize, we use:

FV = PV(1 + r)t

FV = $1,260,000(1.0840)33 = $18,056,409.94


FV = PV(1 + r)t


r = (FV / PV)1 / t – 1r = ($43,125 / $1)1/113 – 1 = .0990 or 9.90%

15. To answer this question, we can use eitherthe FV or the PV formula. Both will give the same answer since theyare the inverse of each other. We will use the FV formula, that is:

FV = PV(1 + r)t

CHAPTER 5 B-114


r = (FV / PV)1 / t – 1r = ($10,311,500 / $12,377,500)1/4 – 1 = – 4.46%

Notice that the interest rate is negative. This occurs when the FVis less than the PV.

B-115 SOLUTIONS

Intermediate


FV = PV(1 + r)t


r = (FV / PV)1 / t – 1

a. PV = $100,000 / (1 + r)30 = $24,099r = ($100,000 / $24,099)1/30 – 1 = .0486 or 4.86%

b. PV = $38,260 / (1 + r)12 = $24,099r = ($38,260 / $24,099)1/12 – 1 = .0393 or 3.93%

c. PV = $100,000 / (1 + r)18 = $38,260r = ($100,000 / $38,260)1/18 – 1 = .0548 or 5.48%


PV = FV / (1 + r)t

PV = $170,000 / (1.12)9 = $61,303.70


FV = PV(1 + r)t

FV = $4,000(1.11)45 = $438,120.97

FV = $4,000(1.11)35 = $154,299.40

Better start early!

19. We need to find the FV of a lump sum. However, the money will onlybe invested for six years, so the number of periods is six.

FV = PV(1 + r)t

FV = $20,000(1.084)6 = $32,449.33

CHAPTER 5 B-116


FV = PV(1 + r)t


t = ln(FV / PV) / ln(1 + r)t = ln($75,000 / $10,000) / ln(1.11) = 19.31

So, the money must be invested for 19.31 years. However, you willnot receive the money for another two years. From now, you’llwait:

2 years + 19.31 years = 21.31 years

Calculator Solutions

1.Enter 10 8% $5,000

N I/Y PV PMT FVSolve for

$10,794.62

$10,794.62 – 9,000 = $1,794.62

2.Enter 11 10% $2,250


$6,419.51

Enter 7 8% $8,752N I/Y PV PMT FV

Solve for

$14,999.39

Enter 14 17% $76,355

B-117 SOLUTIONS


$687,764.17


Solve for

$315,795.75

3.Enter 6 7% $15,451


$10,295.65

CHAPTER 5 B-118


Solve for

$21,914.85


Solve for

$43,516.90


Solve for

$116,631.32

4.Enter 2 $240 $297


11.24%

Enter 10 $360 $1,080N I/Y PV PMT FV

Solve for

11.61%

Enter 15 $39,000 $185,382N I/Y PV PMT FV

Solve for

10.95%


Solve for

9.17%

B-119 SOLUTIONS

5.Enter 9% $560 $1,284


9.63

Enter 10% $810 $4,341N I/Y PV PMT FV

Solve for

17.61

Enter 17% $18,400 $364,518N I/Y PV PMT FV

Solve for

19.02

CHAPTER 5 B-120

Enter 15% $21,500 $173,439N I/Y PV PMT FV

Solve for

14.94

6.Enter 18 $55,000 $290,000


9.68%

7.Enter 7% $1 $2


10.24

Enter 7% $1 $4N I/Y PV PMT FV

Solve for

20.49

8.Enter 7 $200,300 $314,600


6.66%

9.Enter 5.30% $40,000 $170,000


28.02

10.Enter 20 7.4% $650,000,0

00N I/Y PV PMT FV

Solve for

$155,893,400.13

B-121 SOLUTIONS

11.Enter 80 10% $1,000,000


$488.19

12.Enter 105 4.50% $50


$5,083.71

CHAPTER 5 B-122

13.Enter 112 $150 $1,260,000


8.40%

Enter 33 8.40% $1,260,000N I/Y PV PMT FV

Solve for

$18,056,404.94

14.Enter 113 $1 ±$43,125


9.90%

15.Enter 4 $12,377,5

00$10,311,50

0N I/Y PV PMT FV

Solve for

–4.46%

16. a.Enter 30 $24,099 $100,000


4.86%

16. b.Enter 12 $24,099 $38,260


3.93%

16. c.Enter 18 $38,260 $100,000


5.48%

B-123 SOLUTIONS

17.Enter 9 12% $170,000


$61,303.70

18.Enter 45 11% $4,000


$438,120.97


Solve for

$154,299.40

CHAPTER 5 B-124

19.Enter 6 8.40% $20,000


$32,449.33

20.Enter 11% $10,000 $75,000


19.31

From now, you’ll wait 2 + 19.31 = 21.31 years

CHAPTER 6DISCOUNTED CASH FLOW VALUATION


1. The four pieces are the present value (PV), the periodic cash flow (C),the discount rate (r), and the number of payments, or the life of theannuity, t.

2. Assuming positive cash flows, both the present and the future valueswill rise.

3. Assuming positive cash flows, the present value will fall and thefuture value will rise.

4. It’s deceptive, but very common. The basic concept of time value ofmoney is that a dollar today is not worth the same as a dollartomorrow. The deception is particularly irritating given that suchlotteries are usually government sponsored!

5. If the total money is fixed, you want as much as possible as soon aspossible. The team (or, more accurately, the team owner) wants just theopposite.

6. The better deal is the one with equal installments.

7. Yes, they should. APRs generally don’t provide the relevant rate. Theonly advantage is that they are easier to compute, but, with moderncomputing equipment, that advantage is not very important.

8. A freshman does. The reason is that the freshman gets to use the moneyfor much longer before interest starts to accrue. The subsidy is thepresent value (on the day the loan is made) of the interest that wouldhave accrued up until the time it actually begins to accrue.

9. The problem is that the subsidy makes it easier to repay the loan, notobtain it. However, ability to repay the loan depends on future

CHAPTER 6 B-126

employment, not current need. For example, consider a student who iscurrently needy, but is preparing for a career in a high-paying area(such as corporate finance!). Should this student receive the subsidy?How about a student who is currently not needy, but is preparing for arelatively low-paying job (such as becoming a college professor)?

B-127 SOLUTIONS

10. In general, viatical settlements are ethical. In the case of a viaticalsettlement, it is simply an exchange of cash today for payment in thefuture, although the payment depends on the death of the seller. Thepurchaser of the life insurance policy is bearing the risk that theinsured individual will live longer than expected. Although viaticalsettlements are ethical, they may not be the best choice for anindividual. In a Business Week article (October 31, 2005), options wereexamined for a 72 year old male with a life expectancy of 8 years and a$1 million dollar life insurance policy with an annual premium of$37,000. The four options were: 1) Cash the policy today for $100,000.2) Sell the policy in a viatical settlement for $275,000. 3) Reduce thedeath benefit to $375,000, which would keep the policy in force for 12years without premium payments. 4) Stop paying premiums and don’treduce the death benefit. This will run the cash value of the policy tozero in 5 years, but the viatical settlement would be worth $475,000 atthat time. If he died within 5 years, the beneficiaries would receive$1 million. Ultimately, the decision rests on the individual on whatthey perceive as best for themselves. The values that will affect thevalue of the viatical settlement are the discount rate, the face valueof the policy, and the health of the individual selling the policy.


NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiplesteps. Due to space and readability constraints, when these intermediate steps are included in thissolutions manual, rounding may appear to have occurred. However, the final answer for eachproblem is found without rounding during any step in the problem.

Basic

1. To solve this problem, we must find the PV of each cash flow and addthem. To find the PV of a lump sum, we use:

PV = FV / (1 + r)t

PV@10% = $950 / 1.10 + $1,040 / 1.102 + $1,130 / 1.103 + $1,075 / 1.104 =$3,306.37

PV@18% = $950 / 1.18 + $1,040 / 1.182 + $1,130 / 1.183 + $1,075 / 1.184 =$2,794.22

PV@24% = $950 / 1.24 + $1,040 / 1.242 + $1,130 / 1.243 + $1,075 / 1.244 =$2,489.88

CHAPTER 6 B-128

2. To find the PVA, we use the equation:

PVA = C({1 – [1/(1 + r)]t } / r )

At a 5 percent interest rate:

X@5%: PVA = $6,000{[1 – (1/1.05)9 ] / .05 } = $42,646.93

Y@5%: PVA = $8,000{[1 – (1/1.05)6 ] / .05 } = $40,605.54

B-129 SOLUTIONS

And at a 15 percent interest rate:

X@15%: PVA = $6,000{[1 – (1/1.15)9 ] / .15 } = $28,629.50

Y@15%: PVA = $8,000{[1 – (1/1.15)6 ] / .15 } = $30,275.86

Notice that the PV of cash flow X has a greater PV at a 5 percentinterest rate, but a lower PV at a 15 percent interest rate. The reasonis that X has greater total cash flows. At a lower interest rate, thetotal cash flow is more important since the cost of waiting (theinterest rate) is not as great. At a higher interest rate, Y is morevaluable since it has larger cash flows. At the higher interest rate,these bigger cash flows early are more important since the cost ofwaiting (the interest rate) is so much greater.

3. To solve this problem, we must find the FV of each cash flow and addthem. To find the FV of a lump sum, we use:

FV = PV(1 + r)t

FV@8% = $940(1.08)3 + $1,090(1.08)2 + $1,340(1.08) + $1,405 = $5,307.71

FV@11% = $940(1.11)3 + $1,090(1.11)2 + $1,340(1.11) + $1,405 = $5,520.96

FV@24% = $940(1.24)3 + $1,090(1.24)2 + $1,340(1.24) + $1,405 = $6,534.81

Notice we are finding the value at Year 4, the cash flow at Year 4 issimply added to the FV of the other cash flows. In other words, we donot need to compound this cash flow.


PVA = C({1 – [1/(1 + r)]t } / r )

PVA@15 yrs: PVA = $5,300{[1 – (1/1.07)15 ] / .07} = $48,271.94

PVA@40 yrs: PVA = $5,300{[1 – (1/1.07)40 ] / .07} = $70,658.06

PVA@75 yrs: PVA = $5,300{[1 – (1/1.07)75 ] / .07} = $75,240.70

To find the PV of a perpetuity, we use the equation:

PV = C / r

CHAPTER 6 B-130

PV = $5,300 / .07 = $75,714.29

Notice that as the length of the annuity payments increases, thepresent value of the annuity approaches the present value of theperpetuity. The present value of the 75 year annuity and the presentvalue of the perpetuity imply that the value today of all perpetuitypayments beyond 75 years is only $473.59.

B-131 SOLUTIONS

5. Here we have the PVA, the length of the annuity, and the interest rate.We want to calculate the annuity payment. Using the PVA equation:

PVA = C({1 – [1/(1 + r)]t } / r )PVA = $34,000 = $C{[1 – (1/1.0765)15 ] / .0765}

We can now solve this equation for the annuity payment. Doing so, weget:

C = $34,000 / 8.74548 = $3,887.72


PVA = C({1 – [1/(1 + r)]t } / r )PVA = $73,000{[1 – (1/1.085)8 ] / .085} = $411,660.36

7. Here we need to find the FVA. The equation to find the FVA is:

FVA = C{[(1 + r)t – 1] / r}

FVA for 20 years = $4,000[(1.11220 – 1) / .112] = $262,781.16

FVA for 40 years = $4,000[(1.11240 – 1) / .112] = $2,459,072.63

Notice that because of exponential growth, doubling the number ofperiods does not merely double the FVA.

8. Here we have the FVA, the length of the annuity, and the interest rate.We want to calculate the annuity payment. Using the FVA equation:

FVA = C{[(1 + r)t – 1] / r}$90,000 = $C[(1.06810 – 1) / .068]


C = $90,000 / 13.68662 = $6,575.77

9. Here we have the PVA, the length of the annuity, and the interest rate.We want to calculate the annuity payment. Using the PVA equation:

PVA = C({1 – [1/(1 + r)]t } / r)$50,000 = C{[1 – (1/1.075)7 ] / .075}

CHAPTER 6 B-132


C = $50,000 / 5.29660 = $9,440.02

10. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:

PV = C / rPV = $25,000 / .072 = $347,222.22

B-133 SOLUTIONS

11. Here we need to find the interest rate that equates the perpetuity cashflows with the PV of the cash flows. Using the PV of a perpetuityequation:

PV = C / r$375,000 = $25,000 / r

We can now solve for the interest rate as follows:

r = $25,000 / $375,000 = .0667 or 6.67%

12. For discrete compounding, to find the EAR, we use the equation:

EAR = [1 + (APR / m)]m – 1

EAR = [1 + (.08 / 4)]4 – 1 = .0824 or 8.24%

EAR = [1 + (.16 / 12)]12 – 1 = .1723 or 17.23%

EAR = [1 + (.12 / 365)]365 – 1 = .1275 or 12.75%

To find the EAR with continuous compounding, we use the equation:

EAR = eq – 1EAR = e.15 – 1 = .1618 or 16.18%

13. Here we are given the EAR and need to find the APR. Using the equation for discrete compounding:

EAR = [1 + (APR / m)]m – 1

We can now solve for the APR. Doing so, we get:

APR = m[(1 + EAR)1/m – 1]

EAR = .0860 = [1 + (APR / 2)]2 – 1 APR = 2[(1.0860)1/2

– 1] = .0842 or 8.42%

EAR = .1980 = [1 + (APR / 12)]12 – 1 APR = 12[(1.1980)1/12 – 1] = .1820 or 18.20%

EAR = .0940 = [1 + (APR / 52)]52 – 1 APR = 52[(1.0940)1/52 – 1] = .0899 or 8.99%

CHAPTER 6 B-134

Solving the continuous compounding EAR equation:

EAR = eq – 1

We get:

APR = ln(1 + EAR)APR = ln(1 + .1650)APR = .1527 or 15.27%

B-135 SOLUTIONS

14. For discrete compounding, to find the EAR, we use the equation:

EAR = [1 + (APR / m)]m – 1

So, for each bank, the EAR is:

First National: EAR = [1 + (.1420 / 12)]12 – 1 = .1516 or 15.16%

First United: EAR = [1 + (.1450 / 2)]2 – 1 = .1503 or 15.03%

Notice that the higher APR does not necessarily mean the higher EAR.The number of compounding periods within a year will also affect theEAR.

15. The reported rate is the APR, so we need to convert the EAR to an APRas follows:

EAR = [1 + (APR / m)]m – 1

APR = m[(1 + EAR)1/m – 1]APR = 365[(1.16)1/365 – 1] = .1485 or 14.85%

This is deceptive because the borrower is actually paying annualizedinterest of 16% per year, not the 14.85% reported on the loan contract.

16. For this problem, we simply need to find the FV of a lump sum using theequation:

FV = PV(1 + r)t

It is important to note that compounding occurs semiannually. Toaccount for this, we will divide the interest rate by two (the numberof compounding periods in a year), and multiply the number of periodsby two. Doing so, we get:

FV = $2,100[1 + (.084/2)]34 = $8,505.93

17. For this problem, we simply need to find the FV of a lump sum using theequation:

FV = PV(1 + r)t

CHAPTER 6 B-136

It is important to note that compounding occurs daily. To account forthis, we will divide the interest rate by 365 (the number of days in ayear, ignoring leap year), and multiply the number of periods by 365.Doing so, we get:

FV in 5 years = $4,500[1 + (.093/365)]5(365) =$7,163.64

FV in 10 years = $4,500[1 + (.093/365)]10(365) = $11,403.94

FV in 20 years = $4,500[1 + (.093/365)]20(365) = $28,899.97

B-137 SOLUTIONS

18. For this problem, we simply need to find the PV of a lump sum using theequation:

PV = FV / (1 + r)t

It is important to note that compounding occurs daily. To account forthis, we will divide the interest rate by 365 (the number of days in ayear, ignoring leap year), and multiply the number of periods by 365.Doing so, we get:

PV = $58,000 / [(1 + .10/365)7(365)] = $28,804.71

19. The APR is simply the interest rate per period times the number ofperiods in a year. In this case, the interest rate is 30 percent permonth, and there are 12 months in a year, so we get:

APR = 12(30%) = 360%

To find the EAR, we use the EAR formula:

EAR = [1 + (APR / m)]m – 1

EAR = (1 + .30)12 – 1 = 2,229.81%

Notice that we didn’t need to divide the APR by the number ofcompounding periods per year. We do this division to get the interestrate per period, but in this problem we are already given the interestrate per period.

20. We first need to find the annuity payment. We have the PVA, the lengthof the annuity, and the interest rate. Using the PVA equation:

PVA = C({1 – [1/(1 + r)]t } / r)$68,500 = $C[1 – {1 / [1 + (.069/12)]60} / (.069/12)]

Solving for the payment, we get:

C = $68,500 / 50.622252 = $1,353.15

To find the EAR, we use the EAR equation:

EAR = [1 + (APR / m)]m – 1EAR = [1 + (.069 / 12)]12 – 1 = .0712 or 7.12%

CHAPTER 6 B-138

21. Here we need to find the length of an annuity. We know the interestrate, the PV, and the payments. Using the PVA equation:

PVA = C({1 – [1/(1 + r)]t } / r)$18,000 = $500{[1 – (1/1.013)t ] / .013}

B-139 SOLUTIONS

Now we solve for t:

1/1.013t = 1 – {[($18,000)/($500)](.013)}1/1.013t = 0.5321.013t = 1/(0.532) = 1.8797 t = ln 1.8797 / ln 1.013 = 48.86 months

22. Here we are trying to find the interest rate when we know the PV andFV. Using the FV equation:

FV = PV(1 + r)$4 = $3(1 + r) r = 4/3 – 1 = 33.33% per week

The interest rate is 33.33% per week. To find the APR, we multiply thisrate by the number of weeks in a year, so:

APR = (52)33.33% = 1,733.33%

And using the equation to find the EAR:

EAR = [1 + (APR / m)]m – 1EAR = [1 + .3333]52 – 1 = 313,916,515.69%

23. Here we need to find the interest rate that equates the perpetuity cashflows with the PV of the cash flows. Using the PV of a perpetuityequation:

PV = C / r$95,000 = $1,800 / r

We can now solve for the interest rate as follows:

r = $1,800 / $95,000 = .0189 or 1.89% per month

The interest rate is 1.89% per month. To find the APR, we multiply thisrate by the number of months in a year, so:

APR = (12)1.89% = 22.74%

And using the equation to find an EAR:

EAR = [1 + (APR / m)]m – 1

CHAPTER 6 B-140

EAR = [1 + .0189]12 – 1 = 25.26%

24. This problem requires us to find the FVA. The equation to find the FVAis:

FVA = C{[(1 + r)t – 1] / r}FVA = $300[{[1 + (.10/12) ]360 – 1} / (.10/12)] = $678,146.38

B-141 SOLUTIONS

25. In the previous problem, the cash flows are monthly and the compoundingperiod is monthly. This assumption still holds. Since the cash flowsare annual, we need to use the EAR to calculate the future value ofannual cash flows. It is important to remember that you have to makesure the compounding periods of the interest rate is the same as thetiming of the cash flows. In this case, we have annual cash flows, sowe need the EAR since it is the true annual interest rate you willearn. So, finding the EAR:

EAR = [1 + (APR / m)]m – 1EAR = [1 + (.10/12)]12 – 1 = .1047 or 10.47%

Using the FVA equation, we get:

FVA = C{[(1 + r)t – 1] / r}FVA = $3,600[(1.104730 – 1) / .1047] = $647,623.45

26. The cash flows are simply an annuity with four payments per year forfour years, or 16 payments. We can use the PVA equation:

PVA = C({1 – [1/(1 + r)]t } / r)PVA = $2,300{[1 – (1/1.0065)16] / .0065} = $34,843.71

27. The cash flows are annual and the compounding period is quarterly, sowe need to calculate the EAR to make the interest rate comparable withthe timing of the cash flows. Using the equation for the EAR, we get:

EAR = [1 + (APR / m)]m – 1EAR = [1 + (.11/4)]4 – 1 = .1146 or 11.46%

And now we use the EAR to find the PV of each cash flow as a lump sumand add them together:

PV = $725 / 1.1146 + $980 / 1.11462 + $1,360 / 1.11464 = $2,320.36

28. Here the cash flows are annual and the given interest rate is annual,so we can use the interest rate given. We simply find the PV of eachcash flow and add them together.

PV = $1,650 / 1.0845 + $4,200 / 1.08453 + $2,430 / 1.08454 = $6,570.86

Intermediate

CHAPTER 6 B-142

29. The total interest paid by First Simple Bank is the interest rate perperiod times the number of periods. In other words, the interest byFirst Simple Bank paid over 10 years will be:

.07(10) = .7

First Complex Bank pays compound interest, so the interest paid by thisbank will be the FV factor of $1, or:

(1 + r)10

B-143 SOLUTIONS

Setting the two equal, we get:

(.07)(10) = (1 + r)10 – 1

r = 1.71/10 – 1 = .0545 or 5.45%

30. Here we need to convert an EAR into interest rates for differentcompounding periods. Using the equation for the EAR, we get:

EAR = [1 + (APR / m)]m – 1

EAR = .17 = (1 + r)2 – 1; r = (1.17)1/2 – 1 = .0817 or 8.17% persix months

EAR = .17 = (1 + r)4 – 1; r = (1.17)1/4 – 1 = .0400 or 4.00% perquarter

EAR = .17 = (1 + r)12 – 1; r = (1.17)1/12 – 1 = .0132 or 1.32% permonth

Notice that the effective six month rate is not twice the effectivequarterly rate because of the effect of compounding.

31. Here we need to find the FV of a lump sum, with a changing interestrate. We must do this problem in two parts. After the first six months,the balance will be:

FV = $5,000 [1 + (.015/12)]6 = $5,037.62

This is the balance in six months. The FV in another six months will be:

FV = $5,037.62[1 + (.18/12)]6 = $5,508.35

The problem asks for the interest accrued, so, to find the interest, wesubtract the beginning balance

from the FV. The interest accrued is:

Interest = $5,508.35 – 5,000.00 = $508.35

32. We need to find the annuity payment in retirement. Our retirementsavings ends and the retirement withdrawals begin, so the PV of theretirement withdrawals will be the FV of the retirement savings. So, we

CHAPTER 6 B-144

find the FV of the stock account and the FV of the bond account and addthe two FVs.

Stock account: FVA = $700[{[1 + (.11/12) ]360 – 1} / (.11/12)] = $1,963,163.82

Bond account: FVA = $300[{[1 + (.06/12) ]360 – 1} / (.06/12)] = $301,354.51

So, the total amount saved at retirement is:

$1,963,163.82 + 301,354.51 = $2,264,518.33

Solving for the withdrawal amount in retirement using the PVA equation gives us:

PVA = $2,264,518.33 = $C[1 – {1 / [1 + (.09/12)]300} / (.09/12)]C = $2,264,518.33 / 119.1616 = $19,003.763 withdrawal per month

B-145 SOLUTIONS

33. We need to find the FV of a lump sum in one year and two years. It is important that we use the

number of months in compounding since interest is compounded monthly inthis case. So:

FV in one year = $1(1.0117)12 = $1.15

FV in two years = $1(1.0117)24 = $1.32

There is also another common alternative solution. We could find the EAR, and use the number of years as our compounding periods. So we willfind the EAR first:

EAR = (1 + .0117)12 – 1 = .1498 or 14.98%

Using the EAR and the number of years to find the FV, we get:

FV in one year = $1(1.1498)1 = $1.15

FV in two years = $1(1.1498)2 = $1.32

Either method is correct and acceptable. We have simply made sure thatthe interest compounding period is the same as the number of periods weuse to calculate the FV.

34. Here we are finding the annuity payment necessary to achieve the same FV. The interest rate given is a 12 percent APR, with monthly deposits.We must make sure to use the number of months in the equation. So, using the FVA equation:

Starting today:FVA = C[{[1 + (.12/12) ]480 – 1} / (.12/12)]C = $1,000,000 / 11,764.77 = $85.00

Starting in 10 years:FVA = C[{[1 + (.12/12) ]360 – 1} / (.12/12)]C = $1,000,000 / 3,494.96 = $286.13

Starting in 20 years:FVA = C[{[1 + (.12/12) ]240 – 1} / (.12/12)] C = $1,000,000 / 989.255 = $1,010.86

CHAPTER 6 B-146

Notice that a deposit for half the length of time, i.e. 20 years versus40 years, does not mean that the annuity payment is doubled. In thisexample, by reducing the savings period by one-half, the depositnecessary to achieve the same ending value is about twelve times aslarge.

35. Since we are looking to quadruple our money, the PV and FV areirrelevant as long as the FV is three times as large as the PV. Thenumber of periods is four, the number of quarters per year. So:

FV = $3 = $1(1 + r)(12/3) r = .3161 or 31.61%

B-147 SOLUTIONS

36. Since we have an APR compounded monthly and an annual payment, we must first convert the interest rate to an EAR so that the compounding period is the same as the cash flows.

EAR = [1 + (.10 / 12)]12 – 1 = .104713 or 10.4713%

PVA1 = $95,000 {[1 – (1 / 1.104713)2] / .104713} = $163,839.09

PVA2 = $45,000 + $70,000{[1 – (1/1.104713)2] / .104713} = $165,723.54

You would choose the second option since it has a higher PV.

37. We can use the present value of a growing perpetuity equation to findthe value of your deposits today. Doing so, we find:

PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t} PV = $1,000,000{[1/(.08 – .05)] – [1/(.08 – .05)] × [(1 + .05)/(1

+ .08)]30}PV = $19,016,563.18

38. Since your salary grows at 4 percent per year, your salary next yearwill be:

Next year’s salary = $50,000 (1 + .04)Next year’s salary = $52,000

This means your deposit next year will be:

Next year’s deposit = $52,000(.05)Next year’s deposit = $2,600

Since your salary grows at 4 percent, you deposit will also grow at 4percent. We can use the present value of a growing perpetuity equationto find the value of your deposits today. Doing so, we find:

PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t} PV = $2,600{[1/(.11 – .04)] – [1/(.11 – .04)] × [(1 + .04)/(1 + .11)]40}PV = $34,399.45

Now, we can find the future value of this lump sum in 40 years. Wefind:

FV = PV(1 + r)t

CHAPTER 6 B-148

FV = $34,366.45(1 + .11)40

FV = $2,235,994.31

This is the value of your savings in 40 years.

B-149 SOLUTIONS

39. The relationship between the PVA and the interest rate is:

PVA falls as r increases, and PVA rises as r decreasesFVA rises as r increases, and FVA falls as r decreases

The present values of $9,000 per year for 10 years at the various interest rates given are:

PVA@10% = $9,000{[1 – (1/1.10)15] / .10} = $68,454.72

PVA@5% = $9,000{[1 – (1/1.05)15] / .05} = $93,416.92

PVA@15% = $9,000{[1 – (1/1.15)15] / .15} = $52,626.33

40. Here we are given the FVA, the interest rate, and the amount of theannuity. We need to solve for the number of payments. Using the FVAequation:

FVA = $20,000 = $340[{[1 + (.06/12)]t – 1 } / (.06/12)]


1.005t = 1 + [($20,000)/($340)](.06/12) t = ln 1.294118 / ln 1.005 = 51.69 payments

41. Here we are given the PVA, number of periods, and the amount of theannuity. We need to solve for the interest rate. Using the PVAequation:

PVA = $73,000 = $1,450[{1 – [1 / (1 + r)]60}/ r]

To find the interest rate, we need to solve this equation on afinancial calculator, using a spreadsheet, or by trial and error. Ifyou use trial and error, remember that increasing the interest ratelowers the PVA, and decreasing the interest rate increases the PVA.Using a spreadsheet, we find:

r = 0.594%

The APR is the periodic interest rate times the number of periods inthe year, so:

APR = 12(0.594%) = 7.13%

CHAPTER 6 B-150

B-151 SOLUTIONS

42. The amount of principal paid on the loan is the PV of the monthlypayments you make. So, the present value of the $1,150 monthly paymentsis:

PVA = $1,150[(1 – {1 / [1 + (.0635/12)]}360) / (.0635/12)] = $184,817.42

The monthly payments of $1,150 will amount to a principal payment of$184,817.42. The amount of principal you will still owe is:

$240,000 – 184,817.42 = $55,182.58

This remaining principal amount will increase at the interest rate onthe loan until the end of the loan period. So the balloon payment in 30years, which is the FV of the remaining principal will be:

Balloon payment = $55,182.58[1 + (.0635/12)]360 = $368,936.54

43. We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the PV of the cash flows we know are:

PV of Year 1 CF: $1,700 / 1.10 = $1,545.45

PV of Year 3 CF: $2,100 / 1.103 = $1,577.76

PV of Year 4 CF: $2,800 / 1.104 = $1,912.44

So, the PV of the missing CF is:

$6,550 – 1,545.45 – 1,577.76 – 1,912.44 = $1,514.35

The question asks for the value of the cash flow in Year 2, so we must find the future value of this amount. The value of the missing CF is:

$1,514.35(1.10)2 = $1,832.36

44. To solve this problem, we simply need to find the PV of each lump sumand add them together. It is important to note that the first cash flowof $1 million occurs today, so we do not need to discount that cashflow. The PV of the lottery winnings is:

CHAPTER 6 B-152

PV = $1,000,000 + $1,500,000/1.09 + $2,000,000/1.092 + $2,500,000/1.093

+ $3,000,000/1.094 + $3,500,000/1.095 + $4,000,000/1.096 + $4,500,000/1.097 +

$5,000,000/1.098 + $5,500,000/1.099 + $6,000,000/1.0910

PV = $22,812,873.40

45. Here we are finding interest rate for an annuity cash flow. We aregiven the PVA, number of periods, and the amount of the annuity. Weshould also note that the PV of the annuity is not the amount borrowedsince we are making a down payment on the warehouse. The amountborrowed is:

Amount borrowed = 0.80($2,900,000) = $2,320,000

B-153 SOLUTIONS

Using the PVA equation:

PVA = $2,320,000 = $15,000[{1 – [1 / (1 + r)]360}/ r]

Unfortunately this equation cannot be solved to find the interest rateusing algebra. To find the interest rate, we need to solve thisequation on a financial calculator, using a spreadsheet, or by trialand error. If you use trial and error, remember that increasing theinterest rate lowers the PVA, and decreasing the interest rateincreases the PVA. Using a spreadsheet, we find:

r = 0.560%

The APR is the monthly interest rate times the number of months in theyear, so:

APR = 12(0.560%) = 6.72%

And the EAR is:

EAR = (1 + .00560)12 – 1 = .0693 or 6.93%

46. The profit the firm earns is just the PV of the sales price minus thecost to produce the asset. We find the PV of the sales price as the PVof a lump sum:

PV = $165,000 / 1.134 = $101,197.59

And the firm’s profit is:

Profit = $101,197.59 – 94,000.00 = $7,197.59

To find the interest rate at which the firm will break even, we need tofind the interest rate using the PV (or FV) of a lump sum. Using the PVequation for a lump sum, we get:

$94,000 = $165,000 / ( 1 + r)4 r = ($165,000 / $94,000)1/4 – 1 = .1510 or 15.10%

47. We want to find the value of the cash flows today, so we will find the PV of the annuity, and then bring the lump sum PV back to today. The annuity has 18 payments, so the PV of the annuity is:

CHAPTER 6 B-154

PVA = $4,000{[1 – (1/1.10)18] / .10} = $32,805.65

Since this is an ordinary annuity equation, this is the PV one period before the first payment, so it is the PV at t = 7. To find the value today, we find the PV of this lump sum. The value today is:

PV = $32,805.65 / 1.107 = $16,834.48

48. This question is asking for the present value of an annuity, but the interest rate changes during the life of the annuity. We need to find the present value of the cash flows for the last eight years first. ThePV of these cash flows is:

PVA2 = $1,500 [{1 – 1 / [1 + (.07/12)]96} / (.07/12)] = $110,021.35

B-155 SOLUTIONS

Note that this is the PV of this annuity exactly seven years from today. Now we can discount this lump sum to today. The value of this cash flow today is:

PV = $110,021.35 / [1 + (.11/12)]84 = $51,120.33

Now we need to find the PV of the annuity for the first seven years. The value of these cash flows today is:

PVA1 = $1,500 [{1 – 1 / [1 + (.11/12)]84} / (.11/12)] = $87,604.36

The value of the cash flows today is the sum of these two cash flows, so:

PV = $51,120.33 + 87,604.36 = $138,724.68

49. Here we are trying to find the dollar amount invested today that willequal the FVA with a known interest rate, and payments. First we needto determine how much we would have in the annuity account. Finding theFV of the annuity, we get:

FVA = $1,200 [{[ 1 + (.085/12)]180 – 1} / (.085/12)] = $434,143.62

Now we need to find the PV of a lump sum that will give us the same FV.So, using the FV of a lump sum with continuous compounding, we get:

FV = $434,143.62 = PVe.08(15) PV = $434,143.62e–1.20 = $130,761.55

50. To find the value of the perpetuity at t = 7, we first need to use thePV of a perpetuity equation. Using this equation we find:

PV = $3,500 / .062 = $56,451.61

Remember that the PV of a perpetuity (and annuity) equations give thePV one period before the first payment, so, this is the value of theperpetuity at t = 14. To find the value at t = 7, we find the PV of thislump sum as:

PV = $56,451.61 / 1.0627 = $37,051.41

CHAPTER 6 B-156

51. To find the APR and EAR, we need to use the actual cash flows of theloan. In other words, the interest rate quoted in the problem is onlyrelevant to determine the total interest under the terms given. Theinterest rate for the cash flows of the loan is:

PVA = $25,000 = $2,416.67{(1 – [1 / (1 + r)]12 ) / r }

Again, we cannot solve this equation for r, so we need to solve thisequation on a financial calculator, using a spreadsheet, or by trialand error. Using a spreadsheet, we find:

r = 2.361% per month

B-157 SOLUTIONS

So the APR is:

APR = 12(2.361%) = 28.33%

And the EAR is:

EAR = (1.02361)12 – 1 = .3231 or 32.31%

52. The cash flows in this problem are semiannual, so we need the effectivesemiannual rate. The interest rate given is the APR, so the monthlyinterest rate is:

Monthly rate = .10 / 12 = .00833

To get the semiannual interest rate, we can use the EAR equation, but instead of using 12 months as the exponent, we will use 6 months. The effective semiannual rate is:

Semiannual rate = (1.00833)6 – 1 = .0511 or 5.11%

We can now use this rate to find the PV of the annuity. The PV of the annuity is:

PVA @ year 8: $7,000{[1 – (1 / 1.0511)10] / .0511} = $53,776.72

Note, this is the value one period (six months) before the firstpayment, so it is the value at year 8. So, the value at the varioustimes the questions asked for uses this value 8 years from now.

PV @ year 5: $53,776.72 / 1.05116 = $39,888.33

Note, you can also calculate this present value (as well as theremaining present values) using the number of years. To do this, youneed the EAR. The EAR is:

EAR = (1 + .0083)12 – 1 = .1047 or 10.47%

So, we can find the PV at year 5 using the following method as well:

PV @ year 5: $53,776.72 / 1.10473 = $39,888.33

The value of the annuity at the other times in the problem is:

CHAPTER 6 B-158

PV @ year 3: $53,776.72 / 1.051110 = $32,684.88PV @ year 3: $53,776.72 / 1.10475 = $32,684.88

PV @ year 0: $53,776.72 / 1.051116 = $24,243.67PV @ year 0: $53,776.72 / 1.10478 = $24,243.67

53. a. If the payments are in the form of an ordinary annuity, the presentvalue will be:

PVA = C({1 – [1/(1 + r)t]} / r ))PVA = $10,000[{1 – [1 / (1 + .11)]5}/ .11]PVA = $36,958.97

B-159 SOLUTIONS

If the payments are an annuity due, the present value will be:

PVAdue = (1 + r) PVAPVAdue = (1 + .11)$36,958.97PVAdue = $41,024.46

b. We can find the future value of the ordinary annuity as:

FVA = C{[(1 + r)t – 1] / r}FVA = $10,000{[(1 + .11)5 – 1] / .11}FVA = $62,278.01

If the payments are an annuity due, the future value will be:

FVAdue = (1 + r) FVAFVAdue = (1 + .11)$62,278.01FVAdue = $69,128.60

c. Assuming a positive interest rate, the present value of an annuity due will always be larger than the present value of an ordinary annuity. Each cash flow in an annuity due is received one period earlier, which means there is one period less to discount each cash flow. Assuming a positive interest rate, the future value of an ordinary due will always higher than the future value of an ordinaryannuity. Since each cash flow is made one period sooner, each cash flow receives one extra period of compounding.

54. We need to use the PVA due equation, that is:

PVAdue = (1 + r) PVA

Using this equation:

PVAdue = $68,000 = [1 + (.0785/12)] × C[{1 – 1 / [1 + (.0785/12)]60} /(.0785/12)

$67,558.06 = $C{1 – [1 / (1 + .0785/12)60]} / (.0785/12)

C = $1,364.99

Notice, when we find the payment for the PVA due, we simply discountthe PV of the annuity due back one period. We then use this value asthe PV of an ordinary annuity.

CHAPTER 6 B-160

55. The payment for a loan repaid with equal payments is the annuity payment with the loan value as the PV of the annuity. So, the loan payment will be:

PVA = $42,000 = C {[1 – 1 / (1 + .08)5] / .08}

C = $10,519.17

The interest payment is the beginning balance times the interest rate for the period, and the principal payment is the total payment minus the interest payment. The ending balance is the beginning balance minusthe principal payment. The ending balance for a period is the beginningbalance for the next period. The amortization table for an equal payment is:

B-161 SOLUTIONS

YearBeginningBalance

TotalPayment

InterestPayment

PrincipalPayment

Ending Balance

1$42,000.0

0$10,519.1

7 $3,360.00 $7,159.17$34,840.8

32 34,840.83 10,519.17 2,787.27 7,731.90 27,108.923 27,108.92 10,519.17 2,168.71 8,350.46 18,758.474 18,758.47 10,519.17 1,500.68 9,018.49 9,739.975 9,739.97 10,519.17 779.20 9,739.97 0.00

In the third year, $2,168.71 of interest is paid.

Total interest over life of the loan = $3,360 + 2,787.27 + 2,168.71 + 1,500.68 + 779.20

Total interest over life of the loan = $10,595.86

56. This amortization table calls for equal principal payments of $8,400per year. The interest payment is the beginning balance times theinterest rate for the period, and the total payment is the principalpayment plus the interest payment. The ending balance for a period isthe beginning balance for the next period. The amortization table foran equal principal reduction is:

Year

Beginning B

alanceTotal

PaymentInterestPayment

PrincipalPayment

Ending Balance

1$42,000.0

0$11,760.0

0 $3,360.00 $8,400.00$33,600.0

02 33,600.00 11,088.00 2,688.00 8,400.00 25,200.003 25,200.00 10,416.00 2,016.00 8,400.00 16,800.004 16,800.00 9,744.00 1,344.00 8,400.00 8,400.005 8,400.00 9,072.00 672.00 8,400.00 0.00

In the third year, $2,016 of interest is paid.

Total interest over life of the loan = $3,360 + 2,688 + 2,016 + 1,344 +672 = $10,080

Notice that the total payments for the equal principal reduction loan are lower. This is because more principal is repaid early in the loan, which reduces the total interest expense over the life of the loan.

CHAPTER 6 B-162

Challenge

57. The cash flows for this problem occur monthly, and the interest rate given is the EAR. Since the cash flows occur monthly, we must get the effective monthly rate. One way to do this is to find the APR based on monthly compounding, and then divide by 12. So, the pre-retirement APR is:

EAR = .10 = [1 + (APR / 12)]12 – 1; APR = 12[(1.10)1/12 – 1] = .0957 or 9.57%

And the post-retirement APR is:

EAR = .07 = [1 + (APR / 12)]12 – 1; APR = 12[(1.07)1/12 – 1] = .0678 or 6.78%

B-163 SOLUTIONS

First, we will calculate how much he needs at retirement. The amount needed at retirement is the PV of the monthly spending plus the PV of the inheritance. The PV of these two cash flows is:

PVA = $20,000{1 – [1 / (1 + .0678/12)12(25)]} / (.0678/12) = $2,885,496.45

PV = $900,000 / [1 + (.0678/12)]300 = $165,824.26

So, at retirement, he needs:

$2,885,496.45 + 165,824.26 = $3,051,320.71

He will be saving $2,500 per month for the next 10 years until he purchases the cabin. The value of his savings after 10 years will be:

FVA = $2,500[{[ 1 + (.0957/12)]12(10) – 1} / (.0957/12)] = $499,659.64

After he purchases the cabin, the amount he will have left is:

$499,659.64 – 380,000 = $119,659.64

He still has 20 years until retirement. When he is ready to retire, this amount will have grown to:

FV = $119,659.64[1 + (.0957/12)]12(20) = $805,010.23

So, when he is ready to retire, based on his current savings, he will be short:

$3,051,320.71 – 805,010.23 = $2,246,310.48

This amount is the FV of the monthly savings he must make between years10 and 30. So, finding the annuity payment using the FVA equation, we find his monthly savings will need to be:

FVA = $2,246,310.48 = C[{[ 1 + (.1048/12)]12(20) – 1} / (.1048/12)] C = $3,127.44

58. To answer this question, we should find the PV of both options, and compare them. Since we are purchasing the car, the lowest PV is the best option. The PV of the leasing is simply the PV of the lease payments, plus the $99. The interest rate we would use for the leasing

CHAPTER 6 B-164

option is the same as the interest rate of the loan. The PV of leasing is:

PV = $99 + $450{1 – [1 / (1 + .07/12)12(3)]} / (.07/12) = $14,672.91

The PV of purchasing the car is the current price of the car minus the PV of the resale price. The PV of the resale price is:

PV = $23,000 / [1 + (.07/12)]12(3) = $18,654.82

The PV of the decision to purchase is:

$32,000 – 18,654.82 = $13,345.18

B-165 SOLUTIONS

In this case, it is cheaper to buy the car than leasing it since the PVof the purchase cash flows is lower. To find the breakeven resale price, we need to find the resale price that makes the PV of the two options the same. In other words, the PV of the decision to buy should be:

$32,000 – PV of resale price = $14,672.91PV of resale price = $17,327.09

The resale price that would make the PV of the lease versus buy decision is the FV of this value, so:

Breakeven resale price = $17,327.09[1 + (.07/12)]12(3) = $21,363.01

59. To find the quarterly salary for the player, we first need to find thePV of the current contract. The cash flows for the contract are annual,and we are given a daily interest rate. We need to find the EAR so theinterest compounding is the same as the timing of the cash flows. TheEAR is:

EAR = [1 + (.055/365)]365 – 1 = 5.65%

The PV of the current contract offer is the sum of the PV of the cash flows. So, the PV is:

PV = $7,000,000 + $4,500,000/1.0565 + $5,000,000/1.05652 + $6,000,000/1.05653 + $6,800,000/1.05654 + $7,900,000/1.05655 + $8,800,000/1.05656

PV = $38,610,482.57

The player wants the contract increased in value by $1,400,000, so thePV of the new contract will be:

PV = $38,610,482.57 + 1,400,000 = $40,010,482.57

The player has also requested a signing bonus payable today in theamount of $9 million. We can simply subtract this amount from the PV ofthe new contract. The remaining amount will be the PV of the futurequarterly paychecks.

$40,010,482.57 – 9,000,000 = $31,010,482.57

CHAPTER 6 B-166

To find the quarterly payments, first realize that the interest rate weneed is the effective quarterly rate. Using the daily interest rate, wecan find the quarterly interest rate using the EAR equation, with thenumber of days being 91.25, the number of days in a quarter (365 / 4).The effective quarterly rate is:

Effective quarterly rate = [1 + (.055/365)]91.25 – 1 = .01384 or 1.384%

Now we have the interest rate, the length of the annuity, and the PV.Using the PVA equation and solving for the payment, we get:

PVA = $31,010,482.57 = C{[1 – (1/1.01384)24] / .01384} C = $1,527,463.76

B-167 SOLUTIONS

60. To find the APR and EAR, we need to use the actual cash flows of theloan. In other words, the interest rate quoted in the problem is onlyrelevant to determine the total interest under the terms given. Thecash flows of the loan are the $25,000 you must repay in one year, andthe $21,250 you borrow today. The interest rate of the loan is:

$25,000 = $21,250(1 + r)r = ($25,000 / 21,250) – 1 = .1765 or 17.65%

Because of the discount, you only get the use of $21,250, and theinterest you pay on that amount is 17.65%, not 15%.

61. Here we have cash flows that would have occurred in the past and cashflows that would occur in the future. We need to bring both cash flowsto today. Before we calculate the value of the cash flows today, wemust adjust the interest rate so we have the effective monthly interestrate. Finding the APR with monthly compounding and dividing by 12 willgive us the effective monthly rate. The APR with monthly compoundingis:

APR = 12[(1.08)1/12 – 1] = .0772 or 7.72%

To find the value today of the back pay from two years ago, we willfind the FV of the annuity, and then find the FV of the lump sum. Doingso gives us:

FVA = ($47,000/12) [{[ 1 + (.0772/12)]12 – 1} / (.0772/12)] = $48,699.39FV = $48,699.39(1.08) = $52,595.34

Notice we found the FV of the annuity with the effective monthly rate,and then found the FV of the lump sum with the EAR. Alternatively, wecould have found the FV of the lump sum with the effective monthly rateas long as we used 12 periods. The answer would be the same either way.

Now, we need to find the value today of last year’s back pay:

FVA = ($50,000/12) [{[ 1 + (.0772/12)]12 – 1} / (.0772/12)] = $51,807.86

Next, we find the value today of the five year’s future salary:

PVA = ($55,000/12){[{1 – {1 / [1 + (.0772/12)]12(5)}] / (.0772/12)}=$227,539.14

CHAPTER 6 B-168

The value today of the jury award is the sum of salaries, plus thecompensation for pain and suffering, and court costs. The award shouldbe for the amount of:

Award = $52,595.34 + 51,807.86 + 227,539.14 + 100,000 + 20,000 =$451,942.34

As the plaintiff, you would prefer a lower interest rate. In thisproblem, we are calculating both the PV and FV of annuities. A lowerinterest rate will decrease the FVA, but increase the PVA. So, by alower interest rate, we are lowering the value of the back pay. But, weare also increasing the PV of the future salary. Since the futuresalary is larger and has a longer time, this is the more important cashflow to the plaintiff.

B-169 SOLUTIONS

62. Again, to find the interest rate of a loan, we need to look at the cashflows of the loan. Since this loan is in the form of a lump sum, theamount you will repay is the FV of the principal amount, which will be:

Loan repayment amount = $10,000(1.08) = $10,800

The amount you will receive today is the principal amount of the loantimes one minus the points.

Amount received = $10,000(1 – .03) = $9,700

Now, we simply find the interest rate for this PV and FV.

$10,800 = $9,700(1 + r) r = ($10,800 / $9,700) – 1 = .1134 or 11.34%

63. This is the same question as before, with different values. So:

Loan repayment amount = $10,000(1.11) = $11,100

Amount received = $10,000(1 – .02) = $9,800

$11,100 = $9,800(1 + r) r = ($11,100 / $9,800) – 1 = .1327 or 13.27%

The effective rate is not affected by the loan amount since it dropsout when solving for r.

64. First we will find the APR and EAR for the loan with the refundablefee. Remember, we need to use the actual cash flows of the loan to findthe interest rate. With the $2,300 application fee, you will need toborrow $242,300 to have $240,000 after deducting the fee. Solving forthe payment under these circumstances, we get:

PVA = $242,300 = C {[1 – 1/(1.005667)360]/.005667} where .005667 = .068/12C = $1,579.61

We can now use this amount in the PVA equation with the original amountwe wished to borrow, $240,000. Solving for r, we find:

PVA = $240,000 = $1,579.61[{1 – [1 / (1 + r)]360}/ r]

CHAPTER 6 B-170

Solving for r with a spreadsheet, on a financial calculator, or by trialand error, gives:

r = 0.5745% per month

APR = 12(0.5745%) = 6.89%

EAR = (1 + .005745)12 – 1 = 7.12%

B-171 SOLUTIONS

With the nonrefundable fee, the APR of the loan is simply the quotedAPR since the fee is not considered part of the loan. So:

APR = 6.80%

EAR = [1 + (.068/12)]12 – 1 = 7.02%

65. Be careful of interest rate quotations. The actual interest rate of aloan is determined by the cash flows. Here, we are told that the PV ofthe loan is $1,000, and the payments are $41.15 per month for threeyears, so the interest rate on the loan is:

PVA = $1,000 = $41.15[{1 – [1 / (1 + r)]36 } / r ]

Solving for r with a spreadsheet, on a financial calculator, or by trialand error, gives:

r = 2.30% per month

APR = 12(2.30%) = 27.61%

EAR = (1 + .0230)12 – 1 = 31.39%

It’s called add-on interest because the interest amount of the loan isadded to the principal amount of the loan before the loan payments arecalculated.

66. Here we are solving a two-step time value of money problem. Eachquestion asks for a different possible cash flow to fund the sameretirement plan. Each savings possibility has the same FV, that is, thePV of the retirement spending when your friend is ready to retire. Theamount needed when your friend is ready to retire is:

PVA = $105,000{[1 – (1/1.07)20] / .07} = $1,112,371.50

This amount is the same for all three parts of this question.

a. If your friend makes equal annual deposits into the account, this is an annuity with the FVA equal to the amount needed in retirement. Therequired savings each year will be:

FVA = $1,112,371.50 = C[(1.0730 – 1) / .07]C = $11,776.01

CHAPTER 6 B-172

b. Here we need to find a lump sum savings amount. Using the FV fora lump sum equation, we get:

FV = $1,112,371.50 = PV(1.07)30 PV = $146,129.04

B-173 SOLUTIONS

c. In this problem, we have a lump sum savings in addition to an annualdeposit. Since we already know the value needed at retirement, we cansubtract the value of the lump sum savings at retirement to find outhow much your friend is short. Doing so gives us:

FV of trust fund deposit = $150,000(1.07)10 = $295,072.70

So, the amount your friend still needs at retirement is:

FV = $1,112,371.50 – 295,072.70 = $817,298.80

Using the FVA equation, and solving for the payment, we get:

$817,298.80 = C[(1.07 30 – 1) / .07]

C = $8,652.25

This is the total annual contribution, but your friend’s employerwill contribute $1,500 per year, so your friend must contribute:

Friend's contribution = $8,652.25 – 1,500 = $7,152.25

67. We will calculate the number of periods necessary to repay the balance with no fee first. We simply need to use the PVA equation and solve for the number of payments.

Without fee and annual rate = 19.80%:

PVA = $10,000 = $200{[1 – (1/1.0165)t ] / .0165 } where .0165 = .198/12


1/1.0165t = 1 – ($10,000/$200)(.0165)1/1.0165t = .175t = ln (1/.175) / ln 1.0165t = 106.50 months

Without fee and annual rate = 6.20%:

PVA = $10,000 = $200{[1 – (1/1.005167)t ] / .005167 } where .005167 =.062/12

CHAPTER 6 B-174


1/1.005167t = 1 – ($10,000/$200)(.005167)1/1.005167t = .7417t = ln (1/.7417) / ln 1.005167t = 57.99 months

Note that we do not need to calculate the time necessary to repay your current credit card with a fee since no fee will be incurred. The time to repay the new card with a transfer fee is:

B-175 SOLUTIONS

With fee and annual rate = 6.20%:

PVA = $10,200 = $200{ [1 – (1/1.005167)t ] / .005167 } where .005167 = .082/12


1/1.005167t = 1 – ($10,200/$200)(.005167)1/1.005167t = .7365t = ln (1/.7365) / ln 1.005167t = 59.35 months

68. We need to find the FV of the premiums to compare with the cash paymentpromised at age 65. We have to find the value of the premiums at year 6first since the interest rate changes at that time. So:

FV1 = $900(1.12)5 = $1,586.11

FV2 = $900(1.12)4 = $1,416.17

FV3 = $1,000(1.12)3 = $1,404.93

FV4 = $1,000(1.12)2 = $1,254.40

FV5 = $1,100(1.12)1 = $1,232.00

Value at year six = $1,586.11 + 1,416.17 + 1,404.93 + 1,254.40 + 1,232.00 + 1,100 Value at year six = $7,993.60

Finding the FV of this lump sum at the child’s 65th birthday:

FV = $7,993.60(1.08)59 = $749,452.56

The policy is not worth buying; the future value of the deposits is$749,452.56, but the policy contract will pay off $500,000. Thepremiums are worth $249,452.56 more than the policy payoff.

Note, we could also compare the PV of the two cash flows. The PV of thepremiums is:

PV = $900/1.12 + $900/1.122 + $1,000/1.123 + $1,000/1.124 + $1,100/1.125 + $1,100/1.126

CHAPTER 6 B-176

PV = $4,049.81

And the value today of the $500,000 at age 65 is:

PV = $500,000/1.0859 = $5,332.96

PV = $5,332.96/1.126 = $2,701.84

The premiums still have the higher cash flow. At time zero, thedifference is $1,347.97. Whenever you are comparing two or more cashflow streams, the cash flow with the highest value at one time willhave the highest value at any other time.

Here is a question for you: Suppose you invest $1,347.97, thedifference in the cash flows at time zero, for six years at a 12percent interest rate, and then for 59 years at an 8 percent interestrate. How much will it be worth? Without doing calculations, you knowit will be worth $249,452.56, the difference in the cash flows at time65!

69. The monthly payments with a balloon payment loan are calculatedassuming a longer amortization schedule, in this case, 30 years. Thepayments based on a 30-year repayment schedule would be:

PVA = $750,000 = C({1 – [1 / (1 + .081/12)]360} / (.081/12)) C = $5,555.61

Now, at time = 8, we need to find the PV of the payments which have notbeen made. The balloon payment will be:

PVA = $5,555.61({1 – [1 / (1 + .081/12)]12(22)} / (.081/12)) PVA = $683,700.32

70. Here we need to find the interest rate that makes the PVA, the collegecosts, equal to the FVA, the savings. The PV of the college costs are:

PVA = $20,000[{1 – [1 / (1 + r)4]} / r ]

And the FV of the savings is:

FVA = $9,000{[(1 + r)6 – 1 ] / r }

Setting these two equations equal to each other, we get:

B-177 SOLUTIONS

$20,000[{1 – [1 / (1 + r)]4 } / r ] = $9,000{[ (1 + r)6 – 1 ] / r }

Reducing the equation gives us:

(1 + r)6 – 11,000(1 + r)4 + 29,000 = 0

Now we need to find the roots of this equation. We can solve usingtrial and error, a root-solving calculator routine, or a spreadsheet.Using a spreadsheet, we find:

r = 8.07%

71. Here we need to find the interest rate that makes us indifferentbetween an annuity and a perpetuity. To solve this problem, we need tofind the PV of the two options and set them equal to each other. The PVof the perpetuity is:

PV = $20,000 / r

And the PV of the annuity is:

PVA = $28,000[{1 – [1 / (1 + r)]20 } / r ]

CHAPTER 6 B-178

Setting them equal and solving for r, we get:

$20,000 / r = $28,000[ {1 – [1 / (1 + r)]20 } / r ]$20,000 / $28,000 = 1 – [1 / (1 + r)]20 .71431/20 = 1 / (1 + r) r = .0646 or 6.46%

72. The cash flows in this problem occur every two years, so we need tofind the effective two year rate. One way to find the effective twoyear rate is to use an equation similar to the EAR, except use thenumber of days in two years as the exponent. (We use the number of daysin two years since it is daily compounding; if monthly compounding wasassumed, we would use the number of months in two years.) So, theeffective two-year interest rate is:

Effective 2-year rate = [1 + (.10/365)]365(2) – 1 = .2214 or 22.14%

We can use this interest rate to find the PV of the perpetuity. Doing so, we find:

PV = $15,000 /.2214 = $67,760.07

This is an important point: Remember that the PV equation for aperpetuity (and an ordinary annuity) tells you the PV one period beforethe first cash flow. In this problem, since the cash flows are twoyears apart, we have found the value of the perpetuity one period (twoyears) before the first payment, which is one year ago. We need tocompound this value for one year to find the value today. The value ofthe cash flows today is:

PV = $67,760.07(1 + .10/365)365 = $74,885.44

The second part of the question assumes the perpetuity cash flows beginin four years. In this case, when we use the PV of a perpetuityequation, we find the value of the perpetuity two years from today. So,the value of these cash flows today is:

PV = $67,760.07 / (1 + .2214) = $55,478.78

73. To solve for the PVA due:

B-179 SOLUTIONS

PVA = C

(1 +r)+

C(1 +r)2

+....+C

(1 +r)t

PVAdue = C+

C(1 +r )

+....+C

(1 +r )t - 1

PVAdue = (1 +r)( C

(1 +r )+

C(1 +r )2

+....+C

(1 +r )t )

PVAdue = (1 + r) PVA

And the FVA due is:

FVA = C + C(1 + r) + C(1 + r)2 + …. + C(1 + r)t – 1

FVAdue = C(1 + r) + C(1 + r)2 + …. + C(1 + r)t

FVAdue = (1 + r)[C + C(1 + r) + …. + C(1 + r)t – 1]FVAdue = (1 + r)FVA

CHAPTER 6 B-180

74. We need to find the lump sum payment into the retirement account. Thepresent value of the desired amount at retirement is:

PV = FV/(1 + r)t

PV = $2,000,000/(1 + .11)40

PV = $30,768.82

This is the value today. Since the savings are in the form of a growingannuity, we can use the growing annuity equation and solve for thepayment. Doing so, we get:

PV = C {[1 – ((1 + g)/(1 + r))t ] / (r – g)} $30,768.82 = C{[1 – ((1 + .03)/(1 + .11))40 ] / (.11 – .03)}C = $2,591.56

This is the amount you need to save next year. So, the percentage ofyour salary is:

Percentage of salary = $2,591.56/$40,000Percentage of salary = .0648 or 6.48%

Note that this is the percentage of your salary you must save eachyear. Since your salary is increasing at 3 percent, and the savings areincreasing at 3 percent, the percentage of salary will remain constant.

75. a. The APR is the interest rate per week times 52 weeks in a year, so:

APR = 52(7%) = 364%

EAR = (1 + .07)52 – 1 = 32.7253 or 3,273.53%

b. In a discount loan, the amount you receive islowered by the discount, and you repay the full principal. With a 7percent discount, you would receive $9.30 for every $10 in principal,so the weekly interest rate would be:

$10 = $9.30(1 + r)r = ($10 / $9.30) – 1 = .0753 or 7.53%

Note the dollar amount we use is irrelevant. In other words, we coulduse $0.93 and $1, $93 and $100, or any other combination and we wouldget the same interest rate. Now we can find the APR and the EAR:

B-181 SOLUTIONS

APR = 52(7.53%) = 391.40%

EAR = (1 + .0753)52 – 1 = 42.5398 or 4,253.98%

CHAPTER 6 B-182

c.Using the cash flows from the loan, we have the PVA and the annuitypayments and need to find the interest rate, so:

PVA = $68.92 = $25[{1 – [1 / (1 + r)]4}/ r ]

Using a spreadsheet, trial and error, or a financial calculator, we find:

r = 16.75% per week

APR = 52(16.75%) = 870.99%

EAR = 1.167552 – 1 = 3141.7472 or 314,174.72%

76. To answer this, we need to diagram the perpetuity cash flows, whichare: (Note, the subscripts are only to differentiate when the cashflows begin. The cash flows are all the same amount.)

…..C3

C2 C2

C1 C1 C1

Thus, each of the increased cash flows is a perpetuity in itself. So,we can write the cash flows stream as:

C1/R C2/R C3/R C4/R ….

So, we can write the cash flows as the present value of a perpetuity,and a perpetuity of:

C2/R C3/R C4/R ….

The present value of this perpetuity is:

B-183 SOLUTIONS

PV = (C/R) / R = C/R2

So, the present value equation of a perpetuity that increases by C eachperiod is:

PV = C/R + C/R2

CHAPTER 6 B-184

77. We are only concerned with the time it takes money to double, so thedollar amounts are irrelevant. So, we can write the future value of alump sum as:

FV = PV(1 + R)t

$2 = $1(1 + R)t

Solving for t, we find:

ln(2) = t[ln(1 + R)]t = ln(2) / ln(1 + R)

Since R is expressed as a percentage in this case, we can write theexpression as:

t = ln(2) / ln(1 + R/100)

To simplify the equation, we can make use of a Taylor Series expansion:

ln(1 + R) = R – R2/2 + R3/3 – ...

Since R is small, we can truncate the series after the first term:

ln(1 + R) = R

Combine this with the solution for the doubling expression:

t = ln(2) / (R/100)t = 100ln(2) / Rt = 69.3147 / R

This is the exact (approximate) expression, Since 69.3147 is not easilydivisible, and we are only concerned with an approximation, 72 issubstituted.

78. We are only concerned with the time it takes money to double, so thedollar amounts are irrelevant. So, we can write the future value of alump sum with continuously compounded interest as:

$2 = $1eRt2 = eRtRt = ln(2)Rt = .693147

B-185 SOLUTIONS

t = .693147 / R

Since we are using interest rates while the equation uses decimal form,to make the equation correct with percentages, we can multiply by 100:

t = 69.3147 / R

CHAPTER 6 B-186


1.CFo $0 CFo $0 CFo $0C01 $950 C01 $950 C01 $950F01 1 F01 1 F01 1C02 $1,040 C02 $1,040 C02 $1,040F02 1 F02 1 F02 1C03 $1,130 C03 $1,130 C03 $1,130F03 1 F03 1 F03 1C04 $1,075 C04 $1,075 C04 $1,075F04 1 F04 1 F04 1

I = 10 I = 18 I = 24NPV CPT NPV CPT NPV CPT$3,306.37 $2,794.22 $2,489.88

2.Enter 9 5% $6,000


$42,646.93


Solve for

$40,605.54


Solve for

$28,629.50


Solve for

$30,275.86

3.Enter 3 8% $940

N I/Y PV PMT FV

B-187 SOLUTIONS

Solve for

$1,184.13


Solve for

$1,271.38


Solve for

$1,447.20

FV = $1,184.13 + 1,271.38 + 1,447.20 + 1,405 = $5,307.71

CHAPTER 6 B-188

Enter 3 11% $940N I/Y PV PMT FV

Solve for

$1,285.57


Solve for

$1,342.99


Solve for

$1,487.40

FV = $1,285.57 + 1,342.99 + 1,487.40 + 1,405 = $5,520.96


Solve for

$1,792.23


Solve for

$1,675.98


Solve for

$1,661.60

FV = $1,792.23 + 1,675.98 + 1,661.60 + 1,405 = $6,534.81

4.Enter 15 7% $5,300

N I/Y PV PMT FVSolve $48,271.94

B-189 SOLUTIONS

for


Solve for

$70,658.06


Solve for

$75,240.70

CHAPTER 6 B-190

5.Enter 15 7.65% $34,000


$3,887.72

6.Enter 8 8.5% $73,000


$411,660.36

7.Enter 20 11.2% $4,000


$262,781.16

Enter 40 11.2% $4,000N I/Y PV PMT FV

Solve for

$2,459,072.63

8.Enter 10 6.8% $90,000


$6,575.77

9.Enter 7 7.5% $70,000


$9,440.02

12.Enter 8% 4

NOM EFF C/YSolve for

8.24%

B-191 SOLUTIONS

Enter 16% 12NOM EFF C/Y

Solve for

17.23%


Solve for

12.75%

13.Enter 8.6% 2


8.24%

CHAPTER 6 B-192

Enter 19.8% 12NOM EFF C/Y

Solve for

18.20%


Solve for

8.99%

14.Enter 14.2% 12


15.16%


Solve for

15.03%

15.Enter 16% 365


14.85%

16.Enter 17 × 2 8.4%/2 $2,100


$8,505.93

17.Enter 5 365 9.3% / 365 $4,500


$7,163.64

Enter 10 365 9.3% / 365 $4,500

B-193 SOLUTIONS


$11,403.94

Enter 20 365 9.3% / 365 $4,500N I/Y PV PMT FV

Solve for

$28,899.97

18.Enter 7 365 10% / 365 $58,000


$28,804.71

CHAPTER 6 B-194

19.Enter 360% 12


2,229.81%

20.Enter 60 6.9% / 12 $68,500


$1,353.15


Solve for

7.12%

21.Enter 1.3% $18,000 $500


48.86

22.Enter 1,733.33% 52


313,916,515.69%

23.Enter 22.74% 12


25.26%

24.Enter 30 12 10% / 12 $300


$678,146.38

B-195 SOLUTIONS

25.Enter 10.00% 12


10.47%


Solve for

$647,623.45

26.Enter 4 4 0.65% $2,300


$34,843.71

CHAPTER 6 B-196

27.Enter 11.00% 4


11.46%

CFo $0C01 $725F01 1C02 $980F02 1C03 $0F03 1C04 $1,360F04 1

I = 11.46%NPV CPT$2,320.36

28.CFo $0C01 $1,650F01 1C02 $0F02 1C03 $4,200F03 1C04 $2,430F04 1

I = 8.45%NPV CPT$6,570.86

30.Enter 17% 2


16.33%

16.33% / 2 = 8.17%

B-197 SOLUTIONS


Solve for

16.01%

16.01% / 4 = 4.00%


Solve for

15.80%

15.80% / 12 = 1.32%

CHAPTER 6 B-198

31.Enter 6 1.50% / 12 $5,000


$5,037.62

Enter 6 18% / 12 $5,037.62N I/Y PV PMT FV

Solve for

$5,508.35

$5,508.35 – 5,000 = $508.35

32. Stock account:

Enter 360 11% / 12 $700N I/Y PV PMT FV

Solve for

$1,963,163.82

Bond account:

Enter 360 6% / 12 $300N I/Y PV PMT FV

Solve for

$301,354.51

Savings at retirement = $1,963,163.82 + 301,354.51 = $2,264,518.33

Enter 300 9% / 12 $2,264,518.33


$19,003.76

33.Enter 12 1.17% $1


$1.15

B-199 SOLUTIONS

Enter 24 1.17% $1N I/Y PV PMT FV

Solve for

$1.32

34.Enter 480 12% / 12 $1,000,000


$85.00

Enter 360 12% / 12 $1,000,000N I/Y PV PMT FV

Solve for

$286.13

CHAPTER 6 B-200

Enter 240 12% / 12 $1,000,000N I/Y PV PMT FV

Solve for

$1,010.86

35.Enter 12 / 3 $1 $3


31.61%

36.Enter 10.00% 12


10.47%


Solve for

$163,839.09

CFo $45,000C01 $75,000F01 2

I = 10.47%NPV CPT$165,723.94

39.Enter 15 10% $9,000


$68,454.72


Solve $93,426.92

B-201 SOLUTIONS

for


Solve for

$52,626.33

40.Enter 6% / 12 $340 $20,000


51.69

CHAPTER 6 B-202

41.Enter 60 $73,000 $1,450


0.594%

0.594% 12 = 7.13%

42.Enter 360 6.35% / 12 $1,150


$184,817.42

$240,000 – 184,817.42 = $55,182.58

Enter 360 6.35% / 12 $55,182.58N I/Y PV PMT FV

Solve for

$368,936.54

43.

CFo $0C01 $1,700F01 1C02 $0F02 1C03 $2,100F03 1C04 $2,800F04 1

I = 10%NPV CPT$5,035.65

PV of missing CF = $6,550 – 5,035.65 = $1,514.35Value of missing CF:

Enter 2 10% $1,514.35N I/Y PV PMT FV

Solve for

$1,832.36

B-203 SOLUTIONS

CHAPTER 6 B-204

44.

CFo $1,000,000

C01 $1,500,000

F01 1C02 $2,500,00

0F02 1C03 $2,800,00

0F03 1C04 $3,000,00

0F04 1C05 $3,500,00

0F05 1C06 $4,000,00

0F06 1C07 $4,500,00

0F07 1C08 $5,000,00

0F08 1C09 $5,500,00

0F09 1C010 $6,000,00

0I = 9%NPV CPT$22,812,873

45.Enter 360 .80($2,900,

000)$15,000

N I/Y PV PMT FVSolve 0.560%

B-205 SOLUTIONS

for

APR = 0.560% 12 = 6.72%


Solve for

6.93%

46.Enter 4 13% $165,000


$101,197.59

Profit = $101,197.59 – 94,000 = $7,197.59


Solve for

15.10%

CHAPTER 6 B-206

47.Enter 18 10% $4,000


$32,805.65


Solve for

$16,834.48

48.Enter 84 7% / 12 $1,500


$87,604.36

Enter 96 11% / 12 $1,500N I/Y PV PMT FV

Solve for

$110,021.35

Enter 84 7% / 12 $110,021.35


$51,120.33

$87,604.36 + 51,120.33 = $138,724.68

49.Enter 15 × 12 8.5%/12 $1,200


$434,143.62

FV = $434,143.62 = PV e.08(15); PV = $434,143.62e–1.20 = $130,761.55

50. PV@ t = 14: $3,500 / 0.062 = $56,451.61

B-207 SOLUTIONS

Enter 7 6.2% $56,451.61N I/Y PV PMT FV

Solve for

$37,051.41

51.Enter 12 $25,000 $2,416.67


2.361%

APR = 2.361% 12 = 28.33%


Solve for

32.31%

CHAPTER 6 B-208

52. Monthly rate = .10 / 12 = .0083; semiannual rate = (1.0083)6 – 1 = 5.11%


Solve for

$53,776.72


Solve for

$39,888.33


Solve for

$32,684.88


Solve for

$24,243.67

53.a.Enter 5 11% $10,000


$36,958.97

2nd BGN 2nd SET


Solve for

$41,024.46

b.Enter 5 11% $10,000

N I/Y PV PMT FV

B-209 SOLUTIONS

Solve for

$62,278.01

2nd BGN 2nd SET


Solve for

$69,128.60

CHAPTER 6 B-210

54. 2nd BGN 2nd SET

Enter 60 7.85% / 12 $68,000N I/Y PV PMT FV

Solve for

$1,364.99

57. Pre-retirement APR:


Solve for

9.57%

Post-retirement APR:


Solve for

6.78%

At retirement, he needs:

Enter 300 6.78% / 12 $20,000 $900,000N I/Y PV PMT FV

Solve for

$3,051,320.71

In 10 years, his savings will be worth:

Enter 120 7.72% / 12 $2,500N I/Y PV PMT FV

Solve for

$499,659.64

After purchasing the cabin, he will have: $499,659.64 – 380,000 = $119,659.64

Each month between years 10 and 30, he needs to save:

Enter 240 9.57% / 12 $119,659.64 $3,051,320.71±

N I/Y PV PMT FV

B-211 SOLUTIONS

Solve for

$3,127.44

58. PV of purchase:Enter 36 7% / 12 $23,000


$18,654.82

$32,000 – 18,654.82 = $13,345.18

CHAPTER 6 B-212

PV of lease:Enter 36 7% / 12 $450


$14,573.99

$14,573.91 + 99 = $14,672.91Buy the car.

You would be indifferent when the PV of the two cash flows are equal.The present value of the purchase decision must be $14,672.91. Sincethe difference in the two cash flows is $32,000 – 14,672.91 =$17,327.09, this must be the present value of the future resale priceof the car. The break-even resale price of the car is:

Enter 36 7% / 12 $17,327.09N I/Y PV PMT FV

Solve for

$21,363.01

59.Enter 5.50% 365


5.65%

CFo $7,000,000

C01 $4,500,000

F01 1C02 $5,000,00

0F02 1C03 $6,000,00

0F03 1C04 $6,800,00

0F04 1C05 $7,900,00

0F05 1

B-213 SOLUTIONS

C06 $8,800,000

F06 1I = 5.65%NPV CPT$38,610,482.57

New contract value = $38,610,482.57 + 1,400,000 = $40,010,482.57

PV of payments = $40,010,482.57 – 9,000,000 = $31,010,482.57Effective quarterly rate = [1 + (.055/365)]91.25 – 1 = .01384 or 1.384%

Enter 24 1.384% $31,010,482.57


$1,527,463.76

CHAPTER 6 B-214

60.Enter 1 $21,250 $25,000


17.65%

61.Enter 8% 12


7.72%

Enter 12 7.72% / 12 $47,000 /12


$48,699.39


Solve for

$52,595.34

Enter 12 7.72% / 12 $50,000 /12


$51,807.86

Enter 60 7.72% / 12 $55,000 /12


$227,539.14

Award = $52,595.34 + 51,807.86 + 227,539.14 + 100,000 + 20,000 = $451,942.34

B-215 SOLUTIONS

62.Enter 1 $9,700 $10,800


11.34%

63.Enter 1 $9,800 $11,200


14.29%

64. Refundable fee: With the $2,300 application fee, you will need to borrow $242,300 to have

$240,000 after deducting the fee. Solve for the payment under these circumstances.

Enter 30 12 6.80% / 12 $242,300N I/Y PV PMT FV

Solve for

$1,579.61

CHAPTER 6 B-216

Enter 30 12 $240,000 $1,579.61N I/Y PV PMT FV

Solve for

0.5745%

APR = 0.5745% 12 = 6.89%


Solve for

7.12%

Without refundable fee: APR = 6.80%


Solve for

7.02%

65.Enter 36 $1,000 $41.15


2.30%

APR = 2.30% 12 = 27.61%


Solve for

31.39%

66. What she needs at age 65:


Solve for

$1,112,371.50

a.Enter 30 7% $1,112,371

B-217 SOLUTIONS

.50N I/Y PV PMT FV

Solve for

$11,776.01

b.Enter 30 7% $1,112,371

.50N I/Y PV PMT FV

Solve for

$146,129.04

c.Enter 10 7% $150,000


$295,072.70

CHAPTER 6 B-218

At 65, she is short: $1,112,371.50 – 295,072.50 = $817,298.80

Enter 30 7% ±$817,298.8

0N I/Y PV PMT FV

Solve for

$8,652.25

Her employer will contribute $1,500 per year, so she must contribute:

$8,652.25 – 1,500 = $7,152.25 per year

67. Without fee:

Enter 19.8% / 12 $10,000 $200N I/Y PV PMT FV

Solve for

106.50


Solve for

57.99

With fee:


Solve for

59.35

68. Value at Year 6:


Solve for

$1,586.11


B-219 SOLUTIONS

Solve for

$1,416.17


Solve for

$1,404.93


Solve for

$1,254.40

CHAPTER 6 B-220


Solve for

$1,232

So, at Year 5, the value is: $1,586.11 + 1,416.17 + 1,404.93 + 1,254.40 + 1,232

+ 1,100 = $7,993.60At Year 65, the value is:


Solve for

$749,452.56

The policy is not worth buying; the future value of the deposits is $749,452.56 but the policy

contract will pay off $500,000.

69.Enter 30 12 8.1% / 12 $750,000


$5,555.61

Enter 22 12 8.1% / 12 $5,555.61N I/Y PV PMT FV

Solve for

$683,700.32

70.CFo $9,000C01 $9,000F01 5C02 $20,000F02 4

IRR CPT8.07%

75. a. APR = 7% 52 = 364%

B-221 SOLUTIONS


Solve for

3,272.53%

b.Enter 1 $9.30 $10.00


7.53%

CHAPTER 6 B-222

APR = 7.53% 52 = 391.40%


Solve for

4,253.98%

c.Enter 4 $68.92 $25


16.75%

APR = 16.75% 52 = 870.99%


Solve for

314,174.72%

CHAPTER 7INTEREST RATES AND BOND VALUATION


1. No. As interest rates fluctuate, the value of a Treasury securitywill fluctuate. Long-term Treasury securities have substantialinterest rate risk.

2. All else the same, the Treasury security will have lower couponsbecause of its lower default risk, so it will have greater interestrate risk.

3. No. If the bid price were higher than the ask price, theimplication would be that a dealer was willing to sell a bond andimmediately buy it back at a higher price. How many suchtransactions would you like to do?

4. Prices and yields move in opposite directions. Since the bid pricemust be lower, the bid yield must be higher.

5. There are two benefits. First, the company can take advantage ofinterest rate declines by calling in an issue and replacing it witha lower coupon issue. Second, a company might wish to eliminate acovenant for some reason. Calling the issue does this. The cost tothe company is a higher coupon. A put provision is desirable froman investor’s standpoint, so it helps the company by reducing thecoupon rate on the bond. The cost to the company is that it mayhave to buy back the bond at an unattractive price.

6. Bond issuers look at outstanding bonds of similar maturity andrisk. The yields on such bonds are used to establish the couponrate necessary for a particular issue to initially sell for parvalue. Bond issuers also simply ask potential purchasers whatcoupon rate would be necessary to attract them. The coupon rate isfixed and simply determines what the bond’s coupon payments will

CHAPTER 7 B-224

be. The required return is what investors actually demand on theissue, and it will fluctuate through time. The coupon rate andrequired return are equal only if the bond sells for exactly atpar.

7. Yes. Some investors have obligations that are denominated indollars; i.e., they are nominal. Their primary concern is that aninvestment provide the needed nominal dollar amounts. Pensionfunds, for example, often must plan for pension payments many yearsin the future. If those payments are fixed in dollar terms, then itis the nominal return on an investment that is important.

8. Companies pay to have their bonds rated simply because unratedbonds can be difficult to sell; many large investors are prohibitedfrom investing in unrated issues.

9. Treasury bonds have no credit risk since it is backed by the U.S.government, so a rating is not necessary. Junk bonds often are notrated because there would be no point in an issuer paying a ratingagency to assign its bonds a low rating (it’s like paying someoneto kick you!).

B-225 SOLUTIONS

10. The term structure is based on pure discount bonds. The yield curveis based on coupon-bearing issues.

11. Bond ratings have a subjective factor to them. Split ratingsreflect a difference of opinion among credit agencies.

12. As a general constitutional principle, the federal governmentcannot tax the states without their consent if doing so wouldinterfere with state government functions. At one time, thisprinciple was thought to provide for the tax-exempt status ofmunicipal interest payments. However, modern court rulings make itclear that Congress can revoke the municipal exemption, so the onlybasis now appears to be historical precedent. The fact that thestates and the federal government do not tax each other’ssecurities is referred to as “reciprocal immunity.”

13. Lack of transparency means that a buyer or seller can’t see recenttransactions, so it is much harder to determine what the best bidand ask prices are at any point in time.

14. Companies charge that bond rating agencies are pressuring them topay for bond ratings. When a company pays for a rating, it has theopportunity to make its case for a particular rating. With anunsolicited rating, the company has no input.

15. A 100-year bond looks like a share of preferred stock. Inparticular, it is a loan with a life that almost certainly exceedsthe life of the lender, assuming that the lender is an individual.With a junk bond, the credit risk can be so high that the borroweris almost certain to default, meaning that the creditors are verylikely to end up as part owners of the business. In both cases, the“equity in disguise” has a significant tax advantage.



Basic

CHAPTER 7 B-226

1. The yield to maturity is the required rate of return on a bondexpressed as a nominal annual interest rate. For noncallable bonds,the yield to maturity and required rate of return areinterchangeable terms. Unlike YTM and required return, the couponrate is not a return used as the interest rate in bond cash flowvaluation, but is a fixed percentage of par over the life of thebond used to set the coupon payment amount. For the example given,the coupon rate on the bond is still 10 percent, and the YTM is 8percent.

2. Price and yield move in opposite directions; if interest ratesrise, the price of the bond will fall. This is because the fixedcoupon payments determined by the fixed coupon rate are not asvaluable when interest rates rise—hence, the price of the bonddecreases.

B-227 SOLUTIONS

NOTE: Most problems do not explicitly list a par value for bonds. Even though a bond can haveany par value, in general, corporate bonds in the United States will have a par value of $1,000.We will use this par value in all problems unless a different par value is explicitly stated.

3. The price of any bond is the PV of the interest payment, plus thePV of the par value. Notice this problem assumes an annual coupon.The price of the bond will be:

P = $75({1 – [1/(1 + .0875)]10 } / .0875) + $1,000[1 / (1 + .0875)10]= $918.89

We would like to introduce shorthand notation here. Rather thanwrite (or type, as the case may be) the entire equation for the PVof a lump sum, or the PVA equation, it is common to abbreviate theequations as:

PVIFR,t = 1 / (1 + r)t

which stands for Present Value Interest Factor

PVIFAR,t = ({1 – [1/(1 + r)]t } / r )

which stands for Present Value Interest Factor of an Annuity

These abbreviations are short hand notation for the equations inwhich the interest rate and the number of periods are substitutedinto the equation and solved. We will use this shorthand notationin remainder of the solutions key.

4. Here we need to find the YTM of a bond. The equation for the bondprice is:

P = $934 = $90(PVIFAR%,9) + $1,000(PVIFR%,9)

Notice the equation cannot be solved directly for R. Using aspreadsheet, a financial calculator, or trial and error, we find:

R = YTM = 10.15%

If you are using trial and error to find the YTM of the bond, youmight be wondering how to pick an interest rate to start theprocess. First, we know the YTM has to be higher than the couponrate since the bond is a discount bond. That still leaves a lot of

CHAPTER 7 B-228

interest rates to check. One way to get a starting point is to usethe following equation, which will give you an approximation of theYTM:

Approximate YTM = [Annual interest payment + (Price difference frompar / Years to maturity)] /

[(Price + Par value) / 2]

Solving for this problem, we get:

Approximate YTM = [$90 + ($64 / 9] / [($934 + 1,000) / 2] = 10.04%

This is not the exact YTM, but it is close, and it will give you aplace to start.

B-229 SOLUTIONS

5. Here we need to find the coupon rate of the bond. All we need to dois to set up the bond pricing equation and solve for the couponpayment as follows:

P = $1,045 = C(PVIFA7.5%,13) + $1,000(PVIF7.5%,36)

Solving for the coupon payment, we get:

C = $80.54

The coupon payment is the coupon rate times par value. Using thisrelationship, we get:

Coupon rate = $80.54 / $1,000 = .0805 or 8.05%

6. To find the price of this bond, we need to realize that thematurity of the bond is 10 years. The bond was issued one year ago,with 11 years to maturity, so there are 10 years left on the bond.Also, the coupons are semiannual, so we need to use the semiannualinterest rate and the number of semiannual periods. The price ofthe bond is:

P = $34.50(PVIFA3.7%,20) + $1,000(PVIF3.7%,20) = $965.10

7. Here we are finding the YTM of a semiannual coupon bond. The bondprice equation is:

P = $1,050 = $42(PVIFAR%,20) + $1,000(PVIFR%,20)

Since we cannot solve the equation directly for R, using aspreadsheet, a financial calculator, or trial and error, we find:

R = 3.837%

Since the coupon payments are semiannual, this is the semiannualinterest rate. The YTM is the APR of the bond, so:

YTM = 2 3.837% = 7.67%

8. Here we need to find the coupon rate of the bond. All we need to dois to set up the bond pricing equation and solve for the couponpayment as follows:

CHAPTER 7 B-230

P = $924 = C(PVIFA3.4%,29) + $1,000(PVIF3.4%,29)

Solving for the coupon payment, we get:

C = $29.84

Since this is the semiannual payment, the annual coupon payment is:

2 × $29.84 = $59.68

And the coupon rate is the annual coupon payment divided by parvalue, so:

Coupon rate = $59.68 / $1,000 Coupon rate = .0597 or 5.97%

B-231 SOLUTIONS

9. The approximate relationship between nominal interest rates (R),real interest rates (r), and inflation (h) is:

R = r + h

Approximate r = .07 – .038 =.032 or 3.20%

The Fisher equation, which shows the exact relationship betweennominal interest rates, real interest rates, and inflation is:

(1 + R) = (1 + r)(1 + h)

(1 + .07) = (1 + r)(1 + .038)

Exact r = [(1 + .07) / (1 + .038)] – 1 = .0308 or 3.08%

10. The Fisher equation, which shows the exact relationship betweennominal interest rates, real interest rates, and inflation is:

(1 + R) = (1 + r)(1 + h)

R = (1 + .047)(1 + .03) – 1 = .0784 or 7.84%


(1 + R) = (1 + r)(1 + h)

h = [(1 + .14) / (1 + .09)] – 1 = .0459 or 4.59%


(1 + R) = (1 + r)(1 + h)

r = [(1 + .114) / (1.048)] – 1 = .0630 or 6.30%

13. This is a bond since the maturity is greater than 10 years. Thecoupon rate, located in the first column of the quote is 6.125%.The bid price is:

Bid price = 120:07 = 120 7/32 = 120.21875% $1,000 = $1,202.1875

CHAPTER 7 B-232

The previous day’s ask price is found by:

Previous day’s asked price = Today’s asked price – Change = 1208/32 – (5/32) = 120 3/32

The previous day’s price in dollars was:

Previous day’s dollar price = 120.406% $1,000 = $1,204.06

B-233 SOLUTIONS

14. This is a premium bond because it sells for more than 100% of facevalue. The current yield is:

Current yield = Annual coupon payment / Price = $75/$1,351.5625 =5.978%

The YTM is located under the “Asked Yield” column, so the YTM is4.47%.

The bid-ask spread is the difference between the bid price and theask price, so:

Bid-Ask spread = 135:06 – 135:05 = 1/32

Intermediate

15. Here we are finding the YTM of semiannual coupon bonds forvarious maturity lengths. The bond price equation is:

P = C(PVIFAR%,t) + $1,000(PVIFR%,t)

X: P0 = $80(PVIFA6%,13) + $1,000(PVIF6%,13) = $1,177.05P1 = $80(PVIFA6%,12) + $1,000(PVIF6%,12) = $1,167.68P3 = $80(PVIFA6%,10) + $1,000(PVIF6%,10) = $1,147.20P8 = $80(PVIFA6%,5) + $1,000(PVIF6%,5) = $1,084.25P12 = $80(PVIFA6%,1) + $1,000(PVIF6%,1)

= $1,018.87P13 = $1,000

Y: P0 = $60(PVIFA8%,13) + $1,000(PVIF8%,13) = $841.92P1 = $60(PVIFA8%,12) + $1,000(PVIF8%,12) = $849.28P3 = $60(PVIFA8%,10) + $1,000(PVIF8%,10) = $865.80P8 = $60(PVIFA8%,5) + $1,000(PVIF8%,5) = $920.15P12 = $60(PVIFA8%,1) + $1,000(PVIF8%,1)

= $981.48 P13 = $1,000

All else held equal, the premium over par value for a premium bonddeclines as maturity approaches, and the discount from par valuefor a discount bond declines as maturity approaches. This is called“pull to par.” In both cases, the largest percentage price changesoccur at the shortest maturity lengths.

CHAPTER 7 B-234

Also, notice that the price of each bond when no time is left tomaturity is the par value, even though the purchaser would receivethe par value plus the coupon payment immediately. This is becausewe calculate the clean price of the bond.

B-235 SOLUTIONS

16. Any bond that sells at par has a YTM equal to the coupon rate. Bothbonds sell at par, so the initial YTM on both bonds is the couponrate, 9 percent. If the YTM suddenly rises to 11 percent:

PSam = $45(PVIFA5.5%,6) + $1,000(PVIF5.5%,6) = $950.04

PDave = $45(PVIFA5.5%,40) + $1,000(PVIF5.5%,40) = $839.54

The percentage change in price is calculated as:

Percentage change in price = (New price – Original price) /Original price

PSam% = ($950.04 – 1,000) / $1,000 = – 5.00%

PDave% = ($839.54 – 1,000) / $1,000 = – 16.05%

If the YTM suddenly falls to 7 percent:

PSam = $45(PVIFA3.5%,6) + $1,000(PVIF3.5%,6) = $1,053.29

PDave = $45(PVIFA3.5%,40) + $1,000(PVIF3.5%,40) = $1,213.55

PSam% = ($1,053.29 – 1,000) / $1,000 = + 5.33%

PDave% = ($1,213.55 – 1,000) / $1,000 = + 21.36%

All else the same, the longer the maturity of a bond, the greateris its price sensitivity to changes in interest rates.

17. Initially, at a YTM of 8 percent, the prices of the two bonds are:

PJ = $20(PVIFA4%,18) + $1,000(PVIF4%,18) = $746.81

PK = $60(PVIFA4%,18) + $1,000(PVIF4%,18) = $1,253.19

If the YTM rises from 8 percent to 10 percent:

PJ = $20(PVIFA5%,18) + $1,000(PVIF5%,18) = $649.31

PK = $60(PVIFA5%,18) + $1,000(PVIF5%,18) = $1,116.90

The percentage change in price is calculated as:

CHAPTER 7 B-236

Percentage change in price = (New price – Original price) /Original price

PJ% = ($649.31 – 746.81) / $746.81 = – 13.06%PK% = ($1,116.90 – 1,253.19) / $1,253.19 = – 10.88%

B-237 SOLUTIONS

If the YTM declines from 8 percent to 6 percent:

PJ = $20(PVIFA3%,18) + $1,000(PVIF3%,18) = $862.46

PK = $60(PVIFA3%,18) + $1,000(PVIF3%,18) = $1,412.61

PJ% = ($862.46 – 746.81) / $746.81 = + 15.49%

PK% = ($1,412.61 – 1,253.19) / $1,253.19 = + 12.72%

All else the same, the lower the coupon rate on a bond, the greateris its price sensitivity to changes in interest rates.

18. The bond price equation for this bond is:

P0 = $1,068 = $46(PVIFAR%,18) + $1,000(PVIFR%,18)

Using a spreadsheet, financial calculator, or trial and error wefind:

R = 4.06%

This is the semiannual interest rate, so the YTM is:

YTM = 2 4.06% = 8.12%

The current yield is:

Current yield = Annual coupon payment / Price = $92 / $1,068= .0861 or 8.61%

The effective annual yield is the same as the EAR, so using the EARequation from the previous chapter:

Effective annual yield = (1 + 0.0406)2 – 1 = .0829 or 8.29%

19. The company should set the coupon rate on its new bonds equal tothe required return. The required return can be observed in themarket by finding the YTM on outstanding bonds of the company. So,the YTM on the bonds currently sold in the market is:

P = $930 = $40(PVIFAR%,40) + $1,000(PVIFR%,40)

CHAPTER 7 B-238


R = 4.373%


YTM = 2 4.373% = 8.75%

B-239 SOLUTIONS

20. Accrued interest is the coupon payment for the period times thefraction of the period that has passed since the last couponpayment. Since we have a semiannual coupon bond, the coupon paymentper six months is one-half of the annual coupon payment. There arefour months until the next coupon payment, so two months havepassed since the last coupon payment. The accrued interest for thebond is:

Accrued interest = $74/2 × 2/6 = $12.33

And we calculate the clean price as:

Clean price = Dirty price – Accrued interest = $968 – 12.33 =$955.67

21. Accrued interest is the coupon payment for the period times thefraction of the period that has passed since the last couponpayment. Since we have a semiannual coupon bond, the coupon paymentper six months is one-half of the annual coupon payment. There aretwo months until the next coupon payment, so four months havepassed since the last coupon payment. The accrued interest for thebond is:

Accrued interest = $68/2 × 4/6 = $22.67

And we calculate the dirty price as:

Dirty price = Clean price + Accrued interest = $1,073 + 22.67 =$1,095.67

22. To find the number of years to maturity for the bond, we need tofind the price of the bond. Since we already have the coupon rate,we can use the bond price equation, and solve for the number ofyears to maturity. We are given the current yield of the bond, sowe can calculate the price as:

Current yield = .0755 = $80/P0 P0 = $80/.0755 = $1,059.60

Now that we have the price of the bond, the bond price equation is:

P = $1,059.60 = $80[(1 – (1/1.072)t ) / .072 ] + $1,000/1.072t

CHAPTER 7 B-240

We can solve this equation for t as follows:

$1,059.60(1.072)t = $1,111.11(1.072)t – 1,111.11 + 1,000

111.11 = 51.51(1.072)t

2.1570 = 1.072t

t = log 2.1570 / log 1.072 = 11.06 11 years

The bond has 11 years to maturity.

B-241 SOLUTIONS

23. The bond has 14 years to maturity, so the bond price equation is:

P = $1,089.60 = $36(PVIFAR%,28) + $1,000(PVIFR%,28)


R = 3.116%


YTM = 2 3.116% = 6.23%

The current yield is the annual coupon payment divided by the bondprice, so:

Current yield = $72 / $1,089.60 = .0661 or 6.61%

24. a. The bond price is the present value of the cash flows from abond. The YTM is the interest rate used in valuing the cashflows from a bond.

b. If the coupon rate is higher than the required return on a bond,the bond will sell at a premium, since it provides periodicincome in the form of coupon payments in excess of that requiredby investors on other similar bonds. If the coupon rate is lowerthan the required return on a bond, the bond will sell at adiscount since it provides insufficient coupon payments comparedto that required by investors on other similar bonds. Forpremium bonds, the coupon rate exceeds the YTM; for discountbonds, the YTM exceeds the coupon rate, and for bonds selling atpar, the YTM is equal to the coupon rate.

c. Current yield is defined as the annual coupon payment divided bythe current bond price. For

premium bonds, the current yield exceeds the YTM, for discountbonds the current yield is less than the YTM, and for bondsselling at par value, the current yield is equal to the YTM. Inall cases, the current yield plus the expected one-periodcapital gains yield of the bond must be equal to the requiredreturn.

25. The price of a zero coupon bond is the PV of the par, so:

CHAPTER 7 B-242

a. P0 = $1,000/1.04550 = $110.71

b. In one year, the bond will have 24 years to maturity, so theprice will be:

P1 = $1,000/1.04548 = $120.90

B-243 SOLUTIONS

The interest deduction is the price of the bond at the end ofthe year, minus the price at the beginning of the year, so:

Year 1 interest deduction = $120.90 – 110.71 = $10.19

The price of the bond when it has one year left to maturity willbe:

P24 = $1,000/1.0452 = $915.73

Year 24 interest deduction = $1,000 – 915.73 = $84.27

c. Previous IRS regulations required a straight-line calculation ofinterest. The total interest received by the bondholder is:

Total interest = $1,000 – 110.71 = $889.29

The annual interest deduction is simply the total interestdivided by the maturity of the bond, so the straight-linededuction is:

Annual interest deduction = $889.29 / 25 = $35.57

d. The company will prefer straight-line methods when allowedbecause the valuable interest deductions occur earlier in thelife of the bond.

26. a. The coupon bonds have an 8% coupon which matches the 8% requiredreturn, so they will sell at par. The number of bonds that mustbe sold is the amount needed divided by the bond price, so:

Number of coupon bonds to sell = $30,000,000 / $1,000 = 30,000

The number of zero coupon bonds to sell would be:

Price of zero coupon bonds = $1,000/1.0460 = $95.06

Number of zero coupon bonds to sell = $30,000,000 / $95.06 =315,589

b. The repayment of the coupon bond will be the par value plus thelast coupon payment times the number of bonds issued. So:

CHAPTER 7 B-244

Coupon bonds repayment = 30,000($1,040) = $32,400,000

The repayment of the zero coupon bond will be the par valuetimes the number of bonds issued, so:

Zeroes: repayment = 315,589($1,000) = $315,588,822

B-245 SOLUTIONS

c. The total coupon payment for the coupon bonds will be the numberbonds times the coupon payment. For the cash flow of the couponbonds, we need to account for the tax deductibility of theinterest payments. To do this, we will multiply the total couponpayment times one minus the tax rate. So:

Coupon bonds: (30,000)($80)(1–.35) = $1,560,000 cash outflow

Note that this is cash outflow since the company is making theinterest payment.

For the zero coupon bonds, the first year interest payment isthe difference in the price of the zero at the end of the yearand the beginning of the year. The price of the zeroes in oneyear will be:

P1 = $1,000/1.0458 = $102.82

The year 1 interest deduction per bond will be this price minusthe price at the beginning of the year, which we found in partb, so:

Year 1 interest deduction per bond = $102.82 – 95.06 = $7.76

The total cash flow for the zeroes will be the interestdeduction for the year times the number of zeroes sold, timesthe tax rate. The cash flow for the zeroes in year 1 will be:

Cash flows for zeroes in Year 1 = (315,589)($7.76)(.35) = $856,800.00

Notice the cash flow for the zeroes is a cash inflow. This isbecause of the tax deductibility of the imputed interestexpense. That is, the company gets to write off the interestexpense for the year even though the company did not have a cashflow for the interest expense. This reduces the company’s taxliability, which is a cash inflow.

During the life of the bond, the zero generates cash inflows tothe firm in the form of the interest tax shield of debt. Weshould note an important point here: If you find the PV of thecash flows from the coupon bond and the zero coupon bond, they

CHAPTER 7 B-246

will be the same. This is because of the much larger repaymentamount for the zeroes.

27. We found the maturity of a bond in Problem 22. However, in thiscase, the maturity is indeterminate. A bond selling at par can haveany length of maturity. In other words, when we solve the bondpricing equation as we did in Problem 22, the number of periods canbe any positive number.

28. We first need to find the real interest rate on the savings. Usingthe Fisher equation, the real interest rate is:

(1 + R) = (1 + r)(1 + h)1 + .11 = (1 + r)(1 + .038)r = .0694 or 6.94%

B-247 SOLUTIONS

Now we can use the future value of an annuity equation to find theannual deposit. Doing so, we find:

FVA = C{[(1 + r)t – 1] / r}$1,500,000 = $C[(1.069440 – 1) / .0694]C = $7,637.76

Challenge

29. To find the capital gains yield and the current yield, we need tofind the price of the bond. The current price of Bond P and theprice of Bond P in one year is:

P: P0 = $120(PVIFA7%,5) + $1,000(PVIF7%,5) = $1,116.69

P1 = $120(PVIFA7%,4) + $1,000(PVIF7%,4) = $1,097.19

Current yield = $120 / $1,116.69 = .1075 or 10.75%

The capital gains yield is:

Capital gains yield = (New price – Original price) / Originalprice

Capital gains yield = ($1,097.19 – 1,111.69) / $1,116.69 =–.0175 or –1.75%

The current price of Bond D and the price of Bond D in one year is:

D: P0 = $60(PVIFA7%,5) + $1,000(PVIF7%,5) = $883.31

P1 = $60(PVIFA7%,4) + $1,000(PVIF7%,4) = $902.81

Current yield = $60 / $883.81 = .0679 or 6.79%

Capital gains yield = ($902.81 – 883.31) / $883.31 = +.0221 or+2.21%

All else held constant, premium bonds pay high current income whilehaving price depreciation as maturity nears; discount bonds do notpay high current income but have price appreciation as maturitynears. For either bond, the total return is still 9%, but this

CHAPTER 7 B-248

return is distributed differently between current income andcapital gains.

30. a. The rate of return you expect to earn if you purchase a bond andhold it until maturity is the YTM. The bond price equation forthis bond is:

P0 = $1,060 = $70(PVIFAR%,10) + $1,000(PVIF R%,10)


R = YTM = 6.18%

B-249 SOLUTIONS

b. To find our HPY, we need to find the price of the bond in twoyears. The price of the bond in two years, at the new interestrate, will be:

P2 = $70(PVIFA5.18%,8) + $1,000(PVIF5.18%,8) = $1,116.92

To calculate the HPY, we need to find the interest rate thatequates the price we paid for the bond with the cash flows wereceived. The cash flows we received were $70 each year for twoyears, and the price of the bond when we sold it. The equationto find our HPY is:

P0 = $1,060 = $70(PVIFAR%,2) + $1,116.92(PVIFR%,2)

Solving for R, we get:

R = HPY = 9.17%

The realized HPY is greater than the expected YTM when the bond wasbought because interest rates dropped by 1 percent; bond pricesrise when yields fall.

31. The price of any bond (or financial instrument) is the PV of thefuture cash flows. Even though Bond M makes different couponspayments, to find the price of the bond, we just find the PV of thecash flows. The PV of the cash flows for Bond M is:

PM = $1,100(PVIFA3.5%,16)(PVIF3.5%,12) + $1,400(PVIFA3.5%,12)(PVIF3.5%,28) +$20,000(PVIF3.5%,40)PM = $19,018.78

Notice that for the coupon payments of $1,400, we found the PVA forthe coupon payments, and then discounted the lump sum back totoday.

Bond N is a zero coupon bond with a $20,000 par value, therefore,the price of the bond is the PV of the par, or:

PN = $20,000(PVIF3.5%,40) = $5,051.45

32. To calculate this, we need to set up an equation with the callablebond equal to a weighted average of the noncallable bonds. We willinvest X percent of our money in the first noncallable bond, which

CHAPTER 7 B-250

means our investment in Bond 3 (the other noncallable bond) will be(1 – X). The equation is:

C2 = C1 X + C3(1 – X)8.25 = 6.50 X + 12(1 – X)8.25 = 6.50 X + 12 – 12 XX = 0.68181

So, we invest about 68 percent of our money in Bond 1, and about 32percent in Bond 3. This combination of bonds should have the samevalue as the callable bond, excluding the value of the call. So:

P2 = 0.68181P1 + 0.31819P3

P2 = 0.68181(106.375) + 0.31819(134.96875)P2 = 115.4730

B-251 SOLUTIONS

The call value is the difference between this implied bond value and the actual bond price. So, the call value is:

Call value = 115.4730 – 103.50 = 11.9730

Assuming $1,000 par value, the call value is $119.73.

33. In general, this is not likely to happen, although it can (anddid). The reason this bond has a negative YTM is that it is acallable U.S. Treasury bond. Market participants know this. Giventhe high coupon rate of the bond, it is extremely likely to becalled, which means the bondholder will not receive all the cashflows promised. A better measure of the return on a callable bondis the yield to call (YTC). The YTC calculation is the basicallythe same as the YTM calculation, but the number of periods is thenumber of periods until the call date. If the YTC were calculatedon this bond, it would be positive.

34. To find the present value, we need to find the real weekly interestrate. To find the real return, we need to use the effective annualrates in the Fisher equation. So, we find the real EAR is:

(1 + R) = (1 + r)(1 + h)1 + .084 = (1 + r)(1 + .037)r = .0453 or 4.53%

Now, to find the weekly interest rate, we need to find the APR.Using the equation for discrete compounding:

EAR = [1 + (APR / m)]m – 1

We can solve for the APR. Doing so, we get:

APR = m[(1 + EAR)1/m – 1]APR = 52[(1 + .0453)1/52 – 1]APR = .0443 or 4.43%

So, the weekly interest rate is:

Weekly rate = APR / 52Weekly rate = .0443 / 52Weekly rate = .0009 or 0.09%

CHAPTER 7 B-252

Now we can find the present value of the cost of the roses. Thereal cash flows are an ordinary annuity, discounted at the realinterest rate. So, the present value of the cost of the roses is:

PVA = C({1 – [1/(1 + r)]t } / r) PVA = $5({1 – [1/(1 + .0009)]30(52)} / .0009)

PVA = $4,312.13

B-253 SOLUTIONS

35. To answer this question, we need to find the monthly interest rate,which is the APR divided by 12. We also must be careful to use thereal interest rate. The Fisher equation uses the effective annualrate, so, the real effective annual interest rates, and the monthlyinterest rates for each account are:

Stock account:(1 + R) = (1 + r)(1 + h)1 + .11 = (1 + r)(1 + .04)r = .0673 or 6.73%


Monthly rate = APR / 12Monthly rate = .0653 / 12Monthly rate = .0054 or 0.54%

Bond account:(1 + R) = (1 + r)(1 + h)1 + .07 = (1 + r)(1 + .04)r = .0288 or 2.88%



Now we can find the future value of the retirement account in realterms. The future value of each account will be:

Stock account:FVA = C {(1 + r )t – 1] / r}FVA = $900{[(1 + .0054)360 – 1] / .0054]}FVA = $1,001,704.05

Bond account:FVA = C {(1 + r )t – 1] / r}FVA = $450{[(1 + .0024)360 – 1] / .0024]}

CHAPTER 7 B-254

FVA = $255,475.17

The total future value of the retirement account will be the sum ofthe two accounts, or:

Account value = $1,001,704.05 + 255,475.17Account value = $1,257,179.22

B-255 SOLUTIONS

Now we need to find the monthly interest rate in retirement. We canuse the same procedure that we used to find the monthly interestrates for the stock and bond accounts, so:

(1 + R) = (1 + r)(1 + h)1 + .09 = (1 + r)(1 + .04)r = .0481 or 4.81%



Now we can find the real monthly withdrawal in retirement. Usingthe present value of an annuity equation and solving for thepayment, we find:

PVA = C({1 – [1/(1 + r)]t } / r )$1,257,179.22 = C({1 – [1/(1 + .0039)]300 } / .0039)C = $7,134.82

This is the real dollar amount of the monthly withdrawals. Thenominal monthly withdrawals will increase by the inflation rateeach month. To find the nominal dollar amount of the lastwithdrawal, we can increase the real dollar withdrawal by theinflation rate. We can increase the real withdrawal by theeffective annual inflation rate since we are only interested in thenominal amount of the last withdrawal. So, the last withdrawal innominal terms will be:

FV = PV(1 + r)t

FV = $7,134.82(1 + .04)(30 + 25)

FV = $61,690.29


3.Enter 10 8.75% $75 $1,000

N I/Y PV PMT FVSolve $918.89

CHAPTER 7 B-256

for

4.Enter 9 ±$934 $90 $1,000


10.15%

5.Enter 13 7.5% ±$1,045 $1,000


$80.54

Coupon rate = $80.54 / $1,000 = 8.05%

B-257 SOLUTIONS

6.Enter 20 3.70% $34.50 $1,000


$965.10

7.Enter 20 ±$1,050 $42 $1,000


3.837%

3.837% 2 = 7.67%

8.Enter 29 3.40% ±$924 $1,000


$29.84

($29.84 / $1,000)(2) = 5.97%

15. Bond XP0

Enter 13 6% $80 $1,000N I/Y PV PMT FV

Solve for

$1,177.05

P1


Solve for

$1,167.68

P3


Solve for

$1,147.20

P8


CHAPTER 7 B-258

Solve for

$1,084.25

P12


Solve for

$1,018.87

B-259 SOLUTIONS

Bond YP0


Solve for

$841.92

P1


Solve for

$849.28

P3


Solve for

$865.80

P8


Solve for

$920.15

P12


Solve for

$981.48

16. If both bonds sell at par, the initial YTM on both bonds is thecoupon rate, 9 percent. If the YTM suddenly rises to 11 percent:

PSam

Enter 6 5.5% $45 $1,000N I/Y PV PMT FV

Solve for

$950.04

PSam% = ($950.04 – 1,000) / $1,000 = – 5.00%

PDave

Enter 40 5.5% $45 $1,000

CHAPTER 7 B-260


$839.54

PDave% = ($839.54 – 1,000) / $1,000 = – 16.05%

If the YTM suddenly falls to 7 percent:PSam


Solve for

$1,053.29

PSam% = ($1,053.29 – 1,000) / $1,000 = + 5.33%

B-261 SOLUTIONS

PDave


Solve for

$1,1213.55

PDave% = ($1,213.55 – 1,000) / $1,000 = + 21.36%

All else the same, the longer the maturity of a bond, the greateris its price sensitivity to changes

in interest rates.

17. Initially, at a YTM of 8 percent, the prices of the two bonds are:

PJ


Solve for

$746.81

PK


Solve for

$1,253.19

If the YTM rises from 8 percent to 10 percent:PJ


Solve for

$649.31

PJ% = ($649.31 – 746.81) / $746.81 = – 13.06%

PK


Solve for

$1,116.90

PK% = ($1,116.90 – 1,253.19) / $1,253.19 = – 10.88%

If the YTM declines from 8 percent to 6 percent:PJ

CHAPTER 7 B-262


Solve for

$862.46

PJ% = ($862.46 – 746.81) / $746.81 = + 15.49%

PK


Solve for

$1,412.61

PK% = ($1,412.61 – 1,253.19) / $1,253.19 = + 12.72%

All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to

changes in interest rates.

B-263 SOLUTIONS

18.Enter 18 ±$1,068 $46 $1,000


4.06%

4.06% 2 = 8.12%

Enter 8.12 % 2NOM EFF C/Y

Solve for

8.29%

19. The company should set the coupon rate on its new bonds equal tothe required return; the required return can be observed in themarket by finding the YTM on outstanding bonds of the company.

Enter 40 ±$930 $35 $1,000N I/Y PV PMT FV

Solve for

4.373%

4.373% 2 = 8.75%

22. Current yield = .0755 = $90/P0 ; P0 = $90/.0755 = $1,059.60

Enter 7.2% ±$1,059.60 $80 $1,000N I/Y PV PMT FV

Solve for

11.06

11.06 or 11 years

23.Enter 28 ±$1,089.60 $36 $1,000


3.116%

3.116% × 2 = 6.23%

25. a. Po


CHAPTER 7 B-264

Solve for

$110.71

b. P1


Solve for

$120.90

year 1 interest deduction = $120.90 – 110.71 = $10.19

P19


Solve for

$915.73

year 25 interest deduction = $1,000 – 915.73 = $84.27

B-265 SOLUTIONS

c. Total interest = $1,000 – 110.71 = $889.29Annual interest deduction = $889.29 / 25 = $35.57

d. The company will prefer straight-line method when allowedbecause the valuable interest deductions occur earlier in thelife of the bond.

26. a. The coupon bonds have an 8% coupon rate, which matches the 8%required return, so they will sell at par; # of bonds =$30,000,000/$1,000 = 30,000.

For the zeroes:


Solve for

$95.06

$30,000,000/$95.06 = 315,589 will be issued.

b. Coupon bonds: repayment = 30,000($1,080) = $32,400,000Zeroes: repayment = 315,589($1,000) = $315,588,822

c. Coupon bonds: (30,000)($80)(1 –.35) = $1,560,000 cash outflowZeroes:


Solve for

$102.82

year 1 interest deduction = $102.82 – 95.06 = $7.76(315,589)($7.76)(.35) = $856,800.00 cash inflow

During the life of the bond, the zero generates cash inflows to the firm in the form of the

interest tax shield of debt.

29.Bond PP0


Solve for

$1,116.69

P1

Enter 4 7% $120 $1,000

CHAPTER 7 B-266


$1,097.19

Current yield = $120 / $1,116.69 = 10.75% Capital gains yield = ($1,097.19 – 1,116.69) / $1,116.69 = –

1.75%

Bond DP0


Solve for

$883.31

B-267 SOLUTIONS

P1


Solve for

$902.81

Current yield = $60 / $883.31 = 6.79%Capital gains yield = ($902.81 – 883.31) / $883.31 = 2.21%All else held constant, premium bonds pay high current incomewhile having price depreciation as maturity nears; discount bondsdo not pay high current income but have price appreciation asmaturity nears. For either bond, the total return is still 9%,but this return is distributed differently between current incomeand capital gains.

30.a.Enter 10 ±$1,060 $70 $1,000


6.18%

This is the rate of return you expect to earn on your investment when you purchase the bond.

b.Enter 8 5.18% $70 $1,000


$1,116.92

The HPY is:

Enter 2 ±$1,060 $70 $1,116.92N I/Y PV PMT FV

Solve for

9.17%

The realized HPY is greater than the expected YTM when the bond was bought because interest rates dropped by 1 percent; bond prices rise when yields fall.

31.PM

CFo $0C01 $0

CHAPTER 7 B-268

F01 12C02 $1,100F02 16C03 $1,400F03 11C04 $21,400F04 1

I = 3.5%NPV CPT$19,018.78

PN


Solve for

$5,051.45

CHAPTER 8STOCK VALUATION


1. The value of any investment depends on the present value of its cashflows; i.e., what investors will actually receive. The cash flows froma share of stock are the dividends.

2. Investors believe the company will eventually start paying dividends(or be sold to another company).

3. In general, companies that need the cash will often forgo dividendssince dividends are a cash expense. Young, growing companies withprofitable investment opportunities are one example; another example isa company in financial distress. This question is examined in depth ina later chapter.

4. The general method for valuing a share of stock is to find the presentvalue of all expected future dividends. The dividend growth modelpresented in the text is only valid (i) if dividends are expected tooccur forever, that is, the stock provides dividends in perpetuity, and(ii) if a constant growth rate of dividends occurs forever. A violationof the first assumption might be a company that is expected to ceaseoperations and dissolve itself some finite number of years from now.The stock of such a company would be valued by applying the generalmethod of valuation explained in this chapter. A violation of thesecond assumption might be a start-up firm that isn’t currently payingany dividends, but is expected to eventually start making dividendpayments some number of years from now. This stock would also be valuedby the general dividend valuation method explained in this chapter.

5. The common stock probably has a higher price because the dividend cangrow, whereas it is fixed on the preferred. However, the preferred isless risky because of the dividend and liquidation preference, so it ispossible the preferred could be worth more, depending on thecircumstances.

CHAPTER 8 B-270

6. The two components are the dividend yield and the capital gains yield.For most companies, the capital gains yield is larger. This is easy tosee for companies that pay no dividends. For companies that do paydividends, the dividend yields are rarely over five percent and areoften much less.

7. Yes. If the dividend grows at a steady rate, so does the stock price.In other words, the dividend growth rate and the capital gains yieldare the same.

8. In a corporate election, you can buy votes (by buying shares), so moneycan be used to influence or even determine the outcome. Many wouldargue the same is true in political elections, but, in principle atleast, no one has more than one vote.

9. It wouldn’t seem to be. Investors who don’t like the voting features ofa particular class of stock are under no obligation to buy it.

10. Investors buy such stock because they want it, recognizing that theshares have no voting power. Presumably, investors pay a little lessfor such shares than they would otherwise.

B-271 SOLUTIONS

11. Presumably, the current stock value reflects the risk, timing andmagnitude of all future cash flows, both short-term and long-term. Ifthis is correct, then the statement is false.

12. If this assumption is violated, the two-stage dividend growth modelis not valid. In other words, the price calculated will not be correct.Depending on the stock, it may be more reasonable to assume that thedividends fall from the high growth rate to the low perpetual growthrate over a period of years, rather than in one year.


NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiplesteps. Due to space and readability constraints, when these intermediate steps are included in thissolutions manual, rounding may appear to have occurred. However, the final answer for eachproblem is found without rounding during any step in the problem.

Basic

1. The constant dividend growth model is:

Pt = Dt × (1 + g) / (R – g)

So the price of the stock today is:

P0 = D0 (1 + g) / (R – g) = $1.95 (1.06) / (.11 – .06) = $41.34

The dividend at year 4 is the dividend today times the FVIF for thegrowth rate in dividends and four years, so:

P3 = D3 (1 + g) / (R – g) = D0 (1 + g)4 / (R – g) = $1.95 (1.06)4 / (.11 –.06) = $49.24

We can do the same thing to find the dividend in Year 16, which givesus the price in Year 15, so:

P15 = D15 (1 + g) / (R – g) = D0 (1 + g)16 / (R – g) = $1.95 (1.06)16 /(.11 – .06) = $99.07

There is another feature of the constant dividend growth model: Thestock price grows at the dividend growth rate. So, if we know the stockprice today, we can find the future value for any time in the future wewant to calculate the stock price. In this problem, we want to know the

CHAPTER 8 B-272

stock price in three years, and we have already calculated the stockprice today. The stock price in three years will be:

P3 = P0(1 + g)3 = $41.34(1 + .06)3 = $49.24

And the stock price in 15 years will be:

P15 = P0(1 + g)15 = $41.34(1 + .06)15 = $99.07

2. We need to find the required return of the stock. Using the constantgrowth model, we can solve the equation for R. Doing so, we find:

R = (D1 / P0) + g = ($2.10 / $48.00) + .05 = .0938 or 9.38%

B-273 SOLUTIONS

3. The dividend yield is the dividend next year divided by the currentprice, so the dividend yield is:

Dividend yield = D1 / P0 = $2.10 / $48.00 = .0438 or 4.38%

The capital gains yield, or percentage increase in the stock price, isthe same as the dividend growth rate, so:

Capital gains yield = 5%

4. Using the constant growth model, we find the price of the stock todayis:

P0 = D1 / (R – g) = $3.04 / (.11 – .038) = $42.22

5. The required return of a stock is made up of two parts: The dividendyield and the capital gains yield. So, the required return of thisstock is:

R = Dividend yield + Capital gains yield = .063 + .052 = .1150 or11.50%

6. We know the stock has a required return of 11 percent, and the dividendand capital gains yield are equal, so:

Dividend yield = 1/2(.11) = .055 = Capital gains yield

Now we know both the dividend yield and capital gains yield. Thedividend is simply the stock price times the dividend yield, so:

D1 = .055($47) = $2.59

This is the dividend next year. The question asks for the dividend thisyear. Using the relationship between the dividend this year and thedividend next year:

D1 = D0(1 + g)

We can solve for the dividend that was just paid:

$2.59 = D0(1 + .055)

D0 = $2.59 / 1.055 = $2.45

CHAPTER 8 B-274

7. The price of any financial instrument is the PV of the future cashflows. The future dividends of this stock are an annuity for 11 years,so the price of the stock is the PVA, which will be:

P0 = $9.75(PVIFA10%,11) = $63.33

8. The price a share of preferred stock is the dividend divided by therequired return. This is the same equation as the constant growthmodel, with a dividend growth rate of zero percent. Remember, mostpreferred stock pays a fixed dividend, so the growth rate is zero.Using this equation, we find the price per share of the preferred stockis:

R = D/P0 = $5.50/$108 = .0509 or 5.09%

B-275 SOLUTIONS

9. We can use the constant dividend growth model, which is:

Pt = Dt × (1 + g) / (R – g)

So the price of each company’s stock today is:

Red stock price = $2.35 / (.08 – .05) = $78.33Yellow stock price = $2.35 / (.11 – .05) = $39.17Blue stock price = $2.35 / (.14 – .05) = $26.11

As the required return increases, the stock price decreases. This is afunction of the time value of money: A higher discount rate decreasesthe present value of cash flows. It is also important to note thatrelatively small changes in the required return can have a dramaticimpact on the stock price.

Intermediate

10. This stock has a constant growth rate of dividends, but the requiredreturn changes twice. To find the value of the stock today, we willbegin by finding the price of the stock at Year 6, when both thedividend growth rate and the required return are stable forever. Theprice of the stock in Year 6 will be the dividend in Year 7, divided bythe required return minus the growth rate in dividends. So:

P6 = D6 (1 + g) / (R – g) = D0 (1 + g)7 / (R – g) = $3.50 (1.05)7 / (.10 –.05) = $98.50

Now we can find the price of the stock in Year 3. We need to find theprice here since the required return changes at that time. The price ofthe stock in Year 3 is the PV of the dividends in Years 4, 5, and 6,plus the PV of the stock price in Year 6. The price of the stock inYear 3 is:

P3 = $3.50(1.05)4 / 1.12 + $3.50(1.05)5 / 1.122 + $3.50(1.05)6 /1.123 + $98.50 / 1.123 P3 = $80.81

Finally, we can find the price of the stock today. The price today willbe the PV of the dividends in Years 1, 2, and 3, plus the PV of thestock in Year 3. The price of the stock today is:

CHAPTER 8 B-276

P0 = $3.50(1.05) / 1.14 + $3.50(1.05)2 / (1.14)2 + $3.50(1.05)3 /(1.14)3 + $80.81 / (1.14)3 P0 = $63.47

11. Here we have a stock that pays no dividends for 10 years. Once thestock begins paying dividends, it will have a constant growth rate ofdividends. We can use the constant growth model at that point. It isimportant to remember that general constant dividend growth formula is:

Pt = [Dt × (1 + g)] / (R – g)

This means that since we will use the dividend in Year 10, we will befinding the stock price in Year 9. The dividend growth model is similarto the PVA and the PV of a perpetuity: The equation gives you the PVone period before the first payment. So, the price of the stock in Year9 will be:

P9 = D10 / (R – g) = $10.00 / (.14 – .05) = $111.11

B-277 SOLUTIONS

The price of the stock today is simply the PV of the stock price in thefuture. We simply discount the future stock price at the requiredreturn. The price of the stock today will be:

P0 = $111.11 / 1.149 = $34.17

12. The price of a stock is the PV of the future dividends. This stock ispaying four dividends, so the price of the stock is the PV of thesedividends using the required return. The price of the stock is:

P0 = $10 / 1.11 + $14 / 1.112 + $18 / 1.113 + $22 / 1.114 + $26 / 1.115 =$63.45

13. With supernormal dividends, we find the price of the stock when thedividends level off at a constant growth rate, and then find the PV ofthe future stock price, plus the PV of all dividends during thesupernormal growth period. The stock begins constant growth in Year 4,so we can find the price of the stock in Year 4, at the beginning ofthe constant dividend growth, as:

P4 = D4 (1 + g) / (R – g) = $2.00(1.05) / (.12 – .05) = $30.00

The price of the stock today is the PV of the first four dividends,plus the PV of the Year 3 stock price. So, the price of the stock todaywill be:

P0 = $11.00 / 1.11 + $8.00 / 1.112 + $5.00 / 1.113 + $2.00 / 1.114 +$30.00 / 1.114 = $40.09

14. With supernormal dividends, we find the price of the stock when thedividends level off at a constant growth rate, and then find the PV ofthe futures stock price, plus the PV of all dividends during thesupernormal growth period. The stock begins constant growth in Year 4,so we can find the price of the stock in Year 3, one year before theconstant dividend growth begins as:

P3 = D3 (1 + g) / (R – g) = D0 (1 + g1)3 (1 + g2) / (R – g) P3 = $1.80(1.30)3(1.06) / (.13 – .06) P3 = $59.88

The price of the stock today is the PV of the first three dividends,plus the PV of the Year 3 stock price. The price of the stock todaywill be:

CHAPTER 8 B-278

P0 = $1.80(1.30) / 1.13 + $1.80(1.30)2 / 1.132 + $1.80(1.30)3 / 1.133

+ $59.88 / 1.133 P0 = $48.70

We could also use the two-stage dividend growth model for this problem, which is:

P0 = [D0(1 + g1)/(R – g1)]{1 – [(1 + g1)/(1 + R)]T}+ [(1 + g1)/(1 + R)]T[D0(1 + g1)/(R – g1)]P0 = [$1.80(1.30)/(.13 – .30)][1 – (1.30/1.13)3] + [(1 + .30)/(1 + .13)]3[$1.80(1.06)/(.13 – .06)] P0 = $48.70

15. Here we need to find the dividend next year for a stock experiencingsupernormal growth. We know the stock price, the dividend growth rates,and the required return, but not the dividend. First, we need torealize that the dividend in Year 3 is the current dividend times theFVIF. The dividend in Year 3 will be:

D3 = D0 (1.25)3

B-279 SOLUTIONS

And the dividend in Year 4 will be the dividend in Year 3 times oneplus the growth rate, or:

D4 = D0 (1.25)3 (1.15)

The stock begins constant growth in Year 4, so we can find the price ofthe stock in Year 4 as the dividend in Year 5, divided by the requiredreturn minus the growth rate. The equation for the price of the stockin Year 4 is:

P4 = D4 (1 + g) / (R – g)

Now we can substitute the previous dividend in Year 4 into thisequation as follows:

P4 = D0 (1 + g1)3 (1 + g2) (1 + g3) / (R – g)

P4 = D0 (1.25)3 (1.15) (1.08) / (.13 – .08) = 48.52D0

When we solve this equation, we find that the stock price in Year 4 is48.52 times as large as the dividend today. Now we need to find theequation for the stock price today. The stock price today is the PV ofthe dividends in Years 1, 2, 3, and 4, plus the PV of the Year 4 price.So:

P0 = D0(1.25)/1.13 + D0(1.25)2/1.132 + D0(1.25)3/1.133+D0(1.25)3(1.15)/1.134 + 48.52D0/1.134

We can factor out D0 in the equation, and combine the last two terms.Doing so, we get:

P0 = $76 = D0{1.25/1.13 + 1.252/1.132 + 1.253/1.133 + [(1.25)3(1.15) +48.52] / 1.134}

Reducing the equation even further by solving all of the terms in thebraces, we get:

$76 = $34.79D0

D0 = $76 / $34.79 D0 = $2.18

CHAPTER 8 B-280

This is the dividend today, so the projected dividend for the next yearwill be:

D1 = $2.18(1.25) D1 = $2.73

16. The constant growth model can be applied even if the dividends aredeclining by a constant percentage, just make sure to recognize thenegative growth. So, the price of the stock today will be:

P0 = D0 (1 + g) / (R – g) P0 = $10.46(1 – .04) / [(.115 – (–.04)] P0 = $64.78

17. We are given the stock price, the dividend growth rate, and therequired return, and are asked to find the dividend. Using the constantdividend growth model, we get:

P0 = $64 = D0 (1 + g) / (R – g)

B-281 SOLUTIONS

Solving this equation for the dividend gives us:

D0 = $64(.10 – .045) / (1.045) D0 = $3.37

18. The price of a share of preferred stock is the dividend payment dividedby the required return. We know the dividend payment in Year 20, so wecan find the price of the stock in Year 19, one year before the firstdividend payment. Doing so, we get:

P19 = $20.00 / .064 P19 = $312.50

The price of the stock today is the PV of the stock price in thefuture, so the price today will be:

P0 = $312.50 / (1.064)19 P0 = $96.15

19. The annual dividend paid to stockholders is $1.48, and the dividendyield is 2.1 percent. Using the equation for the dividend yield:

Dividend yield = Dividend / Stock price

We can plug the numbers in and solve for the stock price:

.021 = $1.48 / P0

P0 = $1.48/.021 = $70.48

The “Net Chg” of the stock shows the stock decreased by $0.23 on thisday, so the closing stock price yesterday was:

Yesterday’s closing price = $70.48 + 0.23 = $70.71

To find the net income, we need to find the EPS. The stock quote tellsus the P/E ratio for the stock is 19. Since we know the stock price aswell, we can use the P/E ratio to solve for EPS as follows:

P/E = 19 = Stock price / EPS = $70.48 / EPS

EPS = $70.48 / 19 = $3.71

CHAPTER 8 B-282

We know that EPS is just the total net income divided by the number ofshares outstanding, so:

EPS = NI / Shares = $3.71 = NI / 25,000,000

NI = $3.71(25,000,000) = $92,731,830

20. We can use the two-stage dividend growth model for this problem, which is:

P0 = [D0(1 + g1)/(R – g1)]{1 – [(1 + g1)/(1 + R)]T}+ [(1 + g1)/(1 + R)]T[D0(1 + g2)/(R – g2)]P0 = [$1.25(1.28)/(.13 – .28)][1 – (1.28/1.13)8] + [(1.28)/(1.13)]8[$1.25(1.06)/(.13 – .06)] P0 = $69.55

B-283 SOLUTIONS

21. We can use the two-stage dividend growth model for this problem, which is:

P0 = [D0(1 + g1)/(R – g1)]{1 – [(1 + g1)/(1 + R)]T}+ [(1 + g1)/(1 + R)]T[D0(1 + g2)/(R – g2)]P0 = [$1.74(1.25)/(.12 – .25)][1 – (1.25/1.12)11] + [(1.25)/(1.12)]11[$1.74(1.06)/(.12 – .06)] P0 = $142.14

Challenge

22. We are asked to find the dividend yield and capital gains yield foreach of the stocks. All of the stocks have a 15 percent requiredreturn, which is the sum of the dividend yield and the capital gainsyield. To find the components of the total return, we need to find thestock price for each stock. Using this stock price and the dividend, wecan calculate the dividend yield. The capital gains yield for the stockwill be the total return (required return) minus the dividend yield.

W: P0 = D0(1 + g) / (R – g) = $4.50(1.10)/(.19 – .10) = $55.00

Dividend yield = D1/P0 = $4.50(1.10)/$55.00 = .09 or 9%

Capital gains yield = .19 – .09 = .10 or 10%

X: P0 = D0(1 + g) / (R – g) = $4.50/(.19 – 0) = $23.68

Dividend yield = D1/P0 = $4.50/$23.68 = .19 or 19%

Capital gains yield = .19 – .19 = 0%

Y: P0 = D0(1 + g) / (R – g) = $4.50(1 – .05)/(.19 + .05) = $17.81

Dividend yield = D1/P0 = $4.50(0.95)/$17.81 = .24 or 24%

Capital gains yield = .19 – .24 = –.05 or –5%

Z: P2 = D2(1 + g) / (R – g) = D0(1 + g1)2(1 + g2)/(R – g2) =$4.50(1.20)2(1.12)/(.19 – .12) = $103.68

P0 = $4.50 (1.20) / (1.19) + $4.50 (1.20)2 / (1.19)2 + $103.68 /(1.19)2 = $82.33

CHAPTER 8 B-284

Dividend yield = D1/P0 = $4.50(1.20)/$96.10 = .066 or 6.6%

Capital gains yield = .19 – .066 = .124 or 12.4%

In all cases, the required return is 19%, but the return is distributeddifferently between current income and capital gains. High growthstocks have an appreciable capital gains component but a relativelysmall current income yield; conversely, mature, negative-growth stocksprovide a high current income but also price depreciation over time.

23. a. Using the constant growth model, the price of the stock payingannual dividends will be:

P0 = D0(1 + g) / (R – g) = $3.20(1.06)/(.12 – .06) = $56.53

B-285 SOLUTIONS

b. If the company pays quarterly dividends instead of annual dividends,the quarterly dividend will be one-fourth of annual dividend, or:

Quarterly dividend: $3.20(1.06)/4 = $0.848

To find the equivalent annual dividend, we must assume that thequarterly dividends are reinvested at the required return. We canthen use this interest rate to find the equivalent annual dividend.In other words, when we receive the quarterly dividend, we reinvestit at the required return on the stock. So, the effective quarterlyrate is:

Effective quarterly rate: 1.12.25 – 1 = .0287

The effective annual dividend will be the FVA of the quarterlydividend payments at the effective quarterly required return. Inthis case, the effective annual dividend will be:

Effective D1 = $0.848(FVIFA2.87%,4) = $3.54

Now, we can use the constant growth model to find the current stockprice as:

P0 = $3.54/(.12 – .06) = $59.02

Note that we cannot simply find the quarterly effective requiredreturn and growth rate to find the value of the stock. This wouldassume the dividends increased each quarter, not each year.

24. Here we have a stock with supernormal growth, but the dividend growthchanges every year for the first four years. We can find the price ofthe stock in Year 3 since the dividend growth rate is constant afterthe third dividend. The price of the stock in Year 3 will be thedividend in Year 4, divided by the required return minus the constantdividend growth rate. So, the price in Year 3 will be:

P3 = $2.45(1.20)(1.15)(1.10)(1.05) / (.11 – .05) = $65.08

The price of the stock today will be the PV of the first threedividends, plus the PV of the stock price in Year 3, so:

P0 = $2.45(1.20)/(1.11) + $2.45(1.20)(1.15)/1.112 + $2.45(1.20)(1.15)(1.10)/1.113 + $65.08/1.113

CHAPTER 8 B-286

P0 = $55.70

25. Here we want to find the required return that makes the PV of thedividends equal to the current stock price. The equation for the stockprice is:

P = $2.45(1.20)/(1 + R) + $2.45(1.20)(1.15)/(1 + R)2 + $2.45(1.20)(1.15)(1.10)/(1 + R)3

+ [$2.45(1.20)(1.15)(1.10)(1.05)/(R – .05)]/(1 + R)3 = $63.82

We need to find the roots of this equation. Using spreadsheet, trial and error, or a calculator with a root solving function, we find that:

R = 10.24%

B-287 SOLUTIONS

26. Even though the question concerns a stock with a constant growth rate,we need to begin with the equation for two-stage growth given in thechapter, which is:

P0 =

D0(1+ g1 )R - g1 [1 - (1+ g1

1+ R )t] + Pt

(1+ R )t

We can expand the equation (see Problem 27 for more detail) to thefollowing:

P0 =

D0(1+ g1 )R - g1 [1 - (1+ g1

1+ R )t] + (1+ g1

1+ R )t

D(1+ g2 )R - g2

Since the growth rate is constant, g1 = g2 , so:

P0 = D0(1+ g)

R - g [1 - (1+ g1+ R )

t] + (1+ g1+ R )

t

D(1+ g )R - g

Since we want the first t dividends to constitute one-half of the stockprice, we can set the two terms on the right hand side of the equationequal to each other, which gives us:

D0(1 + g )

R - g [1 - (1+ g1+ R )

t] = (1+ g1+ R )

t

D(1+ g )R - g

Since D0(1 + g )

R - g appears on both sides of the equation, we can eliminatethis, which leaves:

1 – (1 + g1 + R )

t

= (1+ g1+ R )

t

Solving this equation, we get:

1 = (1 + g1 + R )

t

+ (1+ g1+ R )

t

CHAPTER 8 B-288

1 = 2 (1 + g1 + R )

t

1/2 = (1+ g1+ R )

t

B-289 SOLUTIONS

t ln (1 + g1 + R ) = ln(0.5)

t =

ln(0.5)

ln(1+ g1+ R )

This expression will tell you the number of dividends that constituteone-half of the current stock price.

27. To find the value of the stock with two-stage dividend growth, considerthat the present value of the first t dividends is the present value ofa growing annuity. Additionally, to find the price of the stock, weneed to add the present value of the stock price at time t. So, thestock price today is:

P0 = PV of t dividends + PV(Pt)

Using g1 to represent the first growth rate and substituting theequation for the present value of a growing annuity, we get:

P0 = D1

[1 - (1+ g11+ R )

t

R - g1 ] + PV(Pt)

Since the dividend in one year will increase at g1, we can re-write theexpression as:

P0 = D0(1 + g1)[1 - (1+ g1

1+ R )t

R - g1 ] + PV(Pt)

Now we can re-write the equation again as:

P0 =

D0(1+ g1 )R - g1 [1 - (1 + g1

1 + R )t] + PV(Pt)

CHAPTER 8 B-290

To find the price of the stock at time t, we can use the constantdividend growth model, or:

Pt =

Dt + 1

R - g2

The dividend at t + 1 will have grown at g1 for t periods, and at g2 forone period, so:

Dt + 1 = D0(1 + g1)t(1 + g2)

B-291 SOLUTIONS

So, we can re-write the equation as:

Pt =

D(1+ g1 )t(1+ g2)

R - g2

Next, we can find value today of the future stock price as:

PV(Pt) =

D(1+ g1 )t (1+ g2)

R - g2 × 1

(1+ R )t

which can be written as:

PV(Pt) = (1+ g11+ R )

t

×

D(1+ g2)R - g2

Substituting this into the stock price equation, we get:

P0 =

D0(1+ g1 )R - g1 [1 - (1+ g1

1+ R )t] + (1+ g1

1+ R )t

×

D(1+ g2)R - g2

In this equation, the first term on the right hand side is the presentvalue of the first t dividends, and the second term is the present valueof the stock price when constant dividend growth forever begins.

28. To find the expression when the growth rate for the first stage isexactly equal to the required return, consider we can find the presentvalue of the dividends in the first stage as:

PV = D0(1 +g1)

1

(1+R)1 + D0(1 +g1)

2

(1+R)2 + D0(1 +g1)

3

(1+R)3 + …

Since g1 is equal to R, each of the terns reduces to:

PV = D0 + D0 + D0 + ….PV = t × D0

So, the expression for the price of a stock when the first growth rate is exactly equal to the required return is:

CHAPTER 8 B-292

Pt = t × D0 +

D0×(1+g1)t×(1+g2 )

R−g2

CHAPTER 9NET PRESENT VALUE AND OTHER INVESTMENT CRITERIA


1. A payback period less than the project’s life means that the NPV ispositive for a zero discount rate, but nothing more definitive can besaid. For discount rates greater than zero, the payback period willstill be less than the project’s life, but the NPV may be positive,zero, or negative, depending on whether the discount rate is lessthan, equal to, or greater than the IRR. The discounted paybackincludes the effect of the relevant discount rate. If a project’sdiscounted payback period is less than the project’s life, it must bethe case that NPV is positive.

2. If a project has a positive NPV for a certain discount rate, then itwill also have a positive NPV for a zero discount rate; thus, thepayback period must be less than the project life. Since discountedpayback is calculated at the same discount rate as is NPV, if NPV ispositive, the discounted payback period must be less than theproject’s life. If NPV is positive, then the present value of futurecash inflows is greater than the initial investment cost; thus PImust be greater than 1. If NPV is positive for a certain discountrate R, then it will be zero for some larger discount rate R*; thusthe IRR must be greater than the required return.

3. a. Payback period is simply the accounting break-even point of aseries of cash flows. To actually compute the payback period, itis assumed that any cash flow occurring during a given period isrealized continuously throughout the period, and not at a singlepoint in time. The payback is then the point in time for theseries of cash flows when the initial cash outlays are fullyrecovered. Given some predetermined cutoff for the payback period,the decision rule is to accept projects that payback before thiscutoff, and reject projects that take longer to payback.

CHAPTER 9 B-294

b. The worst problem associated with payback period is that itignores the time value of money. In addition, the selection of ahurdle point for payback period is an arbitrary exercise thatlacks any steadfast rule or method. The payback period is biasedtowards short-term projects; it fully ignores any cash flows thatoccur after the cutoff point.

c. Despite its shortcomings, payback is often used because (1) theanalysis is straightforward and simple and (2) accounting numbersand estimates are readily available. Materiality considerationsoften warrant a payback analysis as sufficient; maintenanceprojects are another example where the detailed analysis of othermethods is often not needed. Since payback is biased towardsliquidity, it may be a useful and appropriate analysis method forshort-term projects where cash management is most important.

4. a. The discounted payback is calculated the same as is regularpayback, with the exception that each cash flow in the series isfirst converted to its present value. Thus discounted paybackprovides a measure of financial/economic break-even because ofthis discounting, just as regular payback provides a measure ofaccounting break-even because it does not discount the cash flows.Given some predetermined cutoff for the discounted payback period,the decision rule is to accept projects whose discounted cashflows payback before this cutoff period, and to reject all otherprojects.

B-295 SOLUTIONS

b. The primary disadvantage to using the discounted payback method isthat it ignores all cash flows that occur after the cutoff date,thus biasing this criterion towards short-term projects. As aresult, the method may reject projects that in fact have positiveNPVs, or it may accept projects with large future cash outlaysresulting in negative NPVs. In addition, the selection of a cutoffpoint is again an arbitrary exercise.

c. Discounted payback is an improvement on regular payback because ittakes into account the time value of money. For conventional cashflows and strictly positive discount rates, the discounted paybackwill always be greater than the regular payback period.

5. a. The average accounting return is interpreted as an average measureof the accounting performance of a project over time, computed assome average profit measure attributable to the project divided bysome average balance sheet value for the project. This textcomputes AAR as average net income with respect to average (total)book value. Given some predetermined cutoff for AAR, the decisionrule is to accept projects with an AAR in excess of the targetmeasure, and reject all other projects.

b. AAR is not a measure of cash flows and market value, but a measureof financial statement accounts that often bear little resemblanceto the relevant value of a project. In addition, the selection ofa cutoff is arbitrary, and the time value of money is ignored. Fora financial manager, both the reliance on accounting numbersrather than relevant market data and the exclusion of time valueof money considerations are troubling. Despite these problems, AARcontinues to be used in practice because (1) the accountinginformation is usually available, (2) analysts often useaccounting ratios to analyze firm performance, and (3) managerialcompensation is often tied to the attainment of certain targetaccounting ratio goals.

6. a. NPV is simply the present value of a project’s cash flows. NPVspecifically measures, after considering the time value of money,the net increase or decrease in firm wealth due to the project.The decision rule is to accept projects that have a positive NPV,and reject projects with a negative NPV.

b. NPV is superior to the other methods of analysis presented in thetext because it has no serious flaws. The method unambiguouslyranks mutually exclusive projects, and can differentiate betweenprojects of different scale and time horizon. The only drawback toNPV is that it relies on cash flow and discount rate values that

CHAPTER 9 B-296

are often estimates and not certain, but this is a problem sharedby the other performance criteria as well. A project with NPV =$2,500 implies that the total shareholder wealth of the firm willincrease by $2,500 if the project is accepted.

7. a. The IRR is the discount rate that causes the NPV of a series ofcash flows to be exactly zero. IRR can thus be interpreted as afinancial break-even rate of return; at the IRR, the net value ofthe project is zero. The IRR decision rule is to accept projectswith IRRs greater than the discount rate, and to reject projectswith IRRs less than the discount rate.

b. IRR is the interest rate that causes NPV for a series of cashflows to be zero. NPV is preferred in all situations to IRR; IRRcan lead to ambiguous results if there are non-conventional cashflows, and it also ambiguously ranks some mutually exclusiveprojects. However, for stand-alone projects with conventional cashflows, IRR and NPV are interchangeable techniques.

c. IRR is frequently used because it is easier for many financialmanagers and analysts to rate performance in relative terms, suchas “12%”, than in absolute terms, such as “$46,000.” IRR may be apreferred method to NPV in situations where an appropriatediscount rate is unknown are uncertain; in this situation, IRRwould provide more information about the project than would NPV.

B-297 SOLUTIONS

8. a. The profitability index is the present value of cash inflowsrelative to the project cost. As such, it is a benefit/cost ratio,providing a measure of the relative profitability of a project.The profitability index decision rule is to accept projects with aPI greater than one, and to reject projects with a PI less thanone.

b. PI = (NPV + cost)/cost = 1 + (NPV/cost). If a firm has a basket ofpositive NPV projects and is subject to capital rationing, PI mayprovide a good ranking measure of the projects, indicating the“bang for the buck” of each particular project.

9. For a project with future cash flows that are an annuity:

Payback = I / C

And the IRR is:

0 = – I + C / IRR

Solving the IRR equation for IRR, we get:

IRR = C / I

Notice this is just the reciprocal of the payback. So:

IRR = 1 / PB

For long-lived projects with relatively constant cash flows, thesooner the project pays back, the greater is the IRR.

10. There are a number of reasons. Two of the most important have to dowith transportation costs and exchange rates. Manufacturing in theU.S. places the finished product much closer to the point of sale,resulting in significant savings in transportation costs. It alsoreduces inventories because goods spend less time in transit. Higherlabor costs tend to offset these savings to some degree, at leastcompared to other possible manufacturing locations. Of greatimportance is the fact that manufacturing in the U.S. means that amuch higher proportion of the costs are paid in dollars. Since salesare in dollars, the net effect is to immunize profits to a largeextent against fluctuations in exchange rates. This issue isdiscussed in greater detail in the chapter on international finance.

CHAPTER 9 B-298

11. The single biggest difficulty, by far, is coming up with reliablecash flow estimates. Determining an appropriate discount rate is alsonot a simple task. These issues are discussed in greater depth in thenext several chapters. The payback approach is probably the simplest,followed by the AAR, but even these require revenue and costprojections. The discounted cash flow measures (discounted payback,NPV, IRR, and profitability index) are really only slightly moredifficult in practice.

12. Yes, they are. Such entities generally need to allocate available capital efficiently, just as for-profits do. However, it is frequently the case that the “revenues” from not-for-profit ventures are not tangible. For example, charitable giving has real opportunitycosts, but the benefits are generally hard to measure. To the extent that benefits are measurable, the question of an appropriate requiredreturn remains. Payback rules are commonly used in such cases. Finally, realistic cost/benefit analysis along the lines indicated should definitely be used by the U.S. government and would go a long way toward balancing the budget!

B-299 SOLUTIONS

13. The MIRR is calculated by finding the present value of all cashoutflows, the future value of all cash inflows to the end of theproject, and then calculating the IRR of the two cash flows. As aresult, the cash flows have been discounted or compounded by oneinterest rate (the required return), and then the interest ratebetween the two remaining cash flows is calculated. As such, the MIRRis not a true interest rate. In contrast, consider the IRR. If youtake the initial investment, and calculate the future value at theIRR, you can replicate the future cash flows of the project exactly.

14. The statement is incorrect. It is true that if you calculate thefuture value of all intermediate cash flows to the end of the projectat the required return, then calculate the NPV of this future valueand the initial investment, you will get the same NPV. However, NPVsays nothing about reinvestment of intermediate cash flows. The NPVis the present value of the project cash flows. What is actually donewith those cash flows once they are generated is not relevant. Putdifferently, the value of a project depends on the cash flowsgenerated by the project, not on the future value of those cashflows. The fact that the reinvestment “works” only if you use therequired return as the reinvestment rate is also irrelevant simplybecause reinvestment is not relevant in the first place to the valueof the project.

One caveat: Our discussion here assumes that the cash flows aretruly available once they are generated, meaning that it is up tofirm management to decide what to do with the cash flows. In certaincases, there may be a requirement that the cash flows be reinvested.For example, in international investing, a company may be required toreinvest the cash flows in the country in which they are generatedand not “repatriate” the money. Such funds are said to be “blocked”and reinvestment becomes relevant because the cash flows are nottruly available.

15. The statement is incorrect. It is true that if you calculate thefuture value of all intermediate cash flows to the end of the projectat the IRR, then calculate the IRR of this future value and theinitial investment, you will get the same IRR. However, as in theprevious question, what is done with the cash flows once they aregenerated does not affect the IRR. Consider the following example:

C0 C1 C2 IRRProject A –$100 $10 $110 10%

CHAPTER 9 B-300

Suppose this $100 is a deposit into a bank account. The IRR of thecash flows is 10 percent. Does the IRR change if the Year 1 cash flowis reinvested in the account, or if it is withdrawn and spent onpizza? No. Finally, consider the yield to maturity calculation on abond. If you think about it, the YTM is the IRR on the bond, but nomention of a reinvestment assumption for the bond coupons issuggested. The reason is that reinvestment is irrelevant to the YTMcalculation; in the same way, reinvestment is irrelevant in the IRRcalculation. Our caveat about blocked funds applies here as well.

B-301 SOLUTIONS



Basic

1. To calculate the payback period, we need to find the time that theproject has recovered its initial investment. After three years, theproject has created:

$1,600 + 1,900 + 2,300 = $5,800

in cash flows. The project still needs to create another:

$6,400 – 5,800 = $600

in cash flows. During the fourth year, the cash flows from theproject will be $1,400. So, the payback period will be 3 years, pluswhat we still need to make divided by what we will make during thefourth year. The payback period is:

Payback = 3 + ($600 / $1,400) = 3.43 years

2. To calculate the payback period, we need to find the time that theproject has recovered its initial investment. The cash flows in thisproblem are an annuity, so the calculation is simpler. If the initialcost is $2,400, the payback period is:

Payback = 3 + ($105 / $765) = 3.14 years

There is a shortcut to calculate the future cash flows are anannuity. Just divide the initial cost by the annual cash flow. Forthe $2,400 cost, the payback period is:

Payback = $2,400 / $765 = 3.14 years

For an initial cost of $3,600, the payback period is:

Payback = $3,600 / $765 = 4.71 years

CHAPTER 9 B-302

The payback period for an initial cost of $6,500 is a littletrickier. Notice that the total cash inflows after eight years willbe:

Total cash inflows = 8($765) = $6,120

If the initial cost is $6,500, the project never pays back. Noticethat if you use the shortcut for annuity cash flows, you get:

Payback = $6,500 / $765 = 8.50 years

This answer does not make sense since the cash flows stop after eightyears, so again, we must conclude the payback period is never.

B-303 SOLUTIONS

3. Project A has cash flows of $19,000 in Year 1, so the cash flows areshort by $21,000 of recapturing the initial investment, so thepayback for Project A is:

Payback = 1 + ($21,000 / $25,000) = 1.84 years

Project B has cash flows of:

Cash flows = $14,000 + 17,000 + 24,000 = $55,000

during this first three years. The cash flows are still short by$5,000 of recapturing the initial investment, so the payback forProject B is:

B: Payback = 3 + ($5,000 / $270,000) = 3.019 years

Using the payback criterion and a cutoff of 3 years, accept project Aand reject project B.

4. When we use discounted payback, we need to find the value of all cashflows today. The value today of the project cash flows for the firstfour years is:

Value today of Year 1 cash flow = $4,200/1.14 = $3,684.21Value today of Year 2 cash flow = $5,300/1.142 = $4,078.18Value today of Year 3 cash flow = $6,100/1.143 = $4,117.33Value today of Year 4 cash flow = $7,400/1.144 = $4,381.39

To find the discounted payback, we use these values to find thepayback period. The discounted first year cash flow is $3,684.21, sothe discounted payback for a $7,000 initial cost is:

Discounted payback = 1 + ($7,000 – 3,684.21)/$4,078.18 = 1.81 years

For an initial cost of $10,000, the discounted payback is:

Discounted payback = 2 + ($10,000 – 3,684.21 – 4,078.18)/$4,117.33 =2.54 years

Notice the calculation of discounted payback. We know the paybackperiod is between two and three years, so we subtract the discountedvalues of the Year 1 and Year 2 cash flows from the initial cost.This is the numerator, which is the discounted amount we still need

CHAPTER 9 B-304

to make to recover our initial investment. We divide this amount bythe discounted amount we will earn in Year 3 to get the fractionalportion of the discounted payback.

If the initial cost is $13,000, the discounted payback is:

Discounted payback = 3 + ($13,000 – 3,684.21 – 4,078.18 – 4,117.33) /$4,381.39 = 3.26 years

5. R = 0%: 3 + ($2,100 / $4,300) = 3.49 years discounted payback = regular payback = 3.49 years

R = 5%: $4,300/1.05 + $4,300/1.052 + $4,300/1.053 = $11,709.97$4,300/1.054 = $3,537.62discounted payback = 3 + ($15,000 – 11,709.97) / $3,537.62 =

3.93 years

B-305 SOLUTIONS

R = 19%: $4,300(PVIFA19%,6) = $14,662.04The project never pays back.

6. Our definition of AAR is the average net income divided by theaverage book value. The average net income for this project is:

Average net income = ($1,938,200 + 2,201,600 + 1,876,000 + 1,329,500)/ 4 = $1,836,325

And the average book value is:

Average book value = ($15,000,000 + 0) / 2 = $7,500,000

So, the AAR for this project is:

AAR = Average net income / Average book value = $1,836,325 /$7,500,000 = .2448 or 24.48%

7. The IRR is the interest rate that makes the NPV of the project equal tozero. So, the equation that defines the IRR for this project is:

0 = – $34,000 + $16,000/(1+IRR) + $18,000/(1+IRR)2 + $15,000/(1+IRR)3

Using a spreadsheet, financial calculator, or trial and error to findthe root of the equation, we find that:

IRR = 20.97%

Since the IRR is greater than the required return we would accept theproject.

8. The NPV of a project is the PV of the outflows minus the PV of theinflows. The equation for the NPV of this project at an 11 percentrequired return is:

NPV = – $34,000 + $16,000/1.11 + $18,000/1.112 + $15,000/1.113 =$5,991.49

At an 11 percent required return, the NPV is positive, so we wouldaccept the project.

The equation for the NPV of the project at a 30 percent requiredreturn is:

CHAPTER 9 B-306

NPV = – $34,000 + $16,000/1.30 + $18,000/1.302 + $15,000/1.303 = –$4,213.93

At a 30 percent required return, the NPV is negative, so we wouldreject the project.

9. The NPV of a project is the PV of the outflows minus the PV of theinflows. Since the cash inflows are an annuity, the equation for theNPV of this project at an 8 percent required return is:

NPV = –$138,000 + $28,500(PVIFA8%, 9) = $40,036.31

At an 8 percent required return, the NPV is positive, so we wouldaccept the project.

B-307 SOLUTIONS

The equation for the NPV of the project at a 20 percent requiredreturn is:

NPV = –$138,000 + $28,500(PVIFA20%, 9) = –$23,117.45

At a 20 percent required return, the NPV is negative, so we wouldreject the project.

We would be indifferent to the project if the required return wasequal to the IRR of the project, since at that required return theNPV is zero. The IRR of the project is:

0 = –$138,000 + $28,500(PVIFAIRR, 9)

IRR = 14.59%

10. The IRR is the interest rate that makes the NPV of the project equal tozero. So, the equation that defines the IRR for this project is:

0 = –$19,500 + $9,800/(1+IRR) + $10,300/(1+IRR)2 + $8,600/(1+IRR)3


IRR = 22.64%

11. The NPV of a project is the PV of the outflows minus the PV of theinflows. At a zero discount rate (and only at a zero discount rate),the cash flows can be added together across time. So, the NPV of theproject at a zero percent required return is:

NPV = –$19,500 + 9,800 + 10,300 + 8,600 = $9,200

The NPV at a 10 percent required return is:

NPV = –$19,500 + $9,800/1.1 + $10,300/1.12 + $8,600/1.13 = $4,382.79

The NPV at a 20 percent required return is:

NPV = –$19,500 + $9,800/1.2 + $10,300/1.22 + $8,600/1.23 = $796.30

And the NPV at a 30 percent required return is:

CHAPTER 9 B-308

NPV = –$19,500 + $9,800/1.3 + $10,300/1.32 + $8,600/1.33 = –$1,952.44

Notice that as the required return increases, the NPV of the projectdecreases. This will always be true for projects with conventionalcash flows. Conventional cash flows are negative at the beginning ofthe project and positive throughout the rest of the project.

B-309 SOLUTIONS

12. a. The IRR is the interest rate that makes the NPV of the project equalto zero. The equation for the IRR of Project A is:

0 = –$43,000 + $23,000/(1+IRR) + $17,900/(1+IRR)2 + $12,400/(1+IRR)3 +$9,400/(1+IRR)4

Using a spreadsheet, financial calculator, or trial and error tofind the root of the equation, we find that:

IRR = 20.44%

The equation for the IRR of Project B is:

0 = –$43,000 + $7,000/(1+IRR) + $13,800/(1+IRR)2 + $24,000/(1+IRR)3 +$26,000/(1+IRR)4


IRR = 18.84%

Examining the IRRs of the projects, we see that the IRRA isgreater than the IRRB, so IRR decision rule implies acceptingproject A. This may not be a correct decision; however, becausethe IRR criterion has a ranking problem for mutually exclusiveprojects. To see if the IRR decision rule is correct or not, weneed to evaluate the project NPVs.

b. The NPV of Project A is:

NPVA = –$43,000 + $23,000/1.11+ $17,900/1.112 + $12,400/1.113 + $9,400/1.114 NPVA = $7,507.61

And the NPV of Project B is:

NPVB = –$43,000 + $7,000/1.11 + $13,800/1.112 + $24,000/1.113 +$26,000/1.114 NPVB = $9,182.29

The NPVB is greater than the NPVA, so we should accept Project B.

CHAPTER 9 B-310

c. To find the crossover rate, we subtract the cash flows from oneproject from the cash flows of the other project. Here, we willsubtract the cash flows for Project B from the cash flows ofProject A. Once we find these differential cash flows, we find theIRR. The equation for the crossover rate is:

Crossover rate: 0 = $16,000/(1+R) + $4,100/(1+R)2 – $11,600/(1+R)3

– $16,600/(1+R)4


R = 15.30%

At discount rates above 15.30% choose project A; for discountrates below 15.30% choose project B; indifferent between A and Bat a discount rate of 15.30%.

B-311 SOLUTIONS

13. The IRR is the interest rate that makes the NPV of the project equalto zero. The equation to calculate the IRR of Project X is:

0 = –$15,000 + $8,150/(1+IRR) + $5,050/(1+IRR)2 + $6,800/(1+IRR)3


IRR = 16.57%

For Project Y, the equation to find the IRR is:

0 = –$15,000 + $7,700/(1+IRR) + $5,150/(1+IRR)2 + $7,250/(1+IRR)3


IRR = 16.45%

To find the crossover rate, we subtract the cash flows from oneproject from the cash flows of the other project, and find the IRR ofthe differential cash flows. We will subtract the cash flows fromProject Y from the cash flows from Project X. It is irrelevant whichcash flows we subtract from the other. Subtracting the cash flows,the equation to calculate the IRR for these differential cash flowsis:

Crossover rate: 0 = $450/(1+R) – $100/(1+R)2 – $450/(1+R)3


R = 11.73%

The table below shows the NPV of each project for different requiredreturns. Notice that Project Y always has a higher NPV for discountrates below 11.73 percent, and always has a lower NPV for discountrates above 11.73 percent.

R $NPVX $NPVY

0% $5,000.00 $5,100.005% $3,216.50 $3,267.36

CHAPTER 9 B-312

10% $1,691.59 $1,703.2315% $376.59 $356.7820% –$766.20 –$811.3425% –$1,766.40 –$1,832.00

14. a. The equation for the NPV of the project is:

NPV = –$45,000,000 + $78,000,000/1.1 – $14,000,000/1.12 =$13,482,142.86

The NPV is greater than 0, so we would accept the project.

B-313 SOLUTIONS

b. The equation for the IRR of the project is:

0 = –$45,000,000 + $78,000,000/(1+IRR) – $14,000,000/(1+IRR)2

From Descartes rule of signs, we know there are potentially two IRRssince the cash flows change signs twice. From trial and error, thetwo IRRs are:

IRR = 53.00%, –79.67%

When there are multiple IRRs, the IRR decision rule is ambiguous.Both IRRs are correct, that is, both interest rates make the NPVof the project equal to zero. If we are evaluating whether or notto accept this project, we would not want to use the IRR to makeour decision.

15. The profitability index is defined as the PV of the cash inflowsdivided by the PV of the cash outflows. The equation for theprofitability index at a required return of 10 percent is:

PI = [$7,300/1.1 + $6,900/1.12 + $5,700/1.13] / $14,000 = 1.187

The equation for the profitability index at a required return of 15percent is:

PI = [$7,300/1.15 + $6,900/1.152 + $5,700/1.153] / $14,000 = 1.094

The equation for the profitability index at a required return of 22percent is:

PI = [$7,300/1.22 + $6,900/1.222 + $5,700/1.223] / $14,000 = 0.983

We would accept the project if the required return were 10 percent or15 percent since the PI is greater than one. We would reject theproject if the required return were 22 percent since the PI is lessthan one.

16. a. The profitability index is the PV of the future cash flows dividedby the initial investment. The cash flows for both projects are anannuity, so:

PII = $27,000(PVIFA10%,3 ) / $53,000 = 1.267

CHAPTER 9 B-314

PIII = $9,100(PVIFA10%,3) / $16,000 = 1.414

The profitability index decision rule implies that we acceptproject II, since PIII is greater than the PII.

b. The NPV of each project is:

NPVI = –$53,000 + $27,000(PVIFA10%,3) = $14,145.00

NPVII = –$16,000 + $9,100(PVIFA10%,3) = $6,630.35

The NPV decision rule implies accepting Project I, since the NPVI

is greater than the NPVII.

B-315 SOLUTIONS

c. Using the profitability index to compare mutually exclusiveprojects can be ambiguous when the magnitude of the cash flows forthe two projects are of different scale. In this problem, projectI is roughly 3 times as large as project II and produces a largerNPV, yet the profitability index criterion implies that project IIis more acceptable.

17. a. The payback period for each project is:

A: 3 + ($180,000/$390,000) = 3.46 years

B: 2 + ($9,000/$18,000) = 2.50 years

The payback criterion implies accepting project B, because it paysback sooner than project A.

b. The discounted payback for each project is:

A: $20,000/1.15 + $50,000/1.152 + $50,000/1.153 = $88,074.30 $390,000/1.154 = $222,983.77

Discounted payback = 3 + ($390,000 – 88,074.30)/$222,983.77 =3.95 years

B: $19,000/1.15 + $12,000/1.152 + $18,000/1.153 = $37,430.76 $10,500/1.154 = $6,003.41

Discounted payback = 3 + ($40,000 – 37,430.76)/$6,003.41 = 3.43years

The discounted payback criterion implies accepting project Bbecause it pays back sooner than A.

c. The NPV for each project is:

A: NPV = –$300,000 + $20,000/1.15 + $50,000/1.152 + $50,000/1.153 + $390,000/1.154 NPV = $11,058.07

B: NPV = –$40,000 + $19,000/1.15 + $12,000/1.152 + $18,000/1.153 + $10,500/1.154

NPV = $3,434.16

CHAPTER 9 B-316

NPV criterion implies we accept project A because project A has ahigher NPV than project B.

d. The IRR for each project is:

A: $300,000 = $20,000/(1+IRR) + $50,000/(1+IRR)2 + $50,000/(1+IRR)3 +$390,000/(1+IRR)4


IRR = 16.20%

B-317 SOLUTIONS

B: $40,000 = $19,000/(1+IRR) + $12,000/(1+IRR)2 + $18,000/(1+IRR)3 +$10,500/(1+IRR)4


IRR = 19.50%

IRR decision rule implies we accept project B because IRR for B isgreater than IRR for A.

e. The profitability index for each project is:

A: PI = ($20,000/1.15 + $50,000/1.152 + $50,000/1.153 +$390,000/1.154) / $300,000 = 1.037

B: PI = ($19,000/1.15 + $12,000/1.152 + $18,000/1.153 +$10,500/1.154) / $40,000 = 1.086

Profitability index criterion implies accept project B because itsPI is greater than project A’s.

f. In this instance, the NPV criteria implies that you should acceptproject A, while profitability index, payback period, discountedpayback, and IRR imply that you should accept project B. The finaldecision should be based on the NPV since it does not have theranking problem associated with the other capital budgetingtechniques. Therefore, you should accept project A.

18. At a zero discount rate (and only at a zero discount rate), the cashflows can be added together across time. So, the NPV of the projectat a zero percent required return is:

NPV = –$684,680 + 263,279 + 294,060 + 227,604 + 174,356 = $274,619

If the required return is infinite, future cash flows have no value.Even if the cash flow in one year is $1 trillion, at an infinite rateof interest, the value of this cash flow today is zero. So, if thefuture cash flows have no value today, the NPV of the project issimply the cash flow today, so at an infinite interest rate:

NPV = –$684,680

CHAPTER 9 B-318

The interest rate that makes the NPV of a project equal to zero isthe IRR. The equation for the IRR of this project is:

0 = –$684,680 + $263,279/(1+IRR) + $294,060/(1+IRR)2 + $227,604/(1+IRR)3 + 174,356/(1+IRR)4


IRR = 16.23%

B-319 SOLUTIONS

19. The MIRR for the project with all three approaches is:

Discounting approach:

In the discounting approach, we find the value of all cash outflowsto time 0, while any cash inflows remain at the time at which theyoccur. So, the discounting the cash outflows to time 0, we find:

Time 0 cash flow = –$16,000 – $5,100 / 1.105

Time 0 cash flow = –$19,166.70

So, the MIRR using the discounting approach is:

0 = –$19,166.70 + $6,100/(1+MIRR) + $7,800/(1+MIRR)2 + $8,400/(1+MIRR)3 + 6,500/(1+MIRR)4


MIRR = 18.18%

Reinvestment approach:

In the reinvestment approach, we find the future value of all cashexcept the initial cash flow at the end of the project. So,reinvesting the cash flows to time 5, we find:

Time 5 cash flow = $6,100(1.104) + $7,800(1.103) + $8,400(1.102) + $6,500(1.10) – $5,100Time 5 cash flow = $31,526.81


0 = –$16,000 + $31,526.81/(1+MIRR)5 $31,526.81 / $16,000 = (1+MIRR)5 MIRR = ($31,526.81 / $16,000)1/5 – 1MIRR = .1453 or 14.53%

Combination approach:

In the combination approach, we find the value of all cash outflowsat time 0, and the value of all cash inflows at the end of theproject. So, the value of the cash flows is:

CHAPTER 9 B-320

Time 0 cash flow = –$16,000 – $5,100 / 1.105


Time 5 cash flow = $6,100(1.104) + $7,800(1.103) + $8,400(1.102) + $6,500(1.10) Time 5 cash flow = $36,626.81


0 = –$19,166.70 + $36,626.81/(1+MIRR)5 $36,626.81 / $19,166.70 = (1+MIRR)5 MIRR = ($36,626.81 / $19,166.70)1/5 – 1MIRR = .1383 or 13.83%

B-321 SOLUTIONS

Intermediate

20. With different discounting and reinvestment rates, we need to make sure to use the appropriate interest rate. The MIRR for the project with all three approaches is:

Discounting approach:

In the discounting approach, we find the value of all cash outflowsto time 0 at the discount rate, while any cash inflows remain at thetime at which they occur. So, the discounting the cash outflows totime 0, we find:

Time 0 cash flow = –$16,000 – $5,100 / 1.115



0 = –$19,026.60 + $6,100/(1+MIRR) + $7,800/(1+MIRR)2 + $8,400/(1+MIRR)3 + 6,500/(1+MIRR)4


MIRR = 18.55%

Reinvestment approach:

In the reinvestment approach, we find the future value of all cashexcept the initial cash flow at the end of the project using thereinvestment rate. So, the reinvesting the cash flows to time 5, wefind:

Time 5 cash flow = $6,100(1.084) + $7,800(1.083) + $8,400(1.082) + $6,500(1.08) – $5,100Time 5 cash flow = $29,842.50


0 = –$16,000 + $29,842.50/(1+MIRR)5 $29,842.50 / $16,000 = (1+MIRR)5 MIRR = ($29,842.50 / $16,000)1/5 – 1MIRR = .1328 or 13.28%

CHAPTER 9 B-322

Combination approach:

In the combination approach, we find the value of all cash outflowsat time 0 using the discount rate, and the value of all cash inflowsat the end of the project using the reinvestment rate. So, the valueof the cash flows is:

Time 0 cash flow = –$16,000 – $5,100 / 1.115


Time 5 cash flow = $6,100(1.084) + $7,800(1.083) + $8,400(1.082) + $6,500(1.08) Time 5 cash flow = $34,942.50

B-323 SOLUTIONS


0 = –$19,026.60 + $34,942.50/(1+MIRR)5 $34,942.50 / $19,026.60 = (1+MIRR)5 MIRR = ($34,942.50 / $19,026.60)1/5 – 1MIRR = .1293 or 12.93%

21. Since the NPV index has the cost subtracted in the numerator, NPVindex = PI – 1.

22. a. To have a payback equal to the project’s life, given C is aconstant cash flow for N years:

C = I/N

b. To have a positive NPV, I < C (PVIFAR%, N). Thus, C > I / (PVIFAR%,

N).

c. Benefits = C (PVIFAR%, N) = 2 × costs = 2IC = 2I / (PVIFAR%, N)

Challenge

23. Given the seven year payback, the worst case is that the paybackoccurs at the end of the seventh year. Thus, the worst-case:

NPV = –$724,000 + $724,000/1.127 = –$396,499.17

The best case has infinite cash flows beyond the payback point. Thus,the best-case NPV is infinite.

24. The equation for the IRR of the project is:

0 = –$1,512 + $8,586/(1 + IRR) – $18,210/(1 + IRR)2 + $17,100/(1 +IRR)3 – $6,000/(1 + IRR)4

Using Descartes rule of signs, from looking at the cash flows we knowthere are four IRRs for this project. Even with most computerspreadsheets, we have to do some trial and error. From trial anderror, IRRs of 25%, 33.33%, 42.86%, and 66.67% are found.

CHAPTER 9 B-324

We would accept the project when the NPV is greater than zero. Seefor yourself if that NPV is greater than zero for required returnsbetween 25% and 33.33% or between 42.86% and 66.67%.

25. a. Here the cash inflows of the project go on forever, which is aperpetuity. Unlike ordinary perpetuity cash flows, the cash flowshere grow at a constant rate forever, which is a growingperpetuity. If you remember back to the chapter on stockvaluation, we presented a formula for valuing a stock withconstant growth in dividends. This formula is actually the formulafor a growing perpetuity, so we can use it here. The PV of thefuture cash flows from the project is:

PV of cash inflows = C1/(R – g) PV of cash inflows = $85,000/(.13 – .06) = $1,214,285.71

B-325 SOLUTIONS

NPV is the PV of the outflows minus the PV of the inflows, so theNPV is:

NPV of the project = –$1,400,000 + 1,214,285.71 = –$185,714.29

The NPV is negative, so we would reject the project.

b. Here we want to know the minimum growth rate in cash flowsnecessary to accept the project. The minimum growth rate is thegrowth rate at which we would have a zero NPV. The equation for azero NPV, using the equation for the PV of a growing perpetuityis:

0 = –$1,400,000 + $85,000/(.13 – g)

Solving for g, we get:

g = .0693 or 6.93%

26. The IRR of the project is:

$58,000 = $34,000/(1+IRR) + $45,000/(1+IRR)2


IRR = 22.14%

At an interest rate of 12 percent, the NPV is:

NPV = $58,000 – $34,000/1.12 – $45,000/1.122 NPV = –$8,230.87

At an interest rate of zero percent, we can add cash flows, so theNPV is:

NPV = $58,000 – $34,000 – $45,000 NPV = –$21,000.00

And at an interest rate of 24 percent, the NPV is:

NPV = $58,000 – $34,000/1.24 – $45,000/1.242 NPV = +$1,314.26

CHAPTER 9 B-326

The cash flows for the project are unconventional. Since the initialcash flow is positive and the remaining cash flows are negative, thedecision rule for IRR in invalid in this case. The NPV profile isupward sloping, indicating that the project is more valuable when theinterest rate increases.

B-327 SOLUTIONS

27. The IRR is the interest rate that makes the NPV of the project equalto zero. So, the IRR of the project is:

0 = $20,000 – $26,000 / (1 + IRR) + $13,000 / (1 + IRR)2

Even though it appears there are two IRRs, a spreadsheet, financialcalculator, or trial and error will not give an answer. The reason isthat there is no real IRR for this set of cash flows. If you examinethe IRR equation, what we are really doing is solving for the rootsof the equation. Going back to high school algebra, in this problemwe are solving a quadratic equation. In case you don’t remember, thequadratic equation is:

x = −b±√b2−4ac

2a

In this case, the equation is:

x = −(−26,000)±√(−26,000)2−4(20,000)(13,000)

2(26,000)

The square root term works out to be:

676,000,000 – 1,040,000,000 = –364,000,000

The square root of a negative number is a complex number, so there isno real number solution, meaning the project has no real IRR.

28. First, we need to find the future value of the cash flows for the oneyear in which they are blocked by the government. So, reinvesting each cash inflow for one year, we find:

Year 2 cash flow = $205,000(1.04) = $213,200Year 3 cash flow = $265,000(1.04) = $275,600Year 4 cash flow = $346,000(1.04) = $359,840Year 5 cash flow = $220,000(1.04) = $228,800

So, the NPV of the project is:

NPV = –$450,000 + $213,200/1.112 + $275,600/1.113 + $359,840/1.114 + $228,800/1.115

NPV = –$2,626.33

CHAPTER 9 B-328

And the IRR of the project is:

0 = –$450,000 + $213,200/(1 + IRR)2 + $275,600/(1 + IRR)3 +$359,840/(1 + IRR)4

+ $228,800/(1 + IRR)5


IRR = 10.89%

While this may look like a MIRR calculation, it is not an MIRR,rather it is a standard IRR calculation. Since the cash inflows areblocked by the government, they are not available to the company fora period of one year. Thus, all we are doing is calculating the IRRbased on when the cash flows actually occur for the company.


7.CFo –$34,000C01 $16,000F01 1C02 $18,000F02 1C03 $15,000F03 1

IRR CPT20.97%

8.CFo –$34,000 CFo –$34,000C01 $16,000 C01 $16,000F01 1 F01 1C02 $18,000 C02 $18,000F02 1 F02 1C03 $15,000 C03 $15,000F03 1 F03 1

I = 11% I = 30%NPV CPT NPV CPT

B-329 SOLUTIONS

$5,991.49 –$4,213.93

9.CFo –$138,000 CFo –

$138,000CFo –$138,000

C01 $28,500 C01 $28,500 C01 $28,500F01 9 F01 9 F01 9

I = 8% I = 20% IRR CPT NPV CPT NPV CPT 14.59%$40,036.31 –$23,117.45

CHAPTER 9 B-330

10.CFo –$19,500C01 $9,800F01 1C02 $10,300F02 1C03 $8,600F03 1

IRR CPT 22.64%

11.CFo –$19,500 CFo –$19,500C01 $9,800 C01 $9,800F01 1 F01 1C02 $10,300 C02 $10,300F02 1 F02 1C03 $8,600 C03 $8,600F03 1 F03 1

I = 0% I = 10%NPV CPT NPV CPT$9,200 $4.382.79

CFo –$19,500 CFo –$19,500C01 $9,800 C01 $9,800F01 1 F01 1C02 $10,300 C02 $10,300F02 1 F02 1C03 $8,600 C03 $8,600F03 1 F03 1

I = 20% I = 30% NPV CPT NPV CPT$796.30 –$1,952.44

12. Project ACFo –$43,000 CFo –$43,000C01 $23,000 C01 $23,000F01 1 F01 1C02 $17,900 C02 $17,900

B-331 SOLUTIONS

F02 1 F02 1C03 $12,400 C03 $12,400F03 1 F03 1C04 $9,400 C04 $9,400F04 1 F04 1

IRR CPT I = 11% 20.44% NPV CPT

$7,507.61

CHAPTER 9 B-332

Project BCFo –$43,000 CFo –$43,000C01 $7,000 C01 $7,000F01 1 F01 1C02 $13,800 C02 $13,800F02 1 F02 1C03 $24,000 C03 $24,000F03 1 F03 1C04 $26,000 C04 $26,000F04 1 F04 1

IRR CPT I = 11%18.84% NPV CPT

$9,182.29

Crossover rate

CFo $0C01 $16,000F01 1C02 $4,100F02 1C03 –$11,600F03 1C04 –$16,600F04 1

IRR CPT 15.30%

13. Project XCFo –$15,000 CFo –$15,000 CFo –$15,000C01 $8,150 C01 $8,150 C01 $8,150F01 1 F01 1 F01 1C02 $5,050 C02 $5,050 C02 $5,050F02 1 F02 1 F02 1C03 $6,800 C03 $6,800 C03 $6,800F03 1 F03 1 F03 1

I = 0% I = 15% I = 25%NPV CPT NPV CPT NPV CPT$5,000.00 $376.59 –$1,766.40

B-333 SOLUTIONS

Project YCFo –$15,000 CFo –$15,000 CFo –$15,000C01 $7,700 C01 $7,700 C01 $7,700F01 1 F01 1 F01 1C02 $5,150 C02 $5,150 C02 $5,150F02 1 F02 1 F02 1C03 $7,250 C03 $7,250 C03 $7,250F03 1 F03 1 F03 1

I = 0% I = 15% I = 25%NPV CPT NPV CPT NPV CPT$5,100.00 $356.78 –$1,832.00

Crossover rate

CFo $0C01 $450F01 1C02 –$100F02 1C03 –$450F03 1

IRR CPT 11.73%

14.CFo –

$45,000,000

CFo –$45,000,000

C01 $78,000,000

C01 $78,000,000

F01 1 F01 1C02 –

$14,000,000

C02 –$14,000,000

F02 1 F02 1I = 10% IRR CPTNPV CPT 53.00%$13,482,142.86

CHAPTER 9 B-334

Financial calculators will only give you one IRR, even if there aremultiple IRRs. Using trial and error, or a root solving calculator,the other IRR is –79.67%.

B-335 SOLUTIONS

15.CFo $0 CFo $0 CFo $0C01 $7,300 C01 $7,300 C01 $7,300F01 1 F01 1 F01 1C02 $6,900 C02 $6,900 C02 $6,900F02 1 F02 1 F02 1C03 $5,700 C03 $5,700 C03 $5,700F03 1 F03 1 F03 1

I = 10% I = 15% I = 22%NPV CPT NPV CPT NPV CPT$16,621.34 $15,313.06 $13,758.49

@10%: PI = $16,621.34 / $14,000 = 1.187@15%: PI = $15,313.06 / $14,000 = 1.094@22%: PI = $13,758.49 / $14,000 = 0.983

16. Project ICFo $0 CFo –$53,000C01 $27,000 C01 $27,000F01 3 F01 3

I = 10% I = 10% NPV CPT NPV CPT$67,145.00 $14,145.00

PI = $67,145.00 / $53,000 = 1.267

Project IICFo $0 CFo –$16,000C01 $9,100 C01 $9,100F01 3 F01 3

I = 10% I = 10%NPV CPT NPV CPT$22,630.35 $6,630.35

PI = $22,630.35 / $16,000 = 1.414

17.CF(A) c. d. e.

Cfo –$300,000 CFo –$300,000

CFo $0

C01 $20,000 C01 $20,000 C01 $20,000F01 1 F01 1 F01 1

CHAPTER 9 B-336

C02 $50,000 C02 $50,000 C02 $50,000F02 2 F02 2 F02 2C03 $390,000 C03 $390,000 C03 $390,000F03 1 F03 1 F03 1

I = 15% IRR CPT I = 15%NPV CPT 16.20% NPV CPT$11,058.07 $311,058.07

PI = $311,058.07 / $300,000 = 1.037

B-337 SOLUTIONS

CF(B) c. d. e.CFo –$40,000 CFo –$40,000 CFo $0C01 $19,000 C01 $19,000 C01 $19,000F01 1 F01 1 F01 1C02 $12,000 C02 $12,000 C02 $12,000F02 1 F02 1 F02 1C03 $18,000 C03 $18,000 C03 $18,000F03 1 F03 1 F03 1C04 $10,500 C04 $10,500 C04 $10,500F04 1 F04 1 F04 1

I = 15% IRR CPT I = 15% NPV CPT 19.50% NPV CPT$3,434.16 $43,434.16

PI = $43,434.16 / $40,000 = 1.086

f. In this instance, the NPV criteria implies that you should acceptproject A, while payback period, discounted payback, profitabilityindex, and IRR imply that you should accept project B. The finaldecision should be based on the NPV since it does not have theranking problem associated with the other capital budgetingtechniques. Therefore, you should accept project A.

18.CFo –$684,680 CFo –

$684,680C01 $263,279 C01 $263,279F01 1 F01 1C02 $294,060 C02 $294,060F02 1 F02 1C03 $227,604 C03 $227,604F03 1 F03 1C04 $174,356 C04 $174,356F04 1 F04 1

I = 0% IRR CPT NPV CPT 16.23%$274,619

CHAPTER 10MAKING CAPITAL INVESTMENT DECISIONS


1. In this context, an opportunity cost refers to the value of an assetor other input that will be used in a project. The relevant cost iswhat the asset or input is actually worth today, not, for example,what it cost to acquire.

2. For tax purposes, a firm would choose MACRS because it provides forlarger depreciation deductions earlier. These larger deductionsreduce taxes, but have no other cash consequences. Notice that thechoice between MACRS and straight-line is purely a time value issue;the total depreciation is the same, only the timing differs.

3. It’s probably only a mild over-simplification. Current liabilitieswill all be paid, presumably. The cash portion of current assets willbe retrieved. Some receivables won’t be collected, and some inventorywill not be sold, of course. Counterbalancing these losses is thefact that inventory sold above cost (and not replaced at the end ofthe project’s life) acts to increase working capital. These effectstend to offset one another.

4. Management’s discretion to set the firm’s capital structure isapplicable at the firm level. Since any one particular project couldbe financed entirely with equity, another project could be financedwith debt, and the firm’s overall capital structure remainsunchanged, financing costs are not relevant in the analysis of aproject’s incremental cash flows according to the stand-aloneprinciple.

5. The EAC approach is appropriate when comparing mutually exclusiveprojects with different lives that will be replaced when they wearout. This type of analysis is necessary so that the projects have acommon life span over which they can be compared; in effect, each

B-339 SOLUTIONS

project is assumed to exist over an infinite horizon of N-yearrepeating projects. Assuming that this type of analysis is validimplies that the project cash flows remain the same forever, thusignoring the possible effects of, among other things: (1) inflation,(2) changing economic conditions, (3) the increasing unreliability ofcash flow estimates that occur far into the future, and (4) thepossible effects of future technology improvement that could alterthe project cash flows.

6. Depreciation is a non-cash expense, but it is tax-deductible on theincome statement. Thus depreciation causes taxes paid, an actual cashoutflow, to be reduced by an amount equal to the depreciation taxshield tcD. A reduction in taxes that would otherwise be paid is thesame thing as a cash inflow, so the effects of the depreciation taxshield must be added in to get the total incremental aftertax cashflows.

7. There are two particularly important considerations. The first iserosion. Will the essentialized book simply displace copies of theexisting book that would have otherwise been sold? This is of specialconcern given the lower price. The second consideration iscompetition. Will other publishers step in and produce such aproduct? If so, then any erosion is much less relevant. A particularconcern to book publishers (and producers of a variety of otherproduct types) is that the publisher only makes money from the saleof new books. Thus, it is important to examine whether the new bookwould displace sales of used books (good from the publisher’sperspective) or new books (not good). The concern arises any timethere is an active market for used product.

8. Definitely. The damage to Porsche’s reputation is definitely a factorthe company needed to consider. If the reputation was damaged, thecompany would have lost sales of its existing car lines.

9. One company may be able to produce at lower incremental cost ormarket better. Also, of course, one of the two may have made amistake!

10. Porsche would recognize that the outsized profits would dwindle asmore product comes to market and competition becomes more intense.


CHAPTER 10 B-340


Basic

1. The $6 million acquisition cost of the land six years ago is a sunkcost. The $6.4 million current aftertax value of the land is anopportunity cost if the land is used rather than sold off. The $14.2million cash outlay and $890,000 grading expenses are the initialfixed asset investments needed to get the project going. Therefore,the proper year zero cash flow to use in evaluating this project is

$6,400,000 + 14,200,000 + 890,000 = $21,490,000

2. Sales due solely to the new product line are:

19,000($13,000) = $247,000,000

Increased sales of the motor home line occur because of the newproduct line introduction; thus:

4,500($53,000) = $238,500,000

in new sales is relevant. Erosion of luxury motor coach sales is alsodue to the new mid-size campers; thus:

900($91,000) = $81,900,000 loss in sales

is relevant. The net sales figure to use in evaluating the new lineis thus:

$247,000,000 + 238,500,000 – 81,900,000 = $403,600,000

B-341 SOLUTIONS

3. We need to construct a basic income statement. The income statementis:

Sales $ 830,000Variable costs 498,000Fixed costs 181,000Depreciation 77,000 EBT $ 74,000Taxes@35% 25,900 Net income $ 48,100

4. To find the OCF, we need to complete the income statement as follows:

Sales $ 824,500Costs 538,900 Depreciation 126,500 EBT $ 159,100Taxes@34% 54,094 Net income $ 105,006

The OCF for the company is:

OCF = EBIT + Depreciation – TaxesOCF = $159,100 + 126,500 – 54,094 OCF = $231,506

The depreciation tax shield is the depreciation times the tax rate,so:

Depreciation tax shield = tcDepreciationDepreciation tax shield = .34($126,500) Depreciation tax shield = $43,010

The depreciation tax shield shows us the increase in OCF by beingable to expense depreciation.

5. To calculate the OCF, we first need to calculate net income. Theincome statement is:

Sales $ 108,000Variable costs 51,000Depreciation 6,800 EBT $ 50,200

CHAPTER 10 B-342

Taxes@35% 17,570 Net income $ 32,630

Using the most common financial calculation for OCF, we get:

OCF = EBIT + Depreciation – Taxes OCF = $50,200 + 6,800 – 17,570 OCF = $39,430

B-343 SOLUTIONS

The top-down approach to calculating OCF yields:

OCF = Sales – Costs – Taxes OCF = $108,000 – 51,000 – 17,570 OCF = $39,430

The tax-shield approach is:

OCF = (Sales – Costs)(1 – tC) + tCDepreciation OCF = ($108,000 – 51,000)(1 – .35) + .35(6,800) OCF = $39,430

And the bottom-up approach is:

OCF = Net income + Depreciation OCF = $32,630 + 6,800 OCF = $39,430

All four methods of calculating OCF should always give the sameanswer.

6. The MACRS depreciation schedule is shown in Table 10.7. The endingbook value for any year is the beginning book value minus thedepreciation for the year. Remember, to find the amount ofdepreciation for any year, you multiply the purchase price of theasset times the MACRS percentage for the year. The depreciationschedule for this asset is:

Year

Beginning BookValue MACRS

Depreciation

Ending Bookvalue

1 $1,080,000.00 0.1429$154,332.0

0 $925,668.002 925,668.00 0.2449 264,492.00 661,176.003 661,176.00 0.1749 188,892.00 472,284.004 472,284.00 0.1249 134,892.00 337,392.005 337,392.00 0.0893 96,444.00 240,948.006 240,948.00 0.0892 96,336.00 144,612.007 144,612.00 0.0893 96,444.00 48,168.008 48,168.00 0.0446 48,168.00 0

CHAPTER 10 B-344

7. The asset has an 8 year useful life and we want to find the BV of theasset after 5 years. With straight-line depreciation, thedepreciation each year will be:

Annual depreciation = $548,000 / 8 Annual depreciation = $68,500

So, after five years, the accumulated depreciation will be:

Accumulated depreciation = 5($68,500)Accumulated depreciation = $342,500

B-345 SOLUTIONS

The book value at the end of year five is thus:

BV5 = $548,000 – 342,500 BV5 = $205,500

The asset is sold at a loss to book value, so the depreciation taxshield of the loss is recaptured.

Aftertax salvage value = $105,000 + ($205,500 – 105,000)(0.35) Aftertax salvage value = $140,175

To find the taxes on salvage value, remember to use the equation:

Taxes on salvage value = (BV – MV)tc

This equation will always give the correct sign for a tax inflow(refund) or outflow (payment).

8. To find the BV at the end of four years, we need to find theaccumulated depreciation for the first four years. We could calculatea table as in Problem 6, but an easier way is to add the MACRSdepreciation amounts for each of the first four years and multiplythis percentage times the cost of the asset. We can then subtractthis from the asset cost. Doing so, we get:

BV4 = $7,900,000 – 7,900,000(0.2000 + 0.3200 + 0.1920 + 0.1152) BV4 = $1,365,120

The asset is sold at a gain to book value, so this gain is taxable.

Aftertax salvage value = $1,400,000 + ($1,365,120 – 1,400,000)(.35) Aftertax salvage value = $1,387,792

9. Using the tax shield approach to calculating OCF (Remember theapproach is irrelevant; the final answer will be the same no matterwhich of the four methods you use.), we get:

OCF = (Sales – Costs)(1 – tC) + tCDepreciation OCF = ($2,650,000 – 840,000)(1 – 0.35) + 0.35($3,900,000/3) OCF = $1,631,500

10. Since we have the OCF, we can find the NPV as the initial cash outlayplus the PV of the OCFs, which are an annuity, so the NPV is:

CHAPTER 10 B-346

NPV = –$3,900,000 + $1,631,500(PVIFA12%,3) NPV = $18,587.71

B-347 SOLUTIONS

11. The cash outflow at the beginning of the project will increasebecause of the spending on NWC. At the end of the project, thecompany will recover the NWC, so it will be a cash inflow. The saleof the equipment will result in a cash inflow, but we also mustaccount for the taxes which will be paid on this sale. So, the cashflows for each year of the project will be:

Year Cash Flow0 –

$4,200,000

= –$3,900,000 – 300,000

1 1,631,5002 1,631,5003 2,068,000 = $1,631,500 + 300,000 + 210,000 + (0 –

210,000)(.35)

And the NPV of the project is:

NPV = –$4,200,000 + $1,631,500(PVIFA12%,2) + ($2,068,000 / 1.123) NPV = $29,279.79

12. First we will calculate the annual depreciation for the equipmentnecessary for the project. The depreciation amount each year will be:

Year 1 depreciation = $3,900,000(0.3333) = $1,299,870 Year 2 depreciation = $3,900,000(0.4445) = $1,733,550Year 3 depreciation = $3,900,000(0.1481) = $577,590

So, the book value of the equipment at the end of three years, whichwill be the initial investment minus the accumulated depreciation,is:

Book value in 3 years = $3,900,000 – ($1,299,870 + 1,733,550 +577,590) Book value in 3 years = $288,990

The asset is sold at a loss to book value, so this loss is taxabledeductible.

Aftertax salvage value = $210,000 + ($288,990 – 210,000)(0.35) Aftertax salvage value = $237,647

CHAPTER 10 B-348

To calculate the OCF, we will use the tax shield approach, so thecash flow each year is:

OCF = (Sales – Costs)(1 – tC) + tCDepreciation

Year Cash Flow0 –

$4,200,000 = –$3,900,000 – 300,000

1 1,631,454.50

= ($1,810,000)(.65) + 0.35($1,299,870)

2 1,783,242.50

= ($1,810,000)(.65) + 0.35($1,733,550)

3 1,916,303.00

= ($1,810,000)(.65) + 0.35($577,590) + $237,647 + 300,000

Remember to include the NWC cost in Year 0, and the recovery of theNWC at the end of the project. The NPV of the project with theseassumptions is:

NPV = –$4,200,000 + ($1,631,454.50/1.12) + ($1,783,242.50/1.122) + ($1,916,303.00/1.123) NPV = $42,232.43

B-349 SOLUTIONS

13. First we will calculate the annual depreciation of the new equipment.It will be:

Annual depreciation = $560,000/5 Annual depreciation = $112,000

Now, we calculate the aftertax salvage value. The aftertax salvagevalue is the market price minus (or plus) the taxes on the sale ofthe equipment, so:

Aftertax salvage value = MV + (BV – MV)tc

Very often the book value of the equipment is zero as it is in thiscase. If the book value is zero, the equation for the aftertaxsalvage value becomes:

Aftertax salvage value = MV + (0 – MV)tc Aftertax salvage value = MV(1 – tc)

We will use this equation to find the aftertax salvage value since weknow the book value is zero. So, the aftertax salvage value is:

Aftertax salvage value = $85,000(1 – 0.34) Aftertax salvage value = $56,100

Using the tax shield approach, we find the OCF for the project is:

OCF = $165,000(1 – 0.34) + 0.34($112,000) OCF = $146,980

Now we can find the project NPV. Notice we include the NWC in theinitial cash outlay. The recovery of the NWC occurs in Year 5, alongwith the aftertax salvage value.

NPV = –$560,000 – 29,000 + $146,980(PVIFA10%,5) + [($56,100 + 29,000) /1.105] NPV = $21,010.24

14. First we will calculate the annual depreciation of the new equipment.It will be:

Annual depreciation charge = $720,000/5 Annual depreciation charge = $144,000

CHAPTER 10 B-350

The aftertax salvage value of the equipment is:

Aftertax salvage value = $75,000(1 – 0.35) Aftertax salvage value = $48,750

Using the tax shield approach, the OCF is:

OCF = $260,000(1 – 0.35) + 0.35($144,000) OCF = $219,400

B-351 SOLUTIONS

Now we can find the project IRR. There is an unusual feature that isa part of this project. Accepting this project means that we willreduce NWC. This reduction in NWC is a cash inflow at Year 0. Thisreduction in NWC implies that when the project ends, we will have toincrease NWC. So, at the end of the project, we will have a cashoutflow to restore the NWC to its level before the project. We alsomust include the aftertax salvage value at the end of the project.The IRR of the project is:

NPV = 0 = –$720,000 + 110,000 + $219,400(PVIFAIRR%,5) + [($48,750 –110,000) / (1+IRR)5]

IRR = 21.65%

15. To evaluate the project with a $300,000 cost savings, we need the OCFto compute the NPV. Using the tax shield approach, the OCF is:

OCF = $300,000(1 – 0.35) + 0.35($144,000) = $245,400

NPV = –$720,000 + 110,000 + $245,400(PVIFA20%,5) + [($48,750 – 110,000)/ (1.20)5]NPV = $99,281.22

The NPV with a $240,000 cost savings is:

OCF = $240,000(1 – 0.35) + 0.35($144,000) OCF = $206,400

NPV = –$720,000 + 110,000 + $206,400(PVIFA20%,5) + [($48,750 – 110,000)/ (1.20)5]NPV = –$17,352.66

We would accept the project if cost savings were $300,000, and rejectthe project if the cost savings were $240,000. The required pretaxcost savings that would make us indifferent about the project is thecost savings that results in a zero NPV. The NPV of the project is:

NPV = 0 = –$720,000 + $110,000 + OCF(PVIFA20%,5) + [($48,750 – 110,000)/ (1.20)5]

Solving for the OCF, we find the necessary OCF for zero NPV is:

OCF = $212,202.38

CHAPTER 10 B-352

Using the tax shield approach to calculating OCF, we get:

OCF = $212,202.38 = (S – C)(1 – 0.35) + 0.35($144,000)(S – C) = $248,926.73

The cost savings that will make us indifferent is $248,926.73.

B-353 SOLUTIONS

16. To calculate the EAC of the project, we first need the NPV of theproject. Notice that we include the NWC expenditure at the beginningof the project, and recover the NWC at the end of the project. TheNPV of the project is:

NPV = –$270,000 – 25,000 – $42,000(PVIFA11%,5) + $25,000/1.115 = –$435,391.39

Now we can find the EAC of the project. The EAC is:

EAC = –$435,391.39 / (PVIFA11%,5) = –$117,803.98

17. We will need the aftertax salvage value of the equipment to computethe EAC. Even though the equipment for each product has a differentinitial cost, both have the same salvage value. The aftertax salvagevalue for both is:

Both cases: aftertax salvage value = $40,000(1 – 0.35) = $26,000

To calculate the EAC, we first need the OCF and NPV of each option.The OCF and NPV for Techron I is:

OCF = –$67,000(1 – 0.35) + 0.35($290,000/3) = –9,716.67

NPV = –$290,000 – $9,716.67(PVIFA10%,3) + ($26,000/1.103) = –$294,629.73

EAC = –$294,629.73 / (PVIFA10%,3) = –$118,474.97

And the OCF and NPV for Techron II is:

OCF = –$35,000(1 – 0.35) + 0.35($510,000/5) = $12,950

NPV = –$510,000 + $12,950(PVIFA10%,5) + ($26,000/1.105) = –$444,765.36

EAC = –$444,765.36 / (PVIFA10%,5) = –$117,327.98

The two milling machines have unequal lives, so they can only becompared by expressing both on an equivalent annual basis, which iswhat the EAC method does. Thus, you prefer the Techron II because ithas the lower (less negative) annual cost.

CHAPTER 10 B-354

18. To find the bid price, we need to calculate all other cash flows forthe project, and then solve for the bid price. The aftertax salvagevalue of the equipment is:

Aftertax salvage value = $70,000(1 – 0.35) = $45,500

Now we can solve for the necessary OCF that will give the project azero NPV. The equation for the NPV of the project is:

NPV = 0 = –$940,000 – 75,000 + OCF(PVIFA12%,5) + [($75,000 + 45,500) /1.125]

B-355 SOLUTIONS

Solving for the OCF, we find the OCF that makes the project NPV equalto zero is:

OCF = $946,625.06 / PVIFA12%,5 = $262,603.01

The easiest way to calculate the bid price is the tax shieldapproach, so:

OCF = $262,603.01 = [(P – v)Q – FC ](1 – tc) + tcD$262,603.01 = [(P – $9.25)(185,000) – $305,000 ](1 – 0.35) +0.35($940,000/5) P = $12.54

Intermediate

19. First, we will calculate the depreciation each year, which will be:

D1 = $560,000(0.2000) = $112,000 D2 = $560,000(0.3200) = $179,200D3 = $560,000(0.1920) = $107,520 D4 = $560,000(0.1152) = $64,512

The book value of the equipment at the end of the project is:

BV4 = $560,000 – ($112,000 + 179,200 + 107,520 + 64,512) = $96,768

The asset is sold at a loss to book value, so this creates a taxrefund.After-tax salvage value = $80,000 + ($96,768 – 80,000)(0.35) =$85,868.80

So, the OCF for each year will be:

OCF1 = $210,000(1 – 0.35) + 0.35($112,000) = $172,700OCF2 = $210,000(1 – 0.35) + 0.35($179,200) = $196,220OCF3 = $210,000(1 – 0.35) + 0.35($107,520) = $171,132OCF4 = $210,000(1 – 0.35) + 0.35($64,512) = $159,079.20

Now we have all the necessary information to calculate the projectNPV. We need to be careful with the NWC in this project. Notice theproject requires $20,000 of NWC at the beginning, and $3,000 more inNWC each successive year. We will subtract the $20,000 from theinitial cash flow, and subtract $3,000 each year from the OCF to

CHAPTER 10 B-356

account for this spending. In Year 4, we will add back the totalspent on NWC, which is $29,000. The $3,000 spent on NWC capitalduring Year 4 is irrelevant. Why? Well, during this year the projectrequired an additional $3,000, but we would get the money backimmediately. So, the net cash flow for additional NWC would be zero.With all this, the equation for the NPV of the project is:

NPV = – $560,000 – 20,000 + ($172,700 – 3,000)/1.09 + ($196,220 –3,000)/1.092

+ ($171,132 – 3,000)/1.093 + ($159,079.20 + 29,000+ 85,868.80)/1.094 NPV = $69,811.79

B-357 SOLUTIONS

20. If we are trying to decide between two projects that will not bereplaced when they wear out, the proper capital budgeting method touse is NPV. Both projects only have costs associated with them, notsales, so we will use these to calculate the NPV of each project.Using the tax shield approach to calculate the OCF, the NPV of SystemA is:

OCFA = –$110,000(1 – 0.34) + 0.34($430,000/4) OCFA = –$36,050

NPVA = –$430,000 – $36,050(PVIFA11%,4) NPVA = –$541,843.17

And the NPV of System B is:

OCFB = –$98,000(1 – 0.34) + 0.34($570,000/6) OCFB = –$32,380

NPVB = –$570,000 – $32,380(PVIFA11%,6) NPVB = –$706,984.82

If the system will not be replaced when it wears out, then System Ashould be chosen, because it has the more positive NPV.

21. If the equipment will be replaced at the end of its useful life, thecorrect capital budgeting technique is EAC. Using the NPVs wecalculated in the previous problem, the EAC for each system is:

EACA = –$541,843.17 / (PVIFA11%,4) EACA = –$174,650.33

EACB = – $706,984.82 / (PVIFA11%,6) EACB = –$167,114.64

If the conveyor belt system will be continually replaced, we shouldchoose System B since it has the more positive EAC.

22. To find the bid price, we need to calculate all other cash flows forthe project, and then solve for the bid price. The aftertax salvagevalue of the equipment is:

After-tax salvage value = $540,000(1 – 0.34) After-tax salvage value = $356,400

CHAPTER 10 B-358

Now we can solve for the necessary OCF that will give the project azero NPV. The current aftertax value of the land is an opportunitycost, but we also need to include the aftertax value of the land infive years since we can sell the land at that time. The equation forthe NPV of the project is:

NPV = 0 = –$4,100,000 – 2,700,000 – 600,000 + OCF(PVIFA12%,5) –$50,000(PVIFA12%,4)

+ {($356,400 + 600,000 + 4(50,000) + 3,200,000] /1.125}

B-359 SOLUTIONS

Solving for the OCF, we find the OCF that makes the project NPV equalto zero is:

OCF = $5,079,929.11 / PVIFA12%,5 OCF = $1,409,221.77

The easiest way to calculate the bid price is the tax shieldapproach, so:

OCF = $1,409,221.77 = [(P – v)Q – FC ](1 – tC) + tcD$1,409,221.77 = [(P – $0.005)(100,000,000) – $950,000](1 – 0.34) +0.34($4,100,000/5) P = $0.03163

23. At a given price, taking accelerated depreciation compared tostraight-line depreciation causes the NPV to be higher; similarly, ata given price, lower net working capital investment requirements willcause the NPV to be higher. Thus, NPV would be zero at a lower pricein this situation. In the case of a bid price, you could submit alower price and still break-even, or submit the higher price and makea positive NPV.

24. Since we need to calculate the EAC for each machine, sales areirrelevant. EAC only uses the costs of operating the equipment, notthe sales. Using the bottom up approach, or net income plusdepreciation, method to calculate OCF, we get:

Machine A Machine B

Variable costs –$3,500,000 –$3,000,000

Fixed costs –170,000 –130,000 Depreciation –483,333 –566,667 EBT –$4,153,333 –$3,696,667 Tax 1,453,667 1,293,833 Net income –$2,699,667 –$2,402,833

+ Depreciation

483,333 566,667

OCF –$2,216,333 –$1,836,167

The NPV and EAC for Machine A is:

NPVA = –$2,900,000 – $2,216,333(PVIFA10%,6)

CHAPTER 10 B-360

NPVA = –$12,552,709.46

EACA = – $12,552.709.46 / (PVIFA10%,6) EACA = –$2,882,194.74

And the NPV and EAC for Machine B is:

NPVB = –$5,100,000 – 1,836,167(PVIFA10%,9) NPVB = –$15,674,527.56

EACB = – $15,674,527.56 / (PVIFA10%,9) EACB = –$2,721,733.42

You should choose Machine B since it has a more positive EAC.

B-361 SOLUTIONS

25. A kilowatt hour is 1,000 watts for 1 hour. A 60-watt bulb burning for500 hours per year uses

30,000 watt hours, or 30 kilowatt hours. Since the cost of a kilowatthour is $0.101, the cost per year is:

Cost per year = 30($0.101)Cost per year = $3.03

The 60-watt bulb will last for 1,000 hours, which is 2 years of use at 500 hours per year. So, the NPV of the 60-watt bulb is:

NPV = –$0.50 – $3.03(PVIFA10%,2)NPV = –$5.76

And the EAC is:

EAC = –$5.83 / (PVIFA10%,2) EAC = –$3.32

Now we can find the EAC for the 15-watt CFL. A 15-watt bulb burning for 500 hours per year uses 7,500 watts, or 7.5 kilowatts. And, sincethe cost of a kilowatt hour is $0.101, the cost per year is:

Cost per year = 7.5($0.101)Cost per year = $0.7575

The 15-watt CFL will last for 12,000 hours, which is 24 years of use at 500 hours per year. So, the NPV of the CFL is:

NPV = –$3.50 – $0.7575(PVIFA10%,24)NPV = –$10.31

And the EAC is:

EAC = –$10.85 / (PVIFA10%,24) EAC = –$1.15

Thus, the CFL is much cheaper. But see our next two questions.

26. To solve the EAC algebraically for each bulb, we can set up the variables as follows:

W = light bulb wattage

CHAPTER 10 B-362

C = cost per kilowatt hourH = hours burned per yearP = price the light bulb

The number of watts use by the bulb per hour is:

WPH = W / 1,000

And the kilowatt hours used per year is:

KPY = WPH × H

B-363 SOLUTIONS

The electricity cost per year is therefore:

ECY = KPY × C

The NPV of the decision to but the light bulb is:

NPV = – P – ECY(PVIFAR%,t)

And the EAC is:

EAC = NPV / (PVIFAR%,t)

Substituting, we get:

EAC = [–P – (W / 1,000 × H × C)PVIFAR%,t] / PFIVAR%,t

We need to set the EAC of the two light bulbs equal to each other andsolve for C, the cost per kilowatt hour. Doing so, we find:

[–$0.50 – (60 / 1,000 × 500 × C)PVIFA10%,2] / PVIFA10%,2 = [–$3.50 – (15 / 1,000 × 500 × C)PVIFA10%,24] / PVIFA10%,24

C = $0.004509

So, unless the cost per kilowatt hour is extremely low, it makes sense to use the CFL. But when should you replace the incandescent bulb? See the next question.

27. We are again solving for the breakeven kilowatt hour cost, but now the incandescent bulb has only 500 hours of useful life. In this case, the incandescent bulb has only one year of life left. The breakeven electricity cost under these circumstances is:

[–$0.50 – (60 / 1,000 × 500 × C)PVIFA10%,1] / PVIFA10%,1 = [–$3.50 – (15 / 1,000 × 500 × C)PVIFA10%,24] / PVIFA10%,24

C = –$0.007131

Unless the electricity cost is negative (Not very likely!), it does not make financial sense to replace the incandescent bulb until it burns out.

CHAPTER 10 B-364

28. The debate between incandescent bulbs and CFLs is not just afinancial debate, but an environmental one as well. The numbers belowcorrespond to the numbered items in the question:

1. The extra heat generated by an incandescent bulb is waste, but notnecessarily in a heated structure, especially in northernclimates.

2. Since CFLs last so long, from a financial viewpoint, it might makesense to wait if prices are declining.

3. Because of the nontrivial health and disposal issues, CFLs are notas attractive as our previous analysis suggests.

B-365 SOLUTIONS

4. From a company’s perspective, the cost of replacing workingincandescent bulbs may outweigh the financial benefit. However,since CFLs last longer, the cost of replacing the bulbs will belower in the long run.

5. Because incandescent bulbs use more power, more coal has to beburned, which generates more mercury in the environment,potentially offsetting the mercury concern with CFLs.

6. As in the previous question, if CO2 production is an environmentalconcern, the the lower power consumption from CFLs is a benefit.

7. CFLs require more energy to make, potentially offsetting (at leastpartially) the energy savings from their use. Worker safety andsite contamination are also negatives for CFLs.

8. This fact favors the incandescent bulb because the purchasers willonly receive part of the benefit from the CFL.

9. This fact favors waiting for new technology.

10. This fact also favors waiting for new technology.

While there is always a “best” answer, this question shows that theanalysis of the “best” answer is not always easy and may not bepossible because of incomplete data. As for how to better legislatethe use of CFLs, our analysis suggests that requiring them in newconstruction might make sense. Rental properties in general shouldprobably be required to use CFLs (why rentals?).

Another piece of legislation that makes sense is requiring theproducers of CFLs to supply a disposal kit and proper disposalinstructions with each one sold. Finally, we need much betterresearch on the hazards associated with broken bulbs in the home andworkplace and proper procedures for dealing with broken bulbs.

29. Surprise! You should definitely upgrade the truck. Here’s why. At10 mpg, the truck burns 12,000 / 10 = 1,200 gallons of gas per year.The new truck will burn 12,000 / 12.5 = 960 gallons of gas per year,a savings of 240 gallons per year. The car burns 12,000 / 25 = 480gallons of gas per year, while the new car will burn 12,000 / 40 =300 gallons of gas per year, a savings of 180 gallons per year, soit’s not even close.

CHAPTER 10 B-366

This answer may strike you as counterintuitive, so let’s consider anextreme case. Suppose the car gets 6,000 mpg, and you could upgradeto 12,000 mpg. Should you upgrade? Probably not since you would onlysave one gallon of gas per year. So, the reason you should upgradethe truck is that it uses so much more gas in the first place.

Notice that the answer doesn’t depend on the cost of gasoline,meaning that if you upgrade, you should always upgrade the truck. Infact, it doesn’t depend on the miles driven, as long as the milesdriven are the same.

B-367 SOLUTIONS

30. Surprise! You should definitely upgrade the truck. Here’s why. At 10mpg, the truck burns 12,000 / 10 = 1,200

Challenge

31. We will begin by calculating the aftertax salvage value of theequipment at the end of the project’s life. The aftertax salvagevalue is the market value of the equipment minus any taxes paid (orrefunded), so the aftertax salvage value in four years will be:

Taxes on salvage value = (BV – MV)tC

Taxes on salvage value = ($0 – 400,000)(.38)Taxes on salvage value = –$152,000

Market price $400,000 Tax on sale –152,000

Aftertax salvage value $248,000

Now we need to calculate the operating cash flow each year. Using thebottom up approach to calculating operating cash flow, we find:

Year 0 Year 1 Year 2 Year 3 Year 4

Revenues$2,496,0

00$3,354,0

00$3,042,0

00$2,184,0

00 Fixed costs 425,000 425,000 425,000 425,000

Variable costs 374,400 503,100 456,300 327,600

Depreciation

1,399,860

1,866,900 622,020 311,220

EBT $296,740 $559,000$1,538,6

80$1,120,1

80 Taxes 112,761 212,420 584,698 425,668 Net income $183,979 $346,580 $953,982 $694,512

OCF$1,583,8

39$2,213,4

80$1,576,0

02$1,005,7

32

Capital spending

–$4,200,00

0 $248,000

Land–

1,500,0001,600,00

0

CHAPTER 10 B-368

NWC –125,000 125,000

Total cash flow

–$5,825,00

0$1,583,8

39$2,213,4

80$1,576,0

02$2,978,7

32

Notice the calculation of the cash flow at time 0. The capitalspending on equipment and investment in net working capital are cashoutflows are both cash outflows. The aftertax selling price of theland is also a cash outflow. Even though no cash is actually spent onthe land because the company already owns it, the aftertax cash flowfrom selling the land is an opportunity cost, so we need to includeit in the analysis. The company can sell the land at the end of theproject, so we need to include that value as well. With all theproject cash flows, we can calculate the NPV, which is:

NPV = –$5,825,000 + $1,583,839 / 1.13 + $2,213,480 / 1.132 +$1,576,002 / 1.133

+ $2,978,732 / 1.134

NPV = $229,266.82

The company should accept the new product line.

32. This is an in-depth capital budgeting problem. Probably the easiestOCF calculation for this problem is the bottom up approach, so wewill construct an income statement for each year. Beginning with theinitial cash flow at time zero, the project will require aninvestment in equipment. The project will also require an investmentin NWC. The initial NWC investment is given, and the subsequent NWCinvestment will be 15 percent of the next year’s sales. In this case,it will be Year 1 sales. Realizing we need Year 1 sales to calculatethe required NWC capital at time 0, we find that Year 1 sales will be$35,340,000. So, the cash flow required for the project today willbe:

Capital spending

–$24,000,000

Initial NWC –

1,800,000

Total cash flow

–$25,800,000

B-369 SOLUTIONS

Now we can begin the remaining calculations. Sales figures are givenfor each year, along with the price per unit. The variable costs perunit are used to calculate total variable costs, and fixed costs aregiven at $1,200,000 per year. To calculate depreciation each year, weuse the initial equipment cost of $24 million, times the appropriateMACRS depreciation each year. The remainder of each income statementis calculated below. Notice at the bottom of the income statement weadded back depreciation to get the OCF for each year. The sectionlabeled “Net cash flows” will be discussed below:

Year 1 2 3 4 5

Ending book value

$20,570,400

$14,692,800

$10,495,200

$7,497,600

$5,354,400

Sales$35,340,0

00$39,900,0

00$48,640,0

00$50,920,0

00$33,060,0

00

Variable costs

24,645,000

27,825,000

33,920,000

35,510,000

23,055,000

Fixed costs 1,200,000 1,200,000 1,200,000 1,200,000 1,200,000 Depreciation 3,429,600 5,877,600 4,197,600 2,997,600 2,143,200

EBIT$6,065,40

0$4,997,40

0$9,322,40

0$11,212,4

00$6,661,80

0 Taxes 2,122,890 1,749,090 3,262,840 3,924,340 2,331,630

Net income$3,942,51

0$3,248,31

0$6,059,56

0$7,288,06

0$4,330,17

0 Depreciation 3,429,600 5,877,600 4,197,600 2,997,600 2,143,200

Operating cash flow

$7,372,110

$9,125,910

$10,257,160

$10,285,660

$6,473,370

Net cash flows

Operating cash flow

$7,372,110

$9,125,910

$10,257,160

$10,285,660

$6,473,370

Change in NWC –684,000–

1,311,000 –342,000 2,679,000 1,458,000

Capital spending 0 0 0 0 4,994,040

Total cash flow

$6,688,110

$7,814,910

$9,915,160

$12,964,660

$12,925,410

After we calculate the OCF for each year, we need to account for anyother cash flows. The other cash flows in this case are NWC cashflows and capital spending, which is the aftertax salvage of the

CHAPTER 10 B-370

equipment. The required NWC capital is 15 percent of the increase insales in the next year. We will work through the NWC cash flow forYear 1. The total NWC in Year 1 will be 15 percent of sales increasefrom Year 1 to Year 2, or:

Increase in NWC for Year 1 = .15($39,900,000 – 35,340,000) Increase in NWC for Year 1 = $684,000

Notice that the NWC cash flow is negative. Since the sales areincreasing, we will have to spend more money to increase NWC. In Year4, the NWC cash flow is positive since sales are declining. And, inYear 5, the NWC cash flow is the recovery of all NWC the companystill has in the project.

To calculate the aftertax salvage value, we first need the book valueof the equipment. The book value at the end of the five years will bethe purchase price, minus the total depreciation. So, the ending bookvalue is:

Ending book value = $24,000,000 – ($3,429,600 + 5,877,600 + 4,197,600+ 2,997,600

+ 2,143,200) Ending book value = $5,354,400

The market value of the used equipment is 20 percent of the purchaseprice, or $4.8 million, so the aftertax salvage value will be:

Aftertax salvage value = $4,800,000 + ($5,354,400 – 4,800,000)(.35) Aftertax salvage value = $4,994,040

The aftertax salvage value is included in the total cash flows arecapital spending. Now we have all of the cash flows for the project.The NPV of the project is:

NPV = –$25,800,000 + $6,688,110/1.18 + $7,814,910/1.182 + $9,915,160/1.183

+ $12,964,660/1.184 + $12,925,410/1.185

NPV = $3,851,952.23

And the IRR is:

NPV = 0 = –$25,800,000 + $6,688,110/(1 + IRR) + $7,814,910/(1 + IRR)2

B-371 SOLUTIONS

+ $9,915,160/(1 + IRR)3 + $12,964,660/(1 + IRR)4 + $12,925,410/(1 + IRR)5

IRR = 23.62%

We should accept the project.

33. To find the initial pretax cost savings necessary to buy the newmachine, we should use the tax shield approach to find the OCF. Webegin by calculating the depreciation each year using the MACRSdepreciation schedule. The depreciation each year is:

D1 = $610,000(0.3333) = $203,313D2 = $610,000(0.4444) = $271,145D3 = $610,000(0.1482) = $90,341D4 = $610,000(0.0741) = $45,201

Using the tax shield approach, the OCF each year is:

OCF1 = (S – C)(1 – 0.35) + 0.35($203,313)OCF2 = (S – C)(1 – 0.35) + 0.35($271,145)OCF3 = (S – C)(1 – 0.35) + 0.35($90,341)OCF4 = (S – C)(1 – 0.35) + 0.35($45,201)OCF5 = (S – C)(1 – 0.35)

Now we need the aftertax salvage value of the equipment. The aftertaxsalvage value is:

After-tax salvage value = $40,000(1 – 0.35) = $26,000

To find the necessary cost reduction, we must realize that we cansplit the cash flows each year. The OCF in any given year is the costreduction (S – C) times one minus the tax rate, which is an annuityfor the project life, and the depreciation tax shield. To calculatethe necessary cost reduction, we would require a zero NPV. Theequation for the NPV of the project is:

NPV = 0 = –$610,000 – 55,000 + (S – C)(0.65)(PVIFA12%,5) +0.35($203,313/1.12

+ $271,145/1.122 + $90,341/1.123 + $45,201/1.124) +($55,000 + 26,000)/1.125

Solving this equation for the sales minus costs, we get:

CHAPTER 10 B-372

(S – C)(0.65)(PVIFA12%,5) = $447,288.67(S – C) = $190,895.74

34. a. This problem is basically the same as Problem18, except we are given a sales price. The cash flow at Time 0 for all three parts of this question will be:

Capital spending

–$940,000

Change in NWC –

75,000

Total cash flow

–$1,015,000

We will use the initial cash flow and the salvage value we alreadyfound in that problem. Using the bottom up approach to calculatingthe OCF, we get:

Assume price per unit = $13 and units/year = 185,000Year 1 2 3 4 5Sales $2,405,00

0 $2,405,000$2,405,00

0$2,405,00

0$2,405,00

0Variablecosts 1,711,250 1,711,250 1,711,250 1,711,250 1,711,250Fixed costs 305,000 305,000 305,000 305,000 305,000Depreciation 188,000 188,000 188,000 188,000 188,000EBIT 200,750 200,750 200,750 200,750 200,750Taxes (35%) 70,263 70,263 70,263 70,263 70,263Net Income 130,488 130,488 130,488 130,488 130,488Depreciation 188,000 188,000 188,000 188,000 188,000Operating CF $318,488 $318,488 $318,488 $318,488 $318,488

Year 1 2 3 4 5Operating CF $318,488 $318,488 $318,488 $318,488 $318,488Change in NWC 0 0 0 0 75,000Capitalspending 0 0 0 0 45,500Total CF $318,488 $318,488 $318,488 $318,488 $438,988

With these cash flows, the NPV of the project is:

B-373 SOLUTIONS

NPV = –$940,000 – 75,000 + $318,488(PVIFA12%,5) + [($75,000 + 45,500) /1.125]NPV = $201,451.10

If the actual price is above the bid price that results in a zero NPV,the project will have a positive NPV. As for the cartons sold, if thenumber of cartons sold increases, the NPV will increase, and if thecosts increase, the NPV will decrease.

b. To find the minimum number of cartons sold to still breakeven, weneed to use the tax shield approach to calculating OCF, and solve theproblem similar to finding a bid price. Using the initial cash flowand salvage value we already calculated, the equation for a zero NPVof the project is:

NPV = 0 = –$940,000 – 75,000 + OCF(PVIFA12%,5) + [($75,000 + 45,500) /1.125]

So, the necessary OCF for a zero NPV is:

OCF = $946,625.06 / PVIFA12%,5 = $262,603.01

Now we can use the tax shield approach to solve for the minimumquantity as follows:

OCF = $262,603.01 = [(P – v)Q – FC ](1 – tc) + tcD$262,603.01 = [($13.00 – 9.25)Q – 305,000 ](1 – 0.35) + 0.35($940,000/5)

Q = 162,073

As a check, we can calculate the NPV of the project with this quantity. The calculations are:

Year 1 2 3 4 5Sales $2,106,94

9 $2,106,949 $2,106,949 $2,106,949 $2,106,949Variablecosts 1,499,176 1,499,176 1,499,176 1,499,176 1,499,176Fixed costs 305,000 305,000 305,000 305,000 305,000Depreciation 188,000 188,000 188,000 188,000 188,000EBIT 114,774 114,774 114,774 114,774 114,774Taxes (35%) 40,171 40,171 40,171 40,171 40,171

CHAPTER 10 B-374

Net Income 74,603 74,603 74,603 74,603 74,603Depreciation 188,000 188,000 188,000 188,000 188,000Operating CF $262,603 $262,603 $262,603 $262,603 $262,603


NPV = –$940,000 – 75,000 + $262,603(PVIFA12%,5) + [($75,000 + 45,500) /1.125] $0

Note, the NPV is not exactly equal to zero because we had to roundthe number of cartons sold; you cannot sell one-half of a carton.

c. To find the highest level of fixed costs and still breakeven, weneed to use the tax shield approach to calculating OCF, and solve theproblem similar to finding a bid price. Using the initial cash flowand salvage value we already calculated, the equation for a zero NPVof the project is:

NPV = 0 = –$940,000 – 75,000 + OCF(PVIFA12%,5) + [($75,000 + 45,500) /1.125]OCF = $946,625.06 / PVIFA12%,5 = $262,603.01

Notice this is the same OCF we calculated in part b. Now we can usethe tax shield approach to solve for the maximum level of fixed costsas follows:

OCF = $262,603.01 = [(P–v)Q – FC ](1 – tC) + tCD$262,603.01 = [($13.00 – 9.25)(185,000) – FC](1 – 0.35) +0.35($940,000/5) FC = $390,976.15

As a check, we can calculate the NPV of the project with this level offixed costs. The calculations

are:

Year 1 2 3 4 5Sales $2,405,00 $2,405,000 $2,405,000 $2,405,000 $2,405,000

B-375 SOLUTIONS

0Variablecosts 1,711,250 1,711,250 1,711,250 1,711,250 1,711,250Fixed costs 390,976 390,976 390,976 390,976 390,976Depreciation 188,000 188,000 188,000 188,000 188,000EBIT 114,774 114,774 114,774 114,774 114,774Taxes (35%) 40,171 40,171 40,171 40,171 40,171Net Income 74,603 74,603 74,603 74,603 74,603Depreciation 188,000 188,000 188,000 188,000 188,000Operating CF $262,603 $262,603 $262,603 $262,603 $262,603


NPV = –$940,000 – 75,000 + $262,603(PVIFA12%,5) + [($75,000 + 45,500) /1.125] $0

35. We need to find the bid price for a project, but the project hasextra cash flows. Since we don’t already produce the keyboard, thesales of the keyboard outside the contract are relevant cash flows.Since we know the extra sales number and price, we can calculate thecash flows generated by these sales. The cash flow generated from thesale of the keyboard outside the contract is:

1 2 3 4

Sales$855,00

0$1,710,0

00$2,280,0

00$1,425,0

00

Variable costs 525,0001,050,00

01,400,00

0 875,000

EBT$330,00

0 $660,000 $880,000 $550,000 Tax 132,000 264,000 352,000 220,000

Net income (and OCF)

$198,000 $396,000 $528,000 $330,000

So, the addition to NPV of these market sales is:

CHAPTER 10 B-376

NPV of market sales = $198,000/1.13 + $396,000/1.132 + $528,000/1.133 + $330,000/1.134 NPV of market sales = $1,053,672.99

You may have noticed that we did not include the initial cash outlay,depreciation or fixed costs in the calculation of cash flows from themarket sales. The reason is that it is irrelevant whether or not weinclude these here. Remember, we are not only trying to determine thebid price, but we are also determining whether or not the project isfeasible. In other words, we are trying to calculate the NPV of theproject, not just the NPV of the bid price. We will include thesecash flows in the bid price calculation. The reason we stated earlierthat whether we included these costs in this initial calculation wasirrelevant is that you will come up with the same bid price if youinclude these costs in this calculation, or if you include them inthe bid price calculation.

Next, we need to calculate the aftertax salvage value, which is:

Aftertax salvage value = $275,000(1 – .40) = $165,000

Instead of solving for a zero NPV as is usual in setting a bid price,the company president requires an NPV of $100,000, so we will solvefor a NPV of that amount. The NPV equation for this project is(remember to include the NWC cash flow at the beginning of theproject, and the NWC recovery at the end):

NPV = $100,000 = –$3,400,000 – 95,000 + 1,053,672.99 + OCF (PVIFA13%,4)+ [($165,000 + 95,000) / 1.134]

Solving for the OCF, we get:

OCF = $2,381,864.14 / PVIFA13%,4 = $800,768.90

Now we can solve for the bid price as follows:

OCF = $800,768.90 = [(P – v)Q – FC ](1 – tC) + tCD $800,768.90 = [(P – $175)(17,500) – $600,000](1 – 0.40) +

0.40($3,400,000/4)P = $253.17

36. a. Since the two computers have unequal lives, the correct methodto analyze the decision is the EAC. We will begin with the EAC of

B-377 SOLUTIONS

the new computer. Using the depreciation tax shield approach, theOCF for the new computer system is:

OCF = ($145,000)(1 – .38) + ($780,000 / 5)(.38) = $149,180

Notice that the costs are positive, which represents a cashinflow. The costs are positive in this case since the new computerwill generate a cost savings. The only initial cash flow for thenew computer is cost of $780,000. We next need to calculate theaftertax salvage value, which is:

Aftertax salvage value = $150,000(1 – .38) = $93,000

Now we can calculate the NPV of the new computer as:

NPV = –$780,000 + $149,180(PVIFA12%,5) + $93,000 / 1.125

NPV = –$189,468.79

And the EAC of the new computer is:

EAC = –$189,468.79 / (PVIFA12%,5) = –$52,560.49

Analyzing the old computer, the only OCF is the depreciation tax shield, so:

OCF = $130,000(.38) = $49,400

The initial cost of the old computer is a little trickier. Youmight assume that since we already own the old computer there isno initial cost, but we can sell the old computer, so there is anopportunity cost. We need to account for this opportunity cost. Todo so, we will calculate the aftertax salvage value of the oldcomputer today. We need the book value of the old computer to doso. The book value is not given directly, but we are told that theold computer has depreciation of $130,000 per year for the nextthree years, so we can assume the book value is the total amountof depreciation over the remaining life of the system, or$390,000. So, the aftertax salvage value of the old computer is:

Aftertax salvage value = $210,000 + ($390,000 – 210,000)(.38) = $377,200

CHAPTER 10 B-378

This is the initial cost of the old computer system today becausewe are forgoing the opportunity to sell it today. We next need tocalculate the aftertax salvage value of the computer system in twoyears since we are “buying” it today. The aftertax salvage valuein two years is:

Aftertax salvage value = $60,000 + ($130,000 – 60,000)(.38) = $86,600

Now we can calculate the NPV of the old computer as:

NPV = –$377,200 + $49,400(PVIFA11%,2) + 136,000 / 1.122

NPV = –$224,647.49

And the EAC of the old computer is:

EAC = –$224,674.49 / (PVIFA12%,2) = –$132,939.47

Even if we are going to replace the system in two years no matterwhat our decision today, we should replace it today since the EACis more positive.

B-379 SOLUTIONS

b. If we are only concerned with whether or not to replace themachine now, and are not worrying about what will happen in twoyears, the correct analysis is NPV. To calculate the NPV of thedecision on the computer system now, we need the difference in thetotal cash flows of the old computer system and the new computersystem. From our previous calculations, we can say the cash flowsfor each computer system are:

tNew

computerOld

computer Difference0 –$780,000 –$377,200 –$402,8001 149,180 49,400 99,7802 149,180 136,000 13,1803 149,180 0 149,1804 149,180 0 149,1805 242,180 0 242,180

Since we are only concerned with marginal cash flows, the cashflows of the decision to replace the old computer system with thenew computer system are the differential cash flows. The NPV ofthe decision to replace, ignoring what will happen in two yearsis:

NPV = –$402,800 + $99,780/1.12 + $13,180/1.122 + $149,180/1.143 + $149,180/1.144

+ $242,180/1.145

NPV = $35,205.70

If we are not concerned with what will happen in two years, weshould replace the old computer system.

CHAPTER 11PROJECT ANALYSIS AND EVALUATION


1. Forecasting risk is the risk that a poor decision is made because oferrors in projected cash flows. The danger is greatest with a newproduct because the cash flows are probably harder to predict.

2. With a sensitivity analysis, one variable is examined over a broadrange of values. With a scenario analysis, all variables are examinedfor a limited range of values.

3. It is true that if average revenue is less than average cost, thefirm is losing money. This much of the statement is thereforecorrect. At the margin, however, accepting a project with marginalrevenue in excess of its marginal cost clearly acts to increaseoperating cash flow.

4. It makes wages and salaries a fixed cost, driving up operatingleverage.

5. Fixed costs are relatively high because airlines are relativelycapital intensive (and airplanes are expensive). Skilled employeessuch as pilots and mechanics mean relatively high wages which,because of union agreements, are relatively fixed. Maintenanceexpenses are significant and relatively fixed as well.

6. From the shareholder perspective, the financial break-even point isthe most important. A project can exceed the accounting and cashbreak-even points but still be below the financial break-even point.This causes a reduction in shareholder (your) wealth.

7. The project will reach the cash break-even first, the accountingbreak-even next and finally the financial break-even. For a projectwith an initial investment and sales after, this ordering will alwaysapply. The cash break-even is achieved first since it excludesdepreciation. The accounting break-even is next since it includes

B-381 SOLUTIONS

depreciation. Finally, the financial break-even, which includes thetime value of money, is achieved.

8. Soft capital rationing implies that the firm as a whole isn’t shortof capital, but the division or project does not have the necessarycapital. The implication is that the firm is passing up positive NPVprojects. With hard capital rationing the firm is unable to raisecapital for a project under any circumstances. Probably the mostcommon reason for hard capital rationing is financial distress,meaning bankruptcy is a possibility.

9. The implication is that they will face hard capital rationing.

CHAPTER 11 B-382



Basic

1. a. The total variable cost per unit is the sum of the two variablecosts, so:

Total variable costs per unit = $5.43 + 3.13 Total variable costs per unit = $8.56

b. The total costs include all variable costs and fixed costs. Weneed to make sure we are including all variable costs for thenumber of units produced, so:

Total costs = Variable costs + Fixed costs Total costs = $8.56(280,000) + $720,000 Total costs = $3,116,800

c. The cash breakeven, that is the point where cash flow is zero, is:

QC = $720,000 / ($19.99 – 8.56) QC = 62,992.13 units

And the accounting breakeven is:

QA = ($720,000 + 220,000) / ($19.99 – 8.56) QA = 82,239.72 units

2. The total costs include all variable costs and fixed costs. We needto make sure we are including all variable costs for the number ofunits produced, so:

Total costs = ($24.86 + 14.08)(120,000) + $1,550,000 Total costs = $6,222,800

The marginal cost, or cost of producing one more unit, is the totalvariable cost per unit, so:

B-383 SOLUTIONS

Marginal cost = $24.86 + 14.08 Marginal cost = $38.94

CHAPTER 11 B-384

The average cost per unit is the total cost of production, divided bythe quantity produced, so:

Average cost = Total cost / Total quantity Average cost = $6,222,800/120,000 Average cost = $51.86

Minimum acceptable total revenue = 5,000($38.94) Minimum acceptable total revenue = $194,700

Additional units should be produced only if the cost of producingthose units can be recovered.

3. The base-case, best-case, and worst-case values are shown below.Remember that in the best-case, sales and price increase, while costsdecrease. In the worst-case, sales and price decrease, and costsincrease.

UnitScenario Unit Sales Unit Price Variable CostFixed CostsBase 95,000 $1,900.00 $240.00 $4,800,000Best 109,250 $2,185.00 $204.00 $4,080,000 Worst 80,750 $1,615.00 $276.00 $5,520,000

4. An estimate for the impact of changes in price on the profitabilityof the project can be found from the sensitivity of NPV with respectto price: NPV/P. This measure can be calculated by finding the NPVat any two different price levels and forming the ratio of thechanges in these parameters. Whenever a sensitivity analysis isperformed, all other variables are held constant at their base-casevalues.

5. a. To calculate the accounting breakeven, we first need to find thedepreciation for each year. The depreciation is:

Depreciation = $724,000/8 Depreciation = $90,500 per year


QA = ($780,000 + 90,500)/($43 – 29) QA = 62,179 units

B-385 SOLUTIONS

To calculate the accounting breakeven, we must realize at thispoint (and only this point), the OCF is equal to depreciation. So,the DOL at the accounting breakeven is:

DOL = 1 + FC/OCF = 1 + FC/D DOL = 1 + [$780,000/$90,500] DOL = 9.919

b. We will use the tax shield approach to calculate the OCF. The OCFis:

OCFbase = [(P – v)Q – FC](1 – tc) + tcD OCFbase = [($43 – 29)(90,000) – $780,000](0.65) + 0.35($90,500) OCFbase = $343,675

CHAPTER 11 B-386

Now we can calculate the NPV using our base-case projections.There is no salvage value or NWC, so the NPV is:

NPVbase = –$724,000 + $343,675(PVIFA15%,8) NPVbase = $818,180.22

To calculate the sensitivity of the NPV to changes in the quantitysold, we will calculate the NPV at a different quantity. We willuse sales of 95,000 units. The NPV at this sales level is: OCFnew = [($43 – 29)(95,000) – $780,000](0.65) + 0.35($90,500) OCFnew = $389,175

And the NPV is:

NPVnew = –$724,000 + $389,175(PVIFA15%,8) NPVnew = $1,022,353.35

So, the change in NPV for every unit change in sales is:

NPV/S = ($1,022,353.35 – 818,180.22)/(95,000 – 90,000) NPV/S = +$40.835

If sales were to drop by 500 units, then NPV would drop by:

NPV drop = $40.835(500) = $20,417.31

You may wonder why we chose 95,000 units. Because it doesn’tmatter! Whatever sales number we use, when we calculate the changein NPV per unit sold, the ratio will be the same.

c. To find out how sensitive OCF is to a change in variable costs, wewill compute the OCF at a variable cost of $30. Again, the numberwe choose to use here is irrelevant: We will get the same ratio ofOCF to a one dollar change in variable cost no matter whatvariable cost we use. So, using the tax shield approach, the OCFat a variable cost of $30 is:

OCFnew = [($43 – 30)(90,000) – 780,000](0.65) + 0.35($90,500) OCFnew = $285,175

So, the change in OCF for a $1 change in variable costs is:

B-387 SOLUTIONS

OCF/v = ($285,175,450 – 343,675)/($30 – 29) OCF/v = –$58,500

If variable costs decrease by $1 then, OCF would increase by$58,500

CHAPTER 11 B-388

6. We will use the tax shield approach to calculate the OCF for thebest- and worst-case scenarios. For the best-case scenario, the priceand quantity increase by 10 percent, so we will multiply the basecase numbers by 1.1, a 10 percent increase. The variable and fixedcosts both decrease by 10 percent, so we will multiply the base casenumbers by .9, a 10 percent decrease. Doing so, we get:

OCFbest = {[($43)(1.1) – ($29)(0.9)](90,000)(1.1) – $780,000(0.9)}(0.65) + 0.35($90,500) OCFbest = $939,595

The best-case NPV is:

NPVbest = –$724,000 + $939,595(PVIFA15%,8) NPVbest = $3,492,264.85

For the worst-case scenario, the price and quantity decrease by 10percent, so we will multiply the base case numbers by .9, a 10percent decrease. The variable and fixed costs both increase by 10percent, so we will multiply the base case numbers by 1.1, a 10percent increase. Doing so, we get:

OCFworst = {[($43)(0.9) – ($29)(1.1)](90,000)(0.9) – $780,000(1.1)}(0.65) + 0.35($90,500) OCFworst = –$168,005

The worst-case NPV is:

NPVworst = –$724,000 – $168,005(PVIFA15%,8) NPVworst = –$1,477,892.45

7. The cash breakeven equation is:

QC = FC/(P – v)

And the accounting breakeven equation is:

QA = (FC + D)/(P – v)

Using these equations, we find the following cash and accountingbreakeven points:

B-389 SOLUTIONS

(1): QC = $14M/($3,020 – 2,275) QA =($14M + 6.5M)/($3,020 – 2,275)

QC = 18,792 QA = 27,517

(2): QC = $73,000/($38 – 27) QA =($73,000 + 150,000)/($38 – 27)

QC = 6,636 QA = 20,273

(3): QC = $1,200/($11 – 4) QA

= ($1,200 + 840)/($11 – 4) QC = 171 QA = 291

CHAPTER 11 B-390

8. We can use the accounting breakeven equation:

QA = (FC + D)/(P – v)

to solve for the unknown variable in each case. Doing so, we find:

(1): QA = 112,800 = ($820,000 + D)/($41– 30)

D = $420,800

(2): QA = 165,000 = ($3.2M + 1.15M)/(P –$43)

P = $69.36

(3): QA = 4,385 = ($160,000 +105,000)/($98 – v)

v = $37.57

9. The accounting breakeven for the project is:

QA = [$6,000 + ($18,000/4)]/($57 – 32) QA = 540

And the cash breakeven is:

QC = $9,000/($57 – 32) QC = 360

At the financial breakeven, the project will have a zero NPV. Sincethis is true, the initial cost of the project must be equal to the PVof the cash flows of the project. Using this relationship, we canfind the OCF of the project must be:

NPV = 0 implies $18,000 = OCF(PVIFA12%,4) OCF = $5,926.22

Using this OCF, we can find the financial breakeven is:

QF = ($9,000 + $5,926.22)/($57 – 32) = 597

And the DOL of the project is:

DOL = 1 + ($9,000/$5,926.22) = 2.519

B-391 SOLUTIONS

10. In order to calculate the financial breakeven, we need the OCF of theproject. We can use the cash and accounting breakeven points to findthis. First, we will use the cash breakeven to find the price of theproduct as follows:

QC = FC/(P – v) 13,200 = $140,000/(P – $24) P = $34.61

CHAPTER 11 B-392

Now that we know the product price, we can use the accountingbreakeven equation to find the depreciation. Doing so, we find theannual depreciation must be:

QA = (FC + D)/(P – v) 15,500 = ($140,000 + D)/($34.61 – 24) Depreciation = $24,394

We now know the annual depreciation amount. Assuming straight-linedepreciation is used, the initial investment in equipment must befive times the annual depreciation, or:

Initial investment = 5($24,394) = $121,970

The PV of the OCF must be equal to this value at the financialbreakeven since the NPV is zero, so:

$121,970 = OCF(PVIFA16%,5) OCF = $37,250.69

We can now use this OCF in the financial breakeven equation to findthe financial breakeven sales quantity is:

QF = ($140,000 + 37,250.69)/($34.61 – 24) QF = 16,712

11. We know that the DOL is the percentage change in OCF divided by thepercentage change in quantity sold. Since we have the original andnew quantity sold, we can use the DOL equation to find the percentagechange in OCF. Doing so, we find:

DOL = %OCF / %Q

Solving for the percentage change in OCF, we get:

%OCF = (DOL)(%Q)%OCF = 3.40[(70,000 – 65,000)/65,000]%OCF = .2615 or 26.15%

The new level of operating leverage is lower since FC/OCF is smaller.

12. Using the DOL equation, we find:

B-393 SOLUTIONS

DOL = 1 + FC / OCF3.40 = 1 + $130,000/OCF

OCF = $54,167

The percentage change in quantity sold at 58,000 units is:

%ΔQ = (58,000 – 65,000) / 65,000 %ΔQ = –.1077 or –10.77%

CHAPTER 11 B-394

So, using the same equation as in the previous problem, we find:

%ΔOCF = 3.40(–10.77%) %ΔQ = –36.62%

So, the new OCF level will be:

New OCF = (1 – .3662)($54,167) New OCF = $34,333

And the new DOL will be:

New DOL = 1 + ($130,000/$34,333) New DOL = 4.786

13. The DOL of the project is:

DOL = 1 + ($73,000/$87,500) DOL = 1.8343

If the quantity sold changes to 8,500 units, the percentage change inquantity sold is:

%Q = (8,500 – 8,000)/8,000 %ΔQ = .0625 or 6.25%

So, the OCF at 8,500 units sold is:

%OCF = DOL(%Q) %ΔOCF = 1.8343(.0625) %ΔOCF = .1146 or 11.46%

This makes the new OCF:

New OCF = $87,500(1.1146) New OCF = $97,531

And the DOL at 8,500 units is:

DOL = 1 + ($73,000/$97,531) DOL = 1.7485

B-395 SOLUTIONS

14. We can use the equation for DOL to calculate fixed costs. The fixedcost must be:

DOL = 2.35 = 1 + FC/OCFFC = (2.35 – 1)$41,000 FC = $58,080

If the output rises to 11,000 units, the percentage change inquantity sold is:

%Q = (11,000 – 10,000)/10,000 %ΔQ = .10 or 10.00%

CHAPTER 11 B-396

The percentage change in OCF is:

%OCF = 2.35(.10) %ΔOCF = .2350 or 23.50%

So, the operating cash flow at this level of sales will be:

OCF = $43,000(1.235) OCF = $53,105

If the output falls to 9,000 units, the percentage change in quantitysold is:

%Q = (9,000 – 10,000)/10,000 %ΔQ = –.10 or –10.00%

The percentage change in OCF is:

%OCF = 2.35(–.10) %ΔOCF = –.2350 or –23.50%

So, the operating cash flow at this level of sales will be:

OCF = $43,000(1 – .235) OCF = $32,897

15. Using the equation for DOL, we get:

DOL = 1 + FC/OCF

At 11,000 unitsDOL = 1 + $58,050/$53,105DOL = 2.0931

At 9,000 unitsDOL = 1 + $58,050/$32,895 DOL = 2.7647

Intermediate

16. a. At the accounting breakeven, the IRR is zero percent since theproject recovers the initial investment. The payback period is Nyears, the length of the project since the initial investment is

B-397 SOLUTIONS

exactly recovered over the project life. The NPV at the accountingbreakeven is:

NPV = I [(1/N)(PVIFAR%,N) – 1]

b. At the cash breakeven level, the IRR is –100 percent, the paybackperiod is negative, and the NPV is negative and equal to theinitial cash outlay.

CHAPTER 11 B-398

c. The definition of the financial breakeven is where the NPV of theproject is zero. If this is true, then the IRR of the project isequal to the required return. It is impossible to state thepayback period, except to say that the payback period must be lessthan the length of the project. Since the discounted cash flowsare equal to the initial investment, the undiscounted cash flowsare greater than the initial investment, so the payback must beless than the project life.

17. Using the tax shield approach, the OCF at 110,000 units will be:

OCF = [(P – v)Q – FC](1 – tC) + tC(D) OCF = [($32 – 19)(110,000) – 210,000](0.66) + 0.34($490,000/4) OCF = $846,850

We will calculate the OCF at 111,000 units. The choice of the secondlevel of quantity sold is arbitrary and irrelevant. No matter whatlevel of units sold we choose, we will still get the samesensitivity. So, the OCF at this level of sales is:

OCF = [($32 – 19)(111,000) – 210,000](0.66) + 0.34($490,000/4) OCF = $855,430

The sensitivity of the OCF to changes in the quantity sold is:

Sensitivity = OCF/Q = ($846,850 – 855,430)/(110,000 – 111,000) OCF/Q = +$8.58

OCF will increase by $5.28 for every additional unit sold.

18. At 110,000 units, the DOL is:

DOL = 1 + FC/OCFDOL = 1 + ($210,000/$846,850) DOL = 1.2480

The accounting breakeven is:

QA = (FC + D)/(P – v) QA = [$210,000 + ($490,000/4)]/($32 – 19) QA = 25,576

And, at the accounting breakeven level, the DOL is:

B-399 SOLUTIONS

DOL = 1 + [$210,000/($490,000/4)] DOL = 2.7143

CHAPTER 11 B-400

19. a. The base-case, best-case, and worst-case values are shown below.Remember that in the best-case, sales and price increase, whilecosts decrease. In the worst-case, sales and price decrease, andcosts increase.

Scenario Unit salesVariable cost Fixed costsBase 190 $11,200 $410,000Best 209 $10,080 $369,000Worst 171 $12,320 $451,000

Using the tax shield approach, the OCF and NPV for the base caseestimate is:

OCFbase = [($18,000 – 11,200)(190) – $410,000](0.65) +0.35($1,700,000/4) OCFbase = $722,050

NPVbase = –$1,700,000 + $722,050(PVIFA12%,4) NPVbase = $493,118.10

The OCF and NPV for the worst case estimate are:

OCFworst = [($18,000 – 12,320)(171) – $451,000](0.65) +0.35($1,700,000/4) OCFworst = $486,932

NPVworst = –$1,700,000 + $486,932(PVIFA12%,4) NPVworst = –$221,017.41

And the OCF and NPV for the best case estimate are:

OCFbest = [($18,000 – 10,080)(209) – $369,000](0.65) +0.35($1,700,000/4) OCFbest = $984,832

NPVbest = –$1,700,000 + $984,832(PVIFA12%,4) NPVbest = $1,291,278.83

b. To calculate the sensitivity of the NPV to changes in fixed costswe choose another level of fixed costs. We will use fixed costs of$420,000. The OCF using this level of fixed costs and the otherbase case values with the tax shield approach, we get:

B-401 SOLUTIONS

OCF = [($18,000 – 11,200)(190) – $410,000](0.65) +0.35($1,700,000/4) OCF = $715,550

And the NPV is:

NPV = –$1,700,000 + $715,550(PVIFA12%,4) NPV = $473,375.32

The sensitivity of NPV to changes in fixed costs is:

NPV/FC = ($493,118.10 – 473,375.32)/($410,000 – 420,000) NPV/FC = –$1.974

For every dollar FC increase, NPV falls by $1.974.

CHAPTER 11 B-402

c. The cash breakeven is:

QC = FC/(P – v) QC = $410,000/($18,000 – 11,200) QC = 60

d. The accounting breakeven is:

QA = (FC + D)/(P – v)QA = [$410,000 + ($1,700,000/4)]/($18,000 – 11,200) QA = 123

At the accounting breakeven, the DOL is:

DOL = 1 + FC/OCFDOL = 1 + ($410,000/$425,000) = 1.9647

For each 1% increase in unit sales, OCF will increase by 1.9647%.

20. The marketing study and the research and development are both sunkcosts and should be ignored. We will calculate the sales and variablecosts first. Since we will lose sales of the expensive clubs and gainsales of the cheap clubs, these must be accounted for as erosion. Thetotal sales for the new project will be:

SalesNew clubs $750 51,000 =

$38,250,000Exp. clubs

$1,200 (–11,000) = –13,200,000

Cheap clubs

$420 9,500 =3,990,000

$29,040,000

For the variable costs, we must include the units gained or lost fromthe existing clubs. Note that the variable costs of the expensiveclubs are an inflow. If we are not producing the sets anymore, wewill save these variable costs, which is an inflow. So:

Var. costsNew clubs –$330 51,000 = –

B-403 SOLUTIONS

$16,830,000Exp. clubs

–$650 (–11,000) =7,150,000

Cheap clubs

–$190 9,500 = –1,805,000

–$11,485,000

The pro forma income statement will be:

Sales $29,040,000

Variablecosts

11,485,000

Costs 8,100,000

Depreciation

3,200,000

EBT $6,255,000

Taxes 2,502,000

Net income $3,753,000

CHAPTER 11 B-404

Using the bottom up OCF calculation, we get:

OCF = NI + Depreciation = $3,753,000 + 3,200,000 OCF = $6,953,000

So, the payback period is:

Payback period = 3 + $2,791,000/$6,953,000 Payback period = 3.401 years

The NPV is:

NPV = –$22,400,000 – 1,250,000 + $6,953,000(PVIFA10%,7) +$1,250,000/1.107

NPV = $10,841,563.69

And the IRR is:

IRR = –$22,400,000 – 1,250,000 + $6,953,000(PVIFAIRR%,7) +$1,250,000/IRR7

IRR = 22.64%

21. The best case and worst cases for the variables are:

Base Case Best Case Worst CaseUnit sales (new) 51,000 56,100 45,900Price (new) $750 $825 $675VC (new) $330 $297 $363Fixed costs $8,100,000 $7,290,000 $8,910,000Sales lost (expensive)11,000 9,900 12,100Sales gained (cheap)9,500 10,450 8,550

Best-caseWe will calculate the sales and variable costs first. Since we willlose sales of the expensive clubs and gain sales of the cheap clubs,these must be accounted for as erosion. The total sales for the newproject will be:


$46,282,500Exp. clubs

$1,200 (–9,900) = –11,880,000

B-405 SOLUTIONS

Cheap clubs

$420 10,450 =4,389,000

$38,791,500



$16,661,700Exp. clubs

–$650 (–9,900) =6,435,000

Cheap clubs

–$190 10,450 = –1,985,500

–$12,212,200

CHAPTER 11 B-406


Sales $38,791,500

Variablecosts

12,212,200

Costs 7,290,000

Depreciation

3,200,000

EBT 16,089,300

Taxes 6,435,720



OCF = Net income + Depreciation = $9,653,580 + 3,200,000 OCF = $12,853,580

And the best-case NPV is:

NPV = –$22,400,000 – 1,250,000 + $12,853,580(PVIFA10%,7) +1,250,000/1.107

NPV = $39,568,058.39

Worst-caseWe will calculate the sales and variable costs first. Since we willlose sales of the expensive clubs and gain sales of the cheap clubs,these must be accounted for as erosion. The total sales for the newproject will be:


$30,982,500Exp. clubs

$1,200 (– 12,100) = –14,520,000

Cheap clubs

$420 8,550 =3,591,000

$20,053,500

B-407 SOLUTIONS



$16,661,700Exp. clubs

–$650 (– 12,100) =7,865,000

Cheap clubs

–$190 8,550 = –1,624,500

–$10,421,200


Sales $20,053,500

Variablecosts

10,421,200

Costs 8,910,000

Depreciation

3,200,000

EBT –2,477,70

0Taxes 99

1,080 *assumes a tax credit

Net income –$1,486,6

20

CHAPTER 11 B-408


OCF = NI + Depreciation = –$1,486,620 + 3,200,000 OCF = $1,713,380

And the worst-case NPV is:

NPV = –$22,400,000 – 1,250,000 + $1,713,380(PVIFA10%,7) +1,250,000/1.107

NPV = –$14,667,100.92

22. To calculate the sensitivity of the NPV to changes in the price ofthe new club, we simply need to change the price of the new club. Wewill choose $800, but the choice is irrelevant as the sensitivitywill be the same no matter what price we choose.

We will calculate the sales and variable costs first. Since we willlose sales of the expensive clubs and gain sales of the cheap clubs,these must be accounted for as erosion. The total sales for the newproject will be:


$40,800,000Exp. clubs

$1,200 (–11,000) = –13,200,000

Cheap clubs

$420 9,500 =3,990,000

$31,590,000



$16,830,000Exp. clubs

–$650 (–11,000) =7,150,000

Cheap –$190 9,500 = –

B-409 SOLUTIONS

clubs 1,805,000–$11,485,000


Sales $31,590,000

Variablecosts

11,485,000

Costs 8,100,000

Depreciation

3,200,000

EBT 8,805,000

Taxes 3,522,000


0



CHAPTER 11 B-410

And the NPV is:

NPV = –$22,400,000 – 1,250,000 + $8,483,000(PVIFA10%,7) +1,250,000/1.107

NPV = $18,290,244.48

So, the sensitivity of the NPV to changes in the price of the newclub is:

NPV/P = ($10,841,563.69 – 18,290,244.48)/($750 – 800) NPV/P = $148,973.62

For every dollar increase (decrease) in the price of the clubs, theNPV increases (decreases) by $148,973.62.

To calculate the sensitivity of the NPV to changes in the quantitysold of the new club, we simply need to change the quantity sold. Wewill choose 52,000 units, but the choice is irrelevant as thesensitivity will be the same no matter what quantity we choose.

We will calculate the sales and variable costs first. Since we willlose sales of the expensive clubs and gain sales of the cheap clubs,these must be accounted for as erosion. The total sales for the newproject will be:


$39,000,000Exp. clubs

$1,200 (–11,000) = –13,200,000

Cheap clubs

$420 9,500 =3,990,000

$29,790,000



$17,160,000

B-411 SOLUTIONS

Exp. clubs

–$650 (–11,000) =7,150,000

Cheap clubs

–$190 9,500 = –1,805,000

–$11,815,000


Sales $29,790,000

Variablecosts

11,815,000

Costs 8,100,000

Depreciation

3,200,000

EBT 6,675,000

Taxes 2,670,000


0

CHAPTER 11 B-412



The NPV at this quantity is:

NPV = –$22,400,000 – $1,250,000 + $7,205,000(PVIFA10%,7) + $1,250,000/1.107

NPV = $12,068,405.23

So, the sensitivity of the NPV to changes in the quantity sold is:

NPV/Q = ($10,841,563.69 – 12,068,405.23)/(51,000 – 52,000) NPV/Q = $1,226.84

For an increase (decrease) of one set of clubs sold per year, the NPVincreases (decreases) by $1,226.84.

23. a. First we need to determine the total additional cost of thehybrid. The hybrid costs more to purchase and more each year, sothe total additional cost is:

Total additional cost = $5,450 + 6($400)Total additional cost = $7,850

Next, we need to determine the cost per mile for each vehicle. Thecost per mile is the cost per gallon of gasoline divided by themiles per gallon, or:

Cost per mile for traditional = $3.60/23Cost per mile for traditional = $0.156522

Cost per mile for hybrid = $3.60/25Cost per mile for hybrid = $0.144000

So, the savings per mile driven for the hybrid will be:

Savings per mile = $0.156522 – 0.144000Savings per mile = $0.012522

We can now determine the breakeven point by dividing the total additional cost by the savings per mile, which is:

B-413 SOLUTIONS

Total breakeven miles = $7,850 / $0.012522Total breakeven miles = 626,910

So, the miles you would need to drive per year is the totalbreakeven miles divided by the number of years of ownership, or:

Miles per year = 626,910 miles / 6 yearsMiles per year = 104,485 miles/year

CHAPTER 11 B-414

b. First, we need to determine the total miles driven over the lifeof either vehicle, which will be:

Total miles driven = 6(15,000)Total miles driven = 90,000

Since we know the total additional cost of the hybrid from part a,we can determine the necessary savings per mile to make the hybridfinancially attractive. The necessary cost savings per mile willbe:

Cost savings needed per mile = $7,850 / 90,000Cost savings needed per mile = $0.08722

Now we can find the price per gallon for the miles driven. If welet P be the price per gallon, the necessary price per gallon willbe:

P/23 – P/25 = $0.08722P(1/23 – 1/25) = $0.08722P = $25.08

c. To find the number of miles it is necessary to drive, we need thepresent value of the costs and savings to be equal to zero. If welet MDPY equal the miles driven per year, the breakeven equationfor the hybrid car as:

Cost = 0 = –$5,450 – $400(PVIFA10%,6) + $0.012522(MDPY)(PVIFA10%,6)

The savings per mile driven, $0.012522, is the same as wecalculated in part a. Solving this equation for the number ofmiles driven per year, we find:

$0.012522(MDPY)(PVIFA10%,6) = $7,192.10 MDPY(PVIFA10%,6) = 574,369.44Miles driven per year = 131,879

To find the cost per gallon of gasoline necessary to make thehybrid break even in a financial sense, if we let CSPG equal thecost savings per gallon of gas, the cost equation is:

Cost = 0 = –$5,450 – $400(PVIFA10%,6) + CSPG(15,000)(PVIFA10%,6)

B-415 SOLUTIONS

Solving this equation for the cost savings per gallon of gasnecessary for the hybrid to breakeven from a financial sense, wefind:

CSPG(15,000)(PVIFA10%,6) = $7,192.10 CSPG(PVIFA10%,6) = $0.47947Cost savings per gallon of gas = $0.110091

Now we can find the price per gallon for the miles driven. If welet P be the price per gallon, the necessary price per gallon willbe:

P/23 – P/25 = $0.110091P(1/23 – 1/25) = $0.110091P = $31.65

CHAPTER 11 B-416

d. The implicit assumption in the previous analysis is that each cardepreciates by the same dollar amount.

24. a. The cash flow per plane is the initial cost divided by thebreakeven number of planes, or:

Cash flow per plane = $13,000,000,000 / 249Cash flow per plane = $52,208,835

b. In this case the cash flows are a perpetuity. Since we know thecash flow per plane, we need to determine the annual cash flownecessary to deliver a 20 percent return. Using the perpetuityequation, we find:

PV = C /R$13,000,000,000 = C / .20C = $2,600,000,000

This is the total cash flow, so the number of planes that must besold is the total cash flow divided by the cash flow per plane,or:

Number of planes = $2,600,000,000 / $52,208,835Number of planes = 49.80 or about 50 planes per year

c. In this case the cash flows are an annuity. Since we know the cashflow per plane, we need to determine the annual cash flownecessary to deliver a 20 percent return. Using the present valueof an annuity equation, we find:

PV = C(PVIFA20%,10)$13,000,000,000 = C(PVIFA20%,10)C = $3,100,795,839

This is the total cash flow, so the number of planes that must besold is the total cash flow divided by the cash flow per plane,or:

Number of planes = $3,100,795,839 / $52,208,835Number of planes = 59.39 or about 60 planes per year

Challenge

B-417 SOLUTIONS

25. a. The tax shield definition of OCF is:

OCF = [(P – v)Q – FC ](1 – tC) + tCD

Rearranging and solving for Q, we find:

(OCF – tCD)/(1 – tC) = (P – v)Q – FCQ = {FC + [(OCF – tCD)/(1 – tC)]}/(P – v)

CHAPTER 11 B-418

b. The cash breakeven is:

QC = $500,000/($40,000 – 20,000) QC = 25


QA = {$500,000 + [($700,000 – $700,000(0.38))/0.62]}/($40,000 –20,000)

QA = 60

The financial breakeven is the point at which the NPV is zero, so:

OCFF = $3,500,000/PVIFA20%,5 OCFF = $1,170,328.96

So:

QF = [FC + (OCF – tC × D)/(1 – tC)]/(P – v)QF = {$500,000 + [$1,170,328.96 – .38($700,000)]/(1 – .38)}/($40,000

– 20,000)QF = 97.93 98

c. At the accounting break-even point, the net income is zero. Thisusing the bottom up definition of OCF:

OCF = NI + D

We can see that OCF must be equal to depreciation. So, theaccounting breakeven is:

QA = {FC + [(D – tCD)/(1 – t)]}/(P – v) QA = (FC + D)/(P – v) QA = (FC + OCF)/(P – v)

The tax rate has cancelled out in this case.

26. The DOL is expressed as:

DOL = %OCF / %Q DOL = {[(OCF1 – OCF0)/OCF0] / [(Q1 – Q0)/Q0]}

The OCF for the initial period and the first period is:

B-419 SOLUTIONS

OCF1 = [(P – v)Q1 – FC](1 – tC) + tCD

OCF0 = [(P – v)Q0 – FC](1 – tC) + tCD

The difference between these two cash flows is:

OCF1 – OCF0 = (P – v)(1 – tC)(Q1 – Q0)

CHAPTER 11 B-420

Dividing both sides by the initial OCF we get:

(OCF1 – OCF0)/OCF0 = (P – v)( 1– tC)(Q1 – Q0) / OCF0

Rearranging we get:

[(OCF1 – OCF0)/OCF0][(Q1 – Q0)/Q0] = [(P – v)(1 – tC)Q0]/OCF0 = [OCF0 –tCD + FC(1 – t)]/OCF0

DOL = 1 + [FC(1 – t) – tCD]/OCF0

27. a. Using the tax shield approach, the OCF is:

OCF = [($230 – 185)(35,000) – $450,000](0.62) + 0.38($3,200,000/5)OCF = $940,700

And the NPV is:

NPV = –$3,200,000 – 360,000 + $940,700(PVIFA13%,5) + [$360,000 +$500,000(1 – .38)]/1.135 NPV = $112,308.60

b. In the worst-case, the OCF is:

OCFworst = {[($230)(0.9) – 185](35,000) – $450,000}(0.62) +0.38($3,680,000/5) OCFworst = $478,080

And the worst-case NPV is:

NPVworst = –$3,680,000 – $360,000(1.05) + $478,080(PVIFA13%,5) + [$360,000(1.05) + $500,000(0.85)(1 – .38)]/1.135

NPVworst = –$2,028,301.58

The best-case OCF is:

OCFbest = {[$230(1.1) – 185](35,000) – $450,000}(0.62) +0.38($2,720,000/5) OCFbest = $1,403,320

And the best-case NPV is:

NPVbest = – $2,720,000 – $360,000(0.95) + $1,403,320(PVIFA13%,5) + [$360,000(0.95) + $500,000(1.15)(1 – .38)]/1.135

B-421 SOLUTIONS

NPVbest = $2,252,918.79

28. To calculate the sensitivity to changes in quantity sold, we will choose a quantity of 36,000. The OCF at this level of sale is:

OCF = [($230 – 185)(36,000) – $450,000](0.62) + 0.38($3,200,000/5) OCF = $968,600

CHAPTER 11 B-422

The sensitivity of changes in the OCF to quantity sold is:

OCF/Q = ($968,600 – 940,700)/(36,000 – 35,000) OCF/Q = +$27.90

The NPV at this level of sales is:

NPV = –$3,200,000 – $360,000 + $968,600(PVIFA13%,5) + [$360,000 +$500,000(1 – .38)]/1.135 NPV = $210,439.36

And the sensitivity of NPV to changes in the quantity sold is:

NPV/Q = ($210,439.36 – 112,308.60))/(36,000 – 35,000) NPV/Q = +$98.13

You wouldn’t want the quantity to fall below the point where the NPVis zero. We know the NPV changes $98.13 for every unit sale, so wecan divide the NPV for 35,000 units by the sensitivity to get achange in quantity. Doing so, we get:

$112,308.60 = $98.13(Q) Q = 1,144

For a zero NPV, we need to decrease sales by 1,144 units, so the minimum quantity is:

QMin = 35,000 – 1,144 QMin = 33,856

29. At the cash breakeven, the OCF is zero. Setting the tax shieldequation equal to zero and solving for the quantity, we get:

OCF = 0 = [($230 – 185)QC – $450,000](0.62) + 0.38($3,200,000/5) QC = 1,283

The accounting breakeven is:

QA = [$450,000 + ($3,200,000/5)]/($230 – 185)QA = 24,222

From Problem 28, we know the financial breakeven is 33,856 units.

B-423 SOLUTIONS

30. Using the tax shield approach to calculate the OCF, the DOL is:

DOL = 1 + [$450,000(1 – 0.38) – 0.38($3,200,000/5)]/ $940,700 DOL = 1.03806

Thus a 1% rise leads to a 1.03806% rise in OCF. If Q rises to 36,000,then

The percentage change in quantity is:

Q = (36,000 – 35,000)/35,000 = .02857 or 2.857%

So, the percentage change in OCF is:

%OCF = 2.857%(1.03806) %OCF = 2.9659%

From Problem 26: OCF/OCF = ($968,600 – 940,700)/$940,700 OCF/OCF = 0.029659

In general, if Q rises by 1,000 units, OCF rises by 2.9659%.

CHAPTER 12SOME LESSONS FROM CAPITAL MARKET HISTORY


1. They all wish they had! Since they didn’t, it must have been the casethat the stellar performance was not foreseeable, at least not bymost.

2. As in the previous question, it’s easy to see after the fact that theinvestment was terrible, but it probably wasn’t so easy ahead oftime.

3. No, stocks are riskier. Some investors are highly risk averse, andthe extra possible return doesn’t attract them relative to the extrarisk.

4. On average, the only return that is earned is the required return—investors buy assets with returns in excess of the required return(positive NPV), bidding up the price and thus causing the return tofall to the required return (zero NPV); investors sell assets withreturns less than the required return (negative NPV), driving theprice lower and thus causing the return to rise to the requiredreturn (zero NPV).

5. The market is not weak form efficient.

6. Yes, historical information is also public information; weak formefficiency is a subset of semi-strong form efficiency.

7. Ignoring trading costs, on average, such investors merely earn whatthe market offers; stock investments all have a zero NPV. If tradingcosts exist, then these investors lose by the amount of the costs.

8. Unlike gambling, the stock market is a positive sum game; everybodycan win. Also, speculators provide liquidity to markets and thus helpto promote efficiency.

B-425 SOLUTIONS

9. The EMH only says, within the bounds of increasingly strongassumptions about the information processing of investors, thatassets are fairly priced. An implication of this is that, on average,the typical market participant cannot earn excessive profits from aparticular trading strategy. However, that does not mean that a fewparticular investors cannot outperform the market over a particularinvestment horizon. Certain investors who do well for a period oftime get a lot of attention from the financial press, but the scoresof investors who do not do well over the same period of timegenerally get considerably less attention from the financial press.

10. a. If the market is not weak form efficient, then this informationcould be acted on and a profit earned from following the pricetrend. Under (2), (3), and (4), this information is fullyimpounded in the current price and no abnormal profit opportunityexists.

CHAPTER 12 B-426

b. Under (2), if the market is not semi-strong form efficient, thenthis information could be used to buy the stock “cheap” before therest of the market discovers the financial statement anomaly.Since (2) is stronger than (1), both imply that a profitopportunity exists; under (3) and (4), this information is fullyimpounded in the current price and no profit opportunity exists.

c. Under (3), if the market is not strong form efficient, then thisinformation could be used as a profitable trading strategy, bynoting the buying activity of the insiders as a signal that thestock is underpriced or that good news is imminent. Since (1) and(2) are weaker than (3), all three imply that a profit opportunityexists. Note that this assumes the individual who sees the insidertrading is the only one who sees the trading. If the informationabout the trades made by company management is public information,it will be discounted in the stock price and no profit opportunityexists. Under (4), this information does not signal any profitopportunity for traders; any pertinent information the manager-insiders may have is fully reflected in the current share price.



Basic

1. The return of any asset is the increase in price, plus any dividendsor cash flows, all divided by the initial price. The return of thisstock is:

R = [($102 – 91) + 2.40] / $91 = .1473 or 14.73%

2. The dividend yield is the dividend divided by price at the beginningof the period price, so:

Dividend yield = $2.40 / $91 = .0264 or 2.64%

And the capital gains yield is the increase in price divided by theinitial price, so:

Capital gains yield = ($102 – 91) / $91 = .1209 or 12.09%

B-427 SOLUTIONS

3. Using the equation for total return, we find:

R = [($83 – 91) + 2.40] / $91 = –.0615 or –6.15%

And the dividend yield and capital gains yield are:

Dividend yield = $2.40 / $91 = .0264 or 2.64%

Capital gains yield = ($83 – 91) / $91 = –.0879 or –8.79%

Here’s a question for you: Can the dividend yield ever be negative?No, that would mean you were paying the company for the privilege ofowning the stock. It has happened on bonds.

CHAPTER 12 B-428

4. The total dollar return is the increase in price plus the couponpayment, so:

Total dollar return = $1,070 – 1,040 + 70 = $100

The total percentage return of the bond is:

R = [($1,070 – 1,040) + 70] / $1,040 = .0962 or 9.62%

Notice here that we could have simply used the total dollar return of$100 in the numerator of this equation.

Using the Fisher equation, the real return was:

(1 + R) = (1 + r)(1 + h)

r = (1.0962 / 1.04) – 1 = .0540 or 5.40%

5. The nominal return is the stated return, which is 12.30 percent.Using the Fisher equation, the real return was:

(1 + R) = (1 + r)(1 + h)

r = (1.123)/(1.031) – 1 = .0892 or 8.92%

6. Using the Fisher equation, the real returns for long-term governmentand corporate bonds were:

(1 + R) = (1 + r)(1 + h)

rG = 1.058/1.031 – 1 = .0262 or 2.62%

rC = 1.062/1.031 – 1 = .0301 or 3.01%

7. The average return is the sum of the returns, divided by the number ofreturns. The average return for each stock was:

B-429 SOLUTIONS

Remembering back to “sadistics,” we calculate the variance of each stockas:

σX2

=[∑i=1

N

(xi−x̄ )2 ]/ (N−1 )

σX2

=15−1

{(.08−.078 )2+(.21−.078 )2+(.17−.078 )2+(−.16−.078)2+(.09−.078 )2 }=.020670

σY2

=15−1 {(.16−.146 )2+(.38−.146 )2+(.14−.146 )2+(−.21−.146)2+(.26−.146 )2 }=.048680

The standard deviation is the square root of the variance, so thestandard deviation of each stock is:

X = (.020670)1/2 = .1438 or 14.38%

Y = (.048680)1/2 = .2206 or 22.06%

8. We will calculate the sum of the returns for each asset and theobserved risk premium first. Doing so, we get:

YearLarge co. stock returnT-bill returnRisk premium1970 3.94% 6.50% 2.56%1971 14.30 4.36 9.941972 18.99 4.23 14.761973 –14.69 7.29 –21.981974 –26.47 7.99 –34.461975 37.23 5.87 31.36

33.30 36.24 –2.94

a. The average return for large company stocks over this period was:

Large company stocks average return = 33.30% / 6 = 5.55%

And the average return for T-bills over this period was:

T-bills average return = 36.24% / 6 = 6.04%

CHAPTER 12 B-430

b. Using the equation for variance, we find the variance for largecompany stocks over this period was:

Variance = 1/5[(.0394 – .0555)2 + (.1430 – .0555)2 + (.1899 – .0555)2

+ (–.1469 – .0555)2 + (–.2647 – .0555)2 + (.3723 – .0555)2]

Variance = 0.053967

And the standard deviation for large company stocks over this periodwas:

Standard deviation = (0.053967)1/2 = 0.2323 or 23.23%

Using the equation for variance, we find the variance for T-billsover this period was:

Variance = 1/5[(.0650 – .0604)2 + (.0436 – .0604)2 + (.0423– .0604)2 + (.0729 – .0604)2 +

(.0799 – .0604)2 + (.0587 – .0604)2] Variance = 0.000234

And the standard deviation for T-bills over this period was:


c. The average observed risk premium over this period was:

Average observed risk premium = –2.94% / 6 = –0.49%

The variance of the observed risk premium was:

Variance = 1/5[(–.0256 – (–.0049))2 + (.0994 – (–.0049))2 + (.1476 – (–.0049)))2 +

(–.2198 – (–.0049))2 + (–.3446 – (–.0049))2 + (.3136 – (–.0049))2] Variance = 0.059517

And the standard deviation of the observed risk premium was:


d. Before the fact, for most assets the risk premium will bepositive; investors demand compensation over and above the risk-

B-431 SOLUTIONS

free return to invest their money in the risky asset. After thefact, the observed risk premium can be negative if the asset’snominal return is unexpectedly low, the risk-free return isunexpectedly high, or if some combination of these two eventsoccurs.

9. a. To find the average return, we sum all the returns and divide bythe number of returns, so:

Average return = (.07 –.12 +.11 +.38 +.14)/5 = .1160 or 11.60%

CHAPTER 12 B-432

b. Using the equation to calculate variance, we find:

Variance = 1/4[(.07 – .116)2 + (–.12 – .116)2 + (.11 – .116)2 + (.38 – .116)2 + (.14 – .116)2] Variance = 0.032030

So, the standard deviation is:


10. a. To calculate the average real return, we can use the averagereturn of the asset, and the average inflation in the Fisherequation. Doing so, we find:

(1 + R) = (1 + r)(1 + h)

= (1.160/1.035) – 1 = .0783 or 7.83%

b. The average risk premium is simply the average return of theasset, minus the average risk-free rate, so, the average riskpremium for this asset would be:

RP=R – Rf = .1160 – .042 = .0740 or 7.40%

11. We can find the average real risk-free rate using the Fisherequation. The average real risk-free rate was:

(1 + R) = (1 + r)(1 + h)

rf = (1.042/1.035) – 1 = .0068 or 0.68%

And to calculate the average real risk premium, we can subtract theaverage risk-free rate from the average real return. So, the averagereal risk premium was:

rp=r – rf = 7.83% – 0.68% = 7.15%

12. T-bill rates were highest in the early eighties. This was during aperiod of high inflation and is consistent with the Fisher effect.

B-433 SOLUTIONS

Intermediate

13. To find the real return, we first need to find the nominal return,which means we need the current price of the bond. Going back to thechapter on pricing bonds, we find the current price is:

P1 = $80(PVIFA7%,6) + $1,000(PVIF7%,6) = $1,047.67

So the nominal return is:

R = [($1,047.67 – 1,030) + 80]/$1,030 = .0948 or 9.48%

And, using the Fisher equation, we find the real return is:

1 + R = (1 + r)(1 + h)

r = (1.0948/1.042) – 1 = .0507 or 5.07%

14. Here we know the average stock return, and four of the five returnsused to compute the average return. We can work the average returnequation backward to find the missing return. The average return iscalculated as:

.525 = .07 – .12 + .18 + .19 + RR = .205 or 20.5%

The missing return has to be 20.5 percent. Now we can use theequation for the variance to find:

Variance = 1/4[(.07 – .105)2 + (–.12 – .105)2 + (.18 – .105)2 + (.19– .105)2 + (.205 – .105)2] Variance = 0.018675

And the standard deviation is:


15. The arithmetic average return is the sum of the known returns dividedby the number of returns, so:

Arithmetic average return = (.03 + .38 + .21 – .15 + .29 – .13) / 6 Arithmetic average return = .1050 or 10.50%

CHAPTER 12 B-434

Using the equation for the geometric return, we find:

Geometric average return = [(1 + R1) × (1 + R2) × … × (1 + RT)]1/T – 1Geometric average return = [(1 + .03)(1 + .38)(1 + .21)(1 – .15)(1+ .29)(1 – .13)](1/6) – 1 Geometric average return = .0860 or 8.60%

Remember, the geometric average return will always be less than thearithmetic average return if the returns have any variation.

B-435 SOLUTIONS

16. To calculate the arithmetic and geometric average returns, we must first calculate the return for each year. The return for each year is:

R1 = ($73.66 – 60.18 + 0.60) / $60.18 = .2340 or 23.40%R2 = ($94.18 – 73.66 + 0.64) / $73.66 = .2873 or 28.73%R3 = ($89.35 – 94.18 + 0.72) / $94.18 = –.0436 or –4.36%R4 = ($78.49 – 89.35 + 0.80)/ $89.35 = –.1126 or 11.26%R5 = ($95.05 – 78.49 + 1.20) / $78.49 = .2263 or 12.63%

The arithmetic average return was:

RA = (0.2340 + 0.2873 – 0.0436 – 0.1126 + 0.2263)/5 = 0.1183 or 11.83%

And the geometric average return was:

RG = [(1 + .2340)(1 + .2873)(1 – .0436)(1 –.1126)(1 + .2263)]1/5 – 1 =0.1058 or 10.58%

17. Looking at the long-term corporate bond return history in Figure12.10, we see that the mean return was 6.2 percent, with a standarddeviation of 8.4 percent. In the normal probability distribution,approximately 2/3 of the observations are within one standarddeviation of the mean. This means that 1/3 of the observations areoutside one standard deviation away from the mean. Or:

Pr(R< –2.2 or R>14.6) 1/3

But we are only interested in one tail here, that is, returns lessthan –2.2 percent, so:

Pr(R< –2.2) 1/6

You can use the z-statistic and the cumulative normal distributiontable to find the answer as well. Doing so, we find:

z = (X – µ)/

z = (–2.2% – 6.2)/8.4% = –1.00

Looking at the z-table, this gives a probability of 15.87%, or:

CHAPTER 12 B-436

Pr(R< –2.2) .1587 or 15.87%

The range of returns you would expect to see 95 percent of the timeis the mean plus or minus 2 standard deviations, or:

95% level: R ± 2 = 6.2% ± 2(8.4%) = –10.60% to 23.00%

B-437 SOLUTIONS

The range of returns you would expect to see 99 percent of the timeis the mean plus or minus 3 standard deviations, or:

99% level: R ± 3 = 6.2% ± 3(8.4%) = –19.00% to 31.40%

18. The mean return for small company stocks was 17.1 percent, with astandard deviation of 32.6 percent. Doubling your money is a 100%return, so if the return distribution is normal, we can use the z-statistic. So:

z = (X – µ)/

z = (100% – 17.1)/32.6% = 2.543 standard deviations above the mean

This corresponds to a probability of 0.55%, or once every 200years. Tripling your money would be:

z = (200% – 17.1)/32.6% = 5.610 standard deviations above the mean.

This corresponds to a probability of about .000001%, or about onceevery 1 million years.

19. It is impossible to lose more than 100 percent of your investment.Therefore, return distributions are truncated on the lower tail at –100 percent.

20. To find the best forecast, we apply Blume’s formula as follows:

R(5) = 5 - 139 × 11.9% +

40 - 539 × 15.3% = 14.95%

R(10) = 10 - 139 × 11.9% +

40 - 1039 × 15.3% = 14.52%

R(20) = 20 - 139 × 11.9% +

40 - 2039 × 15.3% = 13.64%

21. The best forecast for a one year return is the arithmetic average,which is 12.3 percent. The geometric average, found in Table 12.4 is10.4 percent. To find the best forecast for other periods, we applyBlume’s formula as follows:

R(5) = 5 - 182 - 1 × 10.4% +

82 - 582 - 1 × 12.3% = 12.21%

CHAPTER 12 B-438

R(20) = 20 - 182 - 1 × 10.4% +

82 - 2082 - 1 × 12.3% = 11.85%

R(30) = 30 - 182- 1 × 10.4% +

82 - 3082 - 1 × 12.3% = 11.62%

B-439 SOLUTIONS

22. To find the real return we need to use the Fisher equation. Re-writing the Fisher equation to solve for the real return, we get:

r = [(1 + R)/(1 + h)] – 1

So, the real return each year was:

YearT-billreturn Inflation

Realreturn

1973 0.0729

0.0871

–0.0131

1974 0.0799

0.1234

–0.0387

1975 0.0587

0.0694

–0.0100

1976 0.0507

0.0486

0.0020

1977 0.0545

0.0670

–0.0117

1978 0.0764

0.0902

–0.0127

1979 0.1056

0.1329

–0.0241

1980 0.1210

0.1252

–0.0037

0.6197

0.7438

–0.1120

a. The average return for T-bills over this period was:

Average return = 0.619 / 8 Average return = .0775 or 7.75%

And the average inflation rate was:

Average inflation = 0.7438 / 8 Average inflation = .0930 or 9.30%

b. Using the equation for variance, we find the variance for T-billsover this period was:

Variance = 1/7[(.0729 – .0775)2 + (.0799 – .0775)2 + (.0587 – .0775)2 +(.0507 – .0775)2 +

CHAPTER 12 B-440

(.0545 – .0775)2 + (.0764 – .0775)2 + (.1056 – .0775)2 +(.1210 .0775)2]

Variance = 0.000616

And the standard deviation for T-bills was:

Standard deviation = (0.000616)1/2 Standard deviation = 0.0248 or 2.48%

The variance of inflation over this period was:

Variance = 1/7[(.0871 – .0930)2 + (.1234 – .0930)2 + (.0694– .0930)2 + (.0486 – .0930)2 +

(.0670 – .0930)2 + (.0902 – .0930)2 + (.1329 – .0930)2 +(.1252 .0930)2] Variance = 0.000971

And the standard deviation of inflation was:

Standard deviation = (0.000971)1/2 Standard deviation = 0.0312 or 3.12%

B-441 SOLUTIONS

c. The average observed real return over this period was:

Average observed real return = –.1122 / 8 Average observed real return = –.0140 or –1.40%

d. The statement that T-bills have no risk refers to the fact thatthere is only an extremely small chance of the governmentdefaulting, so there is little default risk. Since T-bills areshort term, there is also very limited interest rate risk.However, as this example shows, there is inflation risk, i.e. thepurchasing power of the investment can actually decline over timeeven if the investor is earning a positive return.

Challenge

23. Using the z-statistic, we find:

z = (X – µ)/

z = (0% – 12.3)/20.0% = –0.615

Pr(R=0) 26.93%

24. For each of the questions asked here, we need to use the z-statistic,which is:

z = (X – µ)/

a. z1 = (10% – 6.2)/8.4% = 0.4524

This z-statistic gives us the probability that the return is lessthan 10 percent, but we are looking for the probability the returnis greater than 10 percent. Given that the total probability is100 percent (or 1), the probability of a return greater than 10percent is 1 minus the probability of a return less than 10percent. Using the cumulative normal distribution table, we get:

Pr(R=10%) = 1 – Pr(R=10%) = 1 – .6745 32.55%

For a return greater than 0 percent:

z2 = (0% – 6.2)/8.4% = –0.7381

CHAPTER 12 B-442

Pr(R=10%) = 1 – Pr(R=10%) = 1 – .7698 23.02%

b. The probability that T-bill returns will be greater than 10percent is:

z3 = (10% – 3.8)/3.1% = 2

Pr(R=10%) = 1 – Pr(R=10%) = 1 – .9772 2.28%

B-443 SOLUTIONS

And the probability that T-bill returns will be less than 0 percentis:

z4 = (0% – 3.8)/3.1% = –1.2258

Pr(R=0) 11.01%

c. The probability that the return on long-term corporate bonds willbe less than –4.18 percent is:

z5 = (–4.18% – 6.2)/8.4% = –1.2357

Pr(R=–4.18%) 10.83%

And the probability that T-bill returns will be greater than 10.56percent is:

z6 = (10.56% – 3.8)/3.1% = 2.1806

Pr(R=10.56%) = 1 – Pr(R=10.56%) = 1 – .9823 1.46%

CHAPTER 13RISK, RETURN, AND THE SECURITY MARKET LINE


1. Some of the risk in holding any asset is unique to the asset inquestion. By investing in a variety of assets, this unique portion ofthe total risk can be eliminated at little cost. On the other hand,there are some risks that affect all investments. This portion of thetotal risk of an asset cannot be costlessly eliminated. In otherwords, systematic risk can be controlled, but only by a costlyreduction in expected returns.

2. If the market expected the growth rate in the coming year to be 2percent, then there would be no change in security prices if thisexpectation had been fully anticipated and priced. However, if themarket had been expecting a growth rate other than 2 percent and theexpectation was incorporated into security prices, then thegovernment’s announcement would most likely cause security prices ingeneral to change; prices would drop if the anticipated growth ratehad been more than 2 percent, and prices would rise if theanticipated growth rate had been less than 2 percent.

3. a. systematicb. unsystematicc. both; probably mostly systematicd. unsystematice. unsystematicf. systematic

4. a. a change in systematic risk has occurred; market prices in generalwill most likely decline.

b. no change in unsystematic risk; company price will most likelystay constant.

c. no change in systematic risk; market prices in general will mostlikely stay constant.

B-445 SOLUTIONS

d. a change in unsystematic risk has occurred; company price willmost likely decline.

e. no change in systematic risk; market prices in general will mostlikely stay constant.

5. No to both questions. The portfolio expected return is a weightedaverage of the asset returns, so it must be less than the largestasset return and greater than the smallest asset return.

6. False. The variance of the individual assets is a measure of thetotal risk. The variance on a well-diversified portfolio is afunction of systematic risk only.

7. Yes, the standard deviation can be less than that of every asset inthe portfolio. However, p cannot be less than the smallest betabecause p is a weighted average of the individual asset betas.

8. Yes. It is possible, in theory, to construct a zero beta portfolio ofrisky assets whose return would be equal to the risk-free rate. It isalso possible to have a negative beta; the return would be less thanthe risk-free rate. A negative beta asset would carry a negative riskpremium because of its value as a diversification instrument.

9. Such layoffs generally occur in the context of corporaterestructurings. To the extent that the market views a restructuringas value-creating, stock prices will rise. So, it’s not layoffs perse that are being cheered on. Nonetheless, Wall Street does encouragecorporations to takes actions to create value, even if such actionsinvolve layoffs.

10. Earnings contain information about recent sales and costs. Thisinformation is useful for projecting future growth rates and cashflows. Thus, unexpectedly low earnings often lead market participantsto reduce estimates of future growth rates and cash flows; pricedrops are the result. The reverse is often true for unexpectedlyhigh earnings.



CHAPTER 13 B-446

Basic

1. The portfolio weight of an asset is total investment in that assetdivided by the total portfolio value. First, we will find theportfolio value, which is:

Total value = 180($45) + 140($27) = $11,880

The portfolio weight for each stock is:

WeightA = 180($45)/$11,880 = .6818

WeightB = 140($27)/$11,880 = .3182

2. The expected return of a portfolio is the sum of the weight of eachasset times the expected return of each asset. The total value of theportfolio is:

Total value = $2,950 + 3,700 = $6,650

So, the expected return of this portfolio is:

E(Rp) = ($2,950/$6,650)(0.11) + ($3,700/$6,650)(0.15) = .1323 or13.23%

3. The expected return of a portfolio is the sum of the weight of eachasset times the expected return of each asset. So, the expectedreturn of the portfolio is:

E(Rp) = .60(.09) + .25(.17) + .15(.13) = .1160 or 11.60%

B-447 SOLUTIONS

4. Here we are given the expected return of the portfolio and theexpected return of each asset in the portfolio, and are asked to findthe weight of each asset. We can use the equation for the expectedreturn of a portfolio to solve this problem. Since the total weightof a portfolio must equal 1 (100%), the weight of Stock Y must be oneminus the weight of Stock X. Mathematically speaking, this means:

E(Rp) = .124 = .14wX + .105(1 – wX)

We can now solve this equation for the weight of Stock X as:

.124 = .14wX + .105 – .105wX

.019 = .035wX

wX = 0.542857

So, the dollar amount invested in Stock X is the weight of Stock Xtimes the total portfolio value, or:

Investment in X = 0.542857($10,000) = $5,428.57

And the dollar amount invested in Stock Y is:

Investment in Y = (1 – 0.542857)($10,000) = $4,574.43

5. The expected return of an asset is the sum of the probability of eachreturn occurring times the probability of that return occurring. So,the expected return of the asset is:

E(R) = .25(–.08) + .75(.21) = .1375 or 13.75%

6. The expected return of an asset is the sum of the probability of eachreturn occurring times the probability of that return occurring. So,the expected return of the asset is:

E(R) = .20(–.05) + .50(.12) + .30(.25) = .1250 or 12.50%

7. The expected return of an asset is the sum of the probability of eachreturn occurring times the probability of that return occurring. So,the expected return of each stock asset is:

E(RA) = .15(.05) + .65(.08) + .20(.13) = .0855 or 8.55%

E(RB) = .15(–.17) + .65(.12) + .20(.29) = .1105 or 11.05%

CHAPTER 13 B-448

To calculate the standard deviation, we first need to calculate thevariance. To find the variance, we find the squared deviations fromthe expected return. We then multiply each possible squared deviationby its probability, then add all of these up. The result is thevariance. So, the variance and standard deviation of each stock is:

A2 =.15(.05 – .0855)2 + .65(.08 – .0855)2 + .20(.13 – .0855)2 = .00060

A = (.00060)1/2 = .0246 or 2.46%

B-449 SOLUTIONS

B2 =.15(–.17 – .1105)2 + .65(.12 – .1105)2 + .20(.29 – .1105)2

= .01830

B = (.01830)1/2 = .1353 or 13.53%

8. The expected return of a portfolio is the sum of the weight of eachasset times the expected return of each asset. So, the expectedreturn of the portfolio is:

E(Rp) = .25(.08) + .55(.15) + .20(.24) = .1505 or 15.05%

If we own this portfolio, we would expect to get a return of 15.05percent.

9. a. To find the expected return of the portfolio, we need to find thereturn of the portfolio in each state of the economy. Thisportfolio is a special case since all three assets have the sameweight. To find the expected return in an equally weightedportfolio, we can sum the returns of each asset and divide by thenumber of assets, so the expected return of the portfolio in eachstate of the economy is:

Boom: E(Rp) = (.07 + .15 + .33)/3 = .1833 or 18.33%Bust: E(Rp) = (.13 + .03 .06)/3 = .0333 or 3.33%

To find the expected return of the portfolio, we multiply thereturn in each state of the economy by the probability of thatstate occurring, and then sum. Doing this, we find:

E(Rp) = .35(.1833) + .65(.0333) = .0858 or 8.58%

b. This portfolio does not have an equal weight in each asset. Westill need to find the return of the portfolio in each state ofthe economy. To do this, we will multiply the return of each assetby its portfolio weight and then sum the products to get theportfolio return in each state of the economy. Doing so, we get:

Boom: E(Rp) = .20(.07) +.20(.15) + .60(.33) =.2420 or 24.20%Bust: E(Rp) = .20(.13) +.20(.03) + .60(.06) = –.0040 or –0.40%

And the expected return of the portfolio is:

E(Rp) = .35(.2420) + .65(.004) = .0821 or 8.21%

CHAPTER 13 B-450

To find the variance, we find the squared deviations from theexpected return. We then multiply each possible squared deviationby its probability, than add all of these up. The result is thevariance. So, the variance and standard deviation of the portfoliois:

p2 = .35(.2420 – .0821)2 + .65(.0040 – .0821)2 = .013767

B-451 SOLUTIONS

10. a. This portfolio does not have an equal weight in each asset. Wefirst need to find the return of the portfolio in each state ofthe economy. To do this, we will multiply the return of each assetby its portfolio weight and then sum the products to get theportfolio return in each state of the economy. Doing so, we get:

Boom: E(Rp) = .30(.3) + .40(.45) + .30(.33) = .3690 or 36.90%Good: E(Rp) = .30(.12) + .40(.10) + .30(.15) = .1210 or 12.10%Poor: E(Rp) = .30(.01) + .40(–.15) + .30(–.05) = –.0720 or –7.20%Bust: E(Rp) = .30(–.06) + .40(–.30) + .30(–.09) = –.1650 or –16.50%


E(Rp) = .15(.3690) + .45(.1210) + .35(–.0720) + .05(–.1650)= .0764 or 7.64%

b. To calculate the standard deviation, we first need to calculatethe variance. To find the variance, we find the squared deviationsfrom the expected return. We then multiply each possible squareddeviation by its probability, than add all of these up. The resultis the variance. So, the variance and standard deviation of theportfolio is:

p2 = .15(.3690 – .0764)2 + .45(.1210 – .0764)2 + .35(–.0720

– .0764)2 + .05(–.1650 – .0764)2 p

2 = .02436

p = (.02436)1/2 = .1561 or 15.61%

11. The beta of a portfolio is the sum of the weight of each asset timesthe beta of each asset. So, the beta of the portfolio is:

p = .25(.84) + .20(1.17) + .15(1.11) + .40(1.36) = 1.15

12. The beta of a portfolio is the sum of the weight of each asset timesthe beta of each asset. If the portfolio is as risky as the market itmust have the same beta as the market. Since the beta of the marketis one, we know the beta of our portfolio is one. We also need toremember that the beta of the risk-free asset is zero. It has to bezero since the asset has no risk. Setting up the equation for thebeta of our portfolio, we get:

CHAPTER 13 B-452

p = 1.0 = 1/3(0) + 1/3(1.38) + 1/3(X)

Solving for the beta of Stock X, we get:

X = 1.62

B-453 SOLUTIONS

13. CAPM states the relationship between the risk of an asset and itsexpected return. CAPM is:

E(Ri) = Rf + [E(RM) – Rf] × i

Substituting the values we are given, we find:

E(Ri) = .052 + (.11 – .052)(1.05) = .1129 or 11.29%

14. We are given the values for the CAPM except for the of the stock.We need to substitute these values into the CAPM, and solve for the of the stock. One important thing we need to realize is that we aregiven the market risk premium. The market risk premium is theexpected return of the market minus the risk-free rate. We must becareful not to use this value as the expected return of the market.Using the CAPM, we find:

E(Ri) = .102 = .045+ .085i

i = 0.67

15. Here we need to find the expected return of the market using theCAPM. Substituting the values given, and solving for the expectedreturn of the market, we find:

E(Ri) = .135 = .055 + [E(RM) – .055](1.17)

E(RM) = .1234 or 12.34%

16. Here we need to find the risk-free rate using the CAPM. Substitutingthe values given, and solving for the risk-free rate, we find:

E(Ri) = .14 = Rf + (.115 – Rf)(1.45)

.14 = Rf + .16675 – 1.45Rf

Rf = .0594 or 5.94%

17. a. Again we have a special case where the portfolio is equallyweighted, so we can sum the returns of each asset and divide bythe number of assets. The expected return of the portfolio is:

E(Rp) = (.16 + .048)/2 = .1040 or 10.40%

CHAPTER 13 B-454

B-455 SOLUTIONS

b. We need to find the portfolio weights that result in a portfoliowith a of 0.95. We know the of the risk-free asset is zero. Wealso know the weight of the risk-free asset is one minus theweight of the stock since the portfolio weights must sum to one,or 100 percent. So:

p = 0.95 = wS(1.35) + (1 – wS)(0) 0.95 = 1.35wS + 0 – 0wS

wS = 0.95/1.35 wS = .7037

And, the weight of the risk-free asset is:

wRf = 1 – .7037 = .2963

c. We need to find the portfolio weights that result in a portfoliowith an expected return of 8 percent. We also know the weight ofthe risk-free asset is one minus the weight of the stock since theportfolio weights must sum to one, or 100 percent. So:

E(Rp) = .08 = .16wS + .048(1 – wS) .08 = .16wS + .048 – .048wS

.032 = .112wS

wS = .2857

So, the of the portfolio will be:

p = .2857(1.35) + (1 – .2857)(0) = 0.386

d. Solving for the of the portfolio as we did in part a, we find:

p = 2.70 = wS(1.35) + (1 – wS)(0)

wS = 2.70/1.35 = 2

wRf = 1 – 2 = –1

The portfolio is invested 200% in the stock and –100% in the risk-free asset. This represents borrowing at the risk-free rate to buymore of the stock.

CHAPTER 13 B-456

18. First, we need to find the of the portfolio. The of the risk-freeasset is zero, and the weight of the risk-free asset is one minus theweight of the stock, the of the portfolio is:

ßp = wW(1.25) + (1 – wW)(0) = 1.25wW

So, to find the of the portfolio for any weight of the stock, wesimply multiply the weight of the stock times its .

B-457 SOLUTIONS

Even though we are solving for the and expected return of aportfolio of one stock and the risk-free asset for differentportfolio weights, we are really solving for the SML. Any combinationof this stock, and the risk-free asset will fall on the SML. For thatmatter, a portfolio of any stock and the risk-free asset, or anyportfolio of stocks, will fall on the SML. We know the slope of theSML line is the market risk premium, so using the CAPM and theinformation concerning this stock, the market risk premium is:

E(RW) = .152 = .053 + MRP(1.25) MRP = .099/1.25 = .0792 or 7.92%

So, now we know the CAPM equation for any stock is:

E(Rp) = .053 + .0793p

The slope of the SML is equal to the market risk premium, which is0.0792. Using these equations to fill in the table, we get the followingresults:

wW E(Rp) ßp

0.00% 5.30% 0.000 25.00% 7.78% 0.313 50.00% 10.25% 0.625 75.00% 12.73% 0.938 100.00% 15.20% 1.250 125.00% 17.68% 1.563 150.00% 20.15% 1.875

19. There are two ways to correctly answer this question. We will workthrough both. First, we can use the CAPM. Substituting in the valuewe are given for each stock, we find:

E(RY) = .08 + .075(1.30) = .1775 or 17.75%

It is given in the problem that the expected return of Stock Y is18.5 percent, but according to the CAPM, the return of the stockbased on its level of risk, the expected return should be 17.75percent. This means the stock return is too high, given its level ofrisk. Stock Y plots above the SML and is undervalued. In other words,

CHAPTER 13 B-458

its price must increase to reduce the expected return to 17.75percent. For Stock Z, we find:

E(RZ) = .08 + .075(0.70) = .1325 or 13.25%

The return given for Stock Z is 12.1 percent, but according to theCAPM the expected return of the stock should be 13.25 percent basedon its level of risk. Stock Z plots below the SML and is overvalued.In other words, its price must decrease to increase the expectedreturn to 13.25 percent.

B-459 SOLUTIONS

We can also answer this question using the reward-to-risk ratio. Allassets must have the same reward-to-risk ratio. The reward-to-riskratio is the risk premium of the asset divided by its . We are giventhe market risk premium, and we know the of the market is one, sothe reward-to-risk ratio for the market is 0.075, or 7.5 percent.Calculating the reward-to-risk ratio for Stock Y, we find:

Reward-to-risk ratio Y = (.185 – .08) / 1.30 = .0808

The reward-to-risk ratio for Stock Y is too high, which means thestock plots above the SML, and the stock is undervalued. Its pricemust increase until its reward-to-risk ratio is equal to the marketreward-to-risk ratio. For Stock Z, we find:

Reward-to-risk ratio Z = (.121 – .08) / .70 = .0586

The reward-to-risk ratio for Stock Z is too low, which means thestock plots below the SML, and the stock is overvalued. Its pricemust decrease until its reward-to-risk ratio is equal to the marketreward-to-risk ratio.

20. We need to set the reward-to-risk ratios of the two assets equal to

each other, which is:

(.185 – Rf)/1.30 = (.121 – Rf)/0.70

We can cross multiply to get:

0.70(.185 – Rf) = 1.30(.121 – Rf)

Solving for the risk-free rate, we find:

0.1295 – 0.70Rf = 0.1573 – 1.30Rf

Rf = .0463 or 4.63%

Intermediate

21. For a portfolio that is equally invested in large-company stocks andlong-term bonds:

Return = (12.30% + 5.80%)/2 = 9.05%

CHAPTER 13 B-460

For a portfolio that is equally invested in small stocks and Treasurybills:

Return = (17.10% + 3.80%)/2 = 10.45%

B-461 SOLUTIONS

22. We know that the reward-to-risk ratios for all assets must be equal.This can be expressed as:

[E(RA) – Rf]/A = [E(RB) – Rf]/ßB

The numerator of each equation is the risk premium of the asset, so:

RPA/A = RPB/B

We can rearrange this equation to get:

B/A = RPB/RPA

If the reward-to-risk ratios are the same, the ratio of the betas ofthe assets is equal to the ratio of the risk premiums of the assets.

23. a. We need to find the return of the portfolio in each state of theeconomy. To do this, we will multiply the return of each asset byits portfolio weight and then sum the products to get theportfolio return in each state of the economy. Doing so, we get:

Boom: E(Rp) = .4(.24) + .4(.36) + .2(.55) = .3500 or 35.00%Normal: E(Rp) = .4(.17) + .4(.13)+ .2(.09) = .1380 or 13.80%Bust: E(Rp) = .4(.00) + .4(–.28) + .2(–.45) = –.2020 or –20.20%


E(Rp) = .35(.35) + .50(.138) + .15(–.202) = .1612 or 16.12%

To calculate the standard deviation, we first need to calculatethe variance. To find the variance, we find the squared deviationsfrom the expected return. We then multiply each possible squareddeviation by its probability, than add all of these up. The resultis the variance. So, the variance and standard deviation of theportfolio is:

2p = .35(.35 – .1612)2 + .50(.138 – .1612)2 + .15(–.202 – .1612)2

2p = .03253

p = (.03253)1/2 = .1804 or 18.04%

CHAPTER 13 B-462

b. The risk premium is the return of a risky asset, minus the risk-free rate. T-bills are often used as the risk-free rate, so:

RPi = E(Rp) – Rf = .1612 – .0380 = .1232 or 12.32%

B-463 SOLUTIONS

c. The approximate expected real return is the expected nominalreturn minus the inflation rate, so:

Approximate expected real return = .1612 – .035 = .1262 or 12.62%

To find the exact real return, we will use the Fisher equation.Doing so, we get:

1 + E(Ri) = (1 + h)[1 + e(ri)] 1.1612 = (1.0350)[1 + e(ri)] e(ri) = (1.1612/1.035) – 1 = .1219 or 12.19%

The approximate real risk premium is the expected return minus therisk-free rate, so:

Approximate expected real risk premium = .1612 – .038 = .1232 or12.32%

The exact expected real risk premium is the approximate expectedreal risk premium, divided by one plus the inflation rate, so:

Exact expected real risk premium = .1168/1.035 = .1190 or 11.90%

24. Since the portfolio is as risky as the market, the of the portfolio must be equal to one. We also know the of the risk-free asset is zero. We can use the equation for the of a portfolio to find the weight of the third stock. Doing so, we find:

p = 1.0 = wA(.85) + wB(1.20) + wC(1.35) + wRf(0)

Solving for the weight of Stock C, we find:

wC = .324074

So, the dollar investment in Stock C must be:

Invest in Stock C = .324074($1,000,000) = $324,074.07

We know the total portfolio value and the investment of two stocks inthe portfolio, so we can find the weight of these two stocks. Theweights of Stock A and Stock B are:

wA = $210,000 / $1,000,000 = .210

CHAPTER 13 B-464

wB = $320,000/$1,000,000 = .320

B-465 SOLUTIONS

We also know the total portfolio weight must be one, so the weight ofthe risk-free asset must be one minus the asset weight we know, or:

1 = wA + wB + wC + wRf = 1 – .210 – .320 – .324074 – wRf

wRf = .145926

So, the dollar investment in the risk-free asset must be:

Invest in risk-free asset = .145926($1,000,000) = $145,925.93

Challenge

25. We are given the expected return of the assets in the portfolio. We also know the sum of the weights of each asset must be equal to one. Using this relationship, we can express the expected return of the portfolio as:

E(Rp) = .185 = wX(.172) + wY(.136) .185 = wX(.172) + (1 – wX)(.136).185 = .172wX + .136 – .136wX

.049 = .036wX

wX = 1.36111

And the weight of Stock Y is:

wY = 1 – 1.36111 wY = –.36111

The amount to invest in Stock Y is:

Investment in Stock Y = –.36111($100,000) Investment in Stock Y = –$36,111.11

A negative portfolio weight means that you short sell the stock. If you are not familiar with short selling, it means you borrow a stock today and sell it. You must then purchase the stock at a later date to repay the borrowed stock. If you short sell a stock, you make a profit if the stock decreases in value.

To find the beta of the portfolio, we can multiply the portfolio weight of each asset times its beta and sum. So, the beta of the portfolio is:

CHAPTER 13 B-466

P = 1.36111(1.40) + (–.36111)(0.95)P = 1.56

.

B-467 SOLUTIONS

26. The amount of systematic risk is measured by the of an asset. Sincewe know the market risk premium and the risk-free rate, if we knowthe expected return of the asset we can use the CAPM to solve for the of the asset. The expected return of Stock I is:

E(RI) = .25(.11) + .50(.29) + .25(.13) = .2050 or 20.50%

Using the CAPM to find the of Stock I, we find:

.2050 = .04 + .08I I = 2.06

The total risk of the asset is measured by its standard deviation, sowe need to calculate the standard deviation of Stock I. Beginning with the calculation of the stock’s variance, we find:

I2 = .25(.11 – .2050)2 + .50(.29 – .2050)2 + .25(.13 – .2050)2

I2 = .00728

I = (.00728)1/2 = .0853 or 8.53%

Using the same procedure for Stock II, we find the expected return tobe:

E(RII) = .25(–.40) + .50(.10) + .25(.56) = .0900

Using the CAPM to find the of Stock II, we find:

.0900 = .04 + .08II II = 0.63

And the standard deviation of Stock II is:

II2 = .25(–.40 – .0900)2 + .50(.10 – .0900)2 + .25(.56 – .0900)2

II2 = .11530

II = (.11530)1/2 = .3396 or 33.96%

Although Stock II has more total risk than I, it has much lesssystematic risk, since its beta is much smaller than I’s. Thus, I hasmore systematic risk, and II has more unsystematic and more totalrisk. Since unsystematic risk can be diversified away, I is actuallythe “riskier” stock despite the lack of volatility in its returns.

CHAPTER 13 B-468

Stock I will have a higher risk premium and a greater expectedreturn.

B-469 SOLUTIONS

27. Here we have the expected return and beta for two assets. We canexpress the returns of the two assets using CAPM. If the CAPM istrue, then the security market line holds as well, which means allassets have the same risk premium. Setting the risk premiums of theassets equal to each other and solving for the risk-free rate, wefind:

(.132 – Rf)/1.35 = (.101 – Rf)/.80.80(.132 – Rf) = 1.35(.101 – Rf).1056 – .8Rf = .13635 – 1.35Rf

.55Rf = .03075Rf = .0559 or 5.59%

Now using CAPM to find the expected return on the market with bothstocks, we find:

.132 = .0559 + 1.35(RM – .0559) .101 = .0559 + .80(RM – .0559)RM = .1123 or 11.23% RM = .1123 or 11.23%

28. a. The expected return of an asset is the sum of the probability ofeach return occurring times the probability of that returnoccurring. So, the expected return of each stock is:

E(RA) = .15(–.08) + .70(.13) + .15(.48) = .1510 or 15.10%

E(RB) = .15(–.05) + .70(.14) + .15(.29) = .1340 or 13.40%

b. We can use the expected returns we calculated to find the slope ofthe Security Market Line. We know that the beta of Stock A is .25greater than the beta of Stock B. Therefore, as beta increasesby .25, the expected return on a security increases by .017(= .1510 – .1340). The slope of the security market line (SML)equals:

CHAPTER 13 B-470

SlopeSML = Rise / RunSlopeSML = Increase in expected return / Increase in betaSlopeSML = (.1510 – .1340) / .25SlopeSML = .0680 or 6.80%

B-471 SOLUTIONS

Since the market’s beta is 1 and the risk-free rate has a beta ofzero, the slope of the Security Market Line equals the expectedmarket risk premium. So, the expected market risk premium must be6.8 percent.

We could also solve this problem using CAPM. The equations for theexpected returns of the two stocks are:

E(RA) = .151 = Rf + (B + .25)(MRP)E(RB) = .134 = Rf + B(MRP)

We can rewrite the CAPM equation for Stock A as:

.151 = Rf + B(MRP) + .25(MRP)

Subtracting the CAPM equation for Stock B from this equationyields:

.017 = .25MRPMRP = .068 or 6.8%

which is the same answer as our previous result.

CHAPTER 14COST OF CAPITAL


1. It is the minimum rate of return the firm must earn overall on itsexisting assets. If it earns more than this, value is created.

2. Book values for debt are likely to be much closer to market valuesthan are equity book values.

3. No. The cost of capital depends on the risk of the project, not thesource of the money.

4. Interest expense is tax-deductible. There is no difference betweenpretax and aftertax equity costs.

5. The primary advantage of the DCF model is its simplicity. The methodis disadvantaged in that (1) the model is applicable only to firmsthat actually pay dividends; many do not; (2) even if a firm does paydividends, the DCF model requires a constant dividend growth rateforever; (3) the estimated cost of equity from this method is verysensitive to changes in g, which is a very uncertain parameter; and(4) the model does not explicitly consider risk, although risk isimplicitly considered to the extent that the market has impounded therelevant risk of the stock into its market price. While the shareprice and most recent dividend can be observed in the market, thedividend growth rate must be estimated. Two common methods ofestimating g are to use analysts’ earnings and payout forecasts or todetermine some appropriate average historical g from the firm’savailable data.

6. Two primary advantages of the SML approach are that the modelexplicitly incorporates the relevant risk of the stock and the methodis more widely applicable than is the dividend discount model model,since the SML doesn’t make any assumptions about the firm’sdividends. The primary disadvantages of the SML method are (1) threeparameters (the risk-free rate, the expected return on the market,and beta) must be estimated, and (2) the method essentially uses

B-473 SOLUTIONS

historical information to estimate these parameters. The risk-freerate is usually estimated to be the yield on very short maturity T-bills and is, hence, observable; the market risk premium is usuallyestimated from historical risk premiums and, hence, is notobservable. The stock beta, which is unobservable, is usuallyestimated either by determining some average historical beta from thefirm and the market’s return data, or by using beta estimatesprovided by analysts and investment firms.

7. The appropriate aftertax cost of debt to the company is the interestrate it would have to pay if it were to issue new debt today. Hence,if the YTM on outstanding bonds of the company is observed, thecompany has an accurate estimate of its cost of debt. If the debt isprivately-placed, the firm could still estimate its cost of debt by(1) looking at the cost of debt for similar firms in similar riskclasses, (2) looking at the average debt cost for firms with the samecredit rating (assuming the firm’s private debt is rated), or (3)consulting analysts and investment bankers. Even if the debt ispublicly traded, an additional complication is when the firm has morethan one issue outstanding; these issues rarely have the same yieldbecause no two issues are ever completely homogeneous.

CHAPTER 14 B-474

8. a. This only considers the dividend yield component of the requiredreturn on equity.

b. This is the current yield only, not the promised yield tomaturity. In addition, it is based on the book value of theliability, and it ignores taxes.

c. Equity is inherently more risky than debt (except, perhaps, in theunusual case where a firm’s assets have a negative beta). For thisreason, the cost of equity exceeds the cost of debt. If taxes areconsidered in this case, it can be seen that at reasonable taxrates, the cost of equity does exceed the cost of debt.

9. RSup = .12 + .75(.08) = .1800 or 18.00%Both should proceed. The appropriate discount rate does not depend onwhich company is investing; it depends on the risk of the project.Since Superior is in the business, it is closer to a pure play.Therefore, its cost of capital should be used. With an 18% cost ofcapital, the project has an NPV of $1 million regardless of who takesit.

10. If the different operating divisions were in much different riskclasses, then separate cost of capital figures should be used for thedifferent divisions; the use of a single, overall cost of capitalwould be inappropriate. If the single hurdle rate were used, riskierdivisions would tend to receive more funds for investment projects,since their return would exceed the hurdle rate despite the fact thatthey may actually plot below the SML and, hence, be unprofitableprojects on a risk-adjusted basis. The typical problem encountered inestimating the cost of capital for a division is that it rarely hasits own securities traded on the market, so it is difficult toobserve the market’s valuation of the risk of the division. Twotypical ways around this are to use a pure play proxy for thedivision, or to use subjective adjustments of the overall firm hurdlerate based on the perceived risk of the division.



Basic

B-475 SOLUTIONS

1. With the information given, we can find the cost of equity using thedividend growth model. Using this model, the cost of equity is:

RE = [$2.40(1.055)/$52] + .055 = .1037 or 10.37%

2. Here we have information to calculate the cost of equity using theCAPM. The cost of equity is:

RE = .053 + 1.05(.12 – .053) = .1234 or 12.34%

3. We have the information available to calculate the cost of equityusing the CAPM and the dividend growth model. Using the CAPM, wefind:

RE = .05 + 0.85(.08) = .1180 or 11.80%

CHAPTER 14 B-476

And using the dividend growth model, the cost of equity is

RE = [$1.60(1.06)/$37] + .06 = .1058 or 10.58%

Both estimates of the cost of equity seem reasonable. If we rememberthe historical return on large capitalization stocks, the estimatefrom the CAPM model is about two percent higher than average, and theestimate from the dividend growth model is about one percent higherthan the historical average, so we cannot definitively say one of theestimates is incorrect. Given this, we will use the average of thetwo, so:

RE = (.1180 + .1058)/2 = .1119 or 11.19%

4. To use the dividend growth model, we first need to find the growthrate in dividends. So, the increase in dividends each year was:

g1 = ($1.12 – 1.05)/$1.05 = .0667 or 6.67% g2 = ($1.19 – 1.12)/$1.12 = .0625 or 6.25%g3 = ($1.30 – 1.19)/$1.19 = .0924 or 9.24%g4 = ($1.43 – 1.30)/$1.30 = .1000 or 10.00%

So, the average arithmetic growth rate in dividends was:

g = (.0667 + .0625 + .0924 + .1000)/4 = .0804 or 8.04%

Using this growth rate in the dividend growth model, we find the costof equity is:

RE = [$1.43(1.0804)/$45.00] + .0804 = .1147 or 11.47%

Calculating the geometric growth rate in dividends, we find:

$1.43 = $1.05(1 + g)4 g = .0803 or 8.03%

The cost of equity using the geometric dividend growth rate is:

RE = [$1.43(1.0803)/$45.00] + .0803 = .1146 or 11.46%

5. The cost of preferred stock is the dividend payment divided by theprice, so:

B-477 SOLUTIONS

RP = $6/$96 = .0625 or 6.25%

6. The pretax cost of debt is the YTM of the company’s bonds, so:

P0 = $1,070 = $35(PVIFAR%,30) + $1,000(PVIFR%,30) R = 3.137%YTM = 2 × 3.137% = 6.27%

And the aftertax cost of debt is:

RD = .0627(1 – .35) = .0408 or 4.08%

CHAPTER 14 B-478

7. a. The pretax cost of debt is the YTM of the company’s bonds, so:

P0 = $950 = $40(PVIFAR%,46) + $1,000(PVIFR%,46) R = 4.249%YTM = 2 × 4.249% = 8.50%

b. The aftertax cost of debt is:

RD = .0850(1 – .35) = .0552 or 5.52%

c. The after-tax rate is more relevant because that is the actualcost to the company.

8. The book value of debt is the total par value of all outstandingdebt, so:

BVD = $80,000,000 + 35,000,000 = $115,000,000

To find the market value of debt, we find the price of the bonds andmultiply by the number of bonds. Alternatively, we can multiply theprice quote of the bond times the par value of the bonds. Doing so,we find:

MVD = .95($80,000,000) + .61($35,000,000) MVD = $76,000,000 + 21,350,000MVD = $97,350,000

The YTM of the zero coupon bonds is:

PZ = $610 = $1,000(PVIFR%,14) R = 3.594%YTM = 2 × 3.594% = 7.19%

So, the aftertax cost of the zero coupon bonds is:

RZ = .0719(1 – .35) = .0467 or 4.67%

The aftertax cost of debt for the company is the weighted average ofthe aftertax cost of debt for all outstanding bond issues. We need touse the market value weights of the bonds. The total aftertax cost ofdebt for the company is:

RD = .0552($76/$97.35) + .0467($21.35/$97.35) = .0534 or 5.34%

B-479 SOLUTIONS

9. a. Using the equation to calculate the WACC, we find:

WACC = .60(.14) + .05(.06) + .35(.08)(1 – .35) = .1052 or 10.52%

b. Since interest is tax deductible and dividends are not, we mustlook at the after-tax cost of debt, which is:

.08(1 – .35) = .0520 or 5.20%

Hence, on an after-tax basis, debt is cheaper than the preferredstock.

CHAPTER 14 B-480

10. Here we need to use the debt-equity ratio to calculate the WACC.Doing so, we find:

WACC = .15(1/1.65) + .09(.65/1.65)(1 – .35) = .1140 or 11.40%

11. Here we have the WACC and need to find the debt-equity ratio of thecompany. Setting up the WACC equation, we find:

WACC = .0890 = .12(E/V) + .079(D/V)(1 – .35)

Rearranging the equation, we find:

.0890(V/E) = .12 + .079(.65)(D/E)

Now we must realize that the V/E is just the equity multiplier, whichis equal to:

V/E = 1 + D/E

.0890(D/E + 1) = .12 + .05135(D/E)

Now we can solve for D/E as:

.06765(D/E) = .031 D/E = .8234

12. a. The book value of equity is the book value per share times thenumber of shares, and the book value of debt is the face value ofthe company’s debt, so:

BVE = 11,000,000($6) = $66,000,000

BVD = $70,000,000 + 55,000,000 = $125,000,000

So, the total value of the company is:

V = $66,000,000 + 125,000,000 = $191,000,000

And the book value weights of equity and debt are:

E/V = $66,000,000/$191,000,000 = .3455

D/V = 1 – E/V = .6545

B-481 SOLUTIONS

b. The market value of equity is the share price times the number ofshares, so:

MVE = 11,000,000($68) = $748,000,000

Using the relationship that the total market value of debt is theprice quote times the par value of the bond, we find the marketvalue of debt is:

MVD = .93($70,000,000) + 1.04($55,000,000) = $122,300,000

CHAPTER 14 B-482

This makes the total market value of the company:

V = $748,000,000 + 122,300,000 = $870,300,000

And the market value weights of equity and debt are:E/V = $748,000,000/$870,300,000 = .8595

D/V = 1 – E/V = .1405

c. The market value weights are more relevant.

13. First, we will find the cost of equity for the company. Theinformation provided allows us to solve for the cost of equity usingthe dividend growth model, so:

RE = [$4.10(1.06)/$68] + .06 = .1239 or 12.39%

Next, we need to find the YTM on both bond issues. Doing so, we find:

P1 = $930 = $35(PVIFAR%,42) + $1,000(PVIFR%,42) R = 3.838% YTM = 3.838% × 2 = 7.68%

P2 = $1,040 = $40(PVIFAR%,12) + $1,000(PVIFR%,12) R = 3.584% YTM = 3.584% × 2 = 7.17%

To find the weighted average aftertax cost of debt, we need theweight of each bond as a percentage of the total debt. We find:

wD1 = .93($70,000,000)/$122,300,000 = .5323

wD2 = 1.04($55,000,000)/$122,300,000 = .4677

Now we can multiply the weighted average cost of debt times one minusthe tax rate to find the weighted average aftertax cost of debt. Thisgives us:

RD = (1 – .35)[(.5323)(.0768) + (.4677)(.0717)] = .0484 or 4.84%

Using these costs we have found and the weight of debt we calculatedearlier, the WACC is:

B-483 SOLUTIONS

WACC = .8595(.1239) + .1405(.0484) = .1133 or 11.33%

14. a. Using the equation to calculate WACC, we find:

WACC = .094 = (1/2.05)(.14) + (1.05/2.05)(1 – .35)RD RD = .0772 or 7.72%

CHAPTER 14 B-484

b. Using the equation to calculate WACC, we find:

WACC = .094 = (1/2.05)RE + (1.05/2.05)(.068) RE = .1213 or 12.13%

15. We will begin by finding the market value of each type of financing.We find:

MVD = 8,000($1,000)(0.92) = $7,360,000 MVE = 250,000($57) = $14,250,000MVP = 15,000($93) = $1,395,000

And the total market value of the firm is:

V = $7,360,000 + 14,250,000 + 1,395,000 = $23,005,000

Now, we can find the cost of equity using the CAPM. The cost ofequity is:

RE = .045 + 1.05(.08) = .1290 or 12.90%

The cost of debt is the YTM of the bonds, so:

P0 = $920 = $32.50(PVIFAR%,40) + $1,000(PVIFR%,40) R = 3.632%YTM = 3.632% × 2 = 7.26%


RD = (1 – .35)(.0726) = .0472 or 4.72%

The cost of preferred stock is:

RP = $5/$93 = .0538 or 5.38%

Now we have all of the components to calculate the WACC. The WACC is:

WACC = .0472(7.36/23.005) + .1290(14.25/23.005) + .0538(1.395/23.005)= .0983 or 9.83%

Notice that we didn’t include the (1 – tC) term in the WACC equation.We used the aftertax cost of debt in the equation, so the term is notneeded here.

B-485 SOLUTIONS

16. a. We will begin by finding the market value of each type offinancing. We find:

MVD = 105,000($1,000)(0.93) = $97,650,000 MVE = 9,000,000($34) = $306,000,000MVP = 250,000($91) = $22,750,000

And the total market value of the firm is:

V = $97,650,000 + 306,000,000 + 22,750,000 = $426,400,000

CHAPTER 14 B-486

So, the market value weights of the company’s financing is:

D/V = $97,650,000/$426,400,000 = .2290P/V = $22,750,000/$426,400,000 = .0534 E/V = $306,000,000/$426,400,000 = .7176

b. For projects equally as risky as the firm itself, the WACC shouldbe used as the discount rate.

First we can find the cost of equity using the CAPM. The cost ofequity is:

RE = .05 + 1.25(.085) = .1563 or 15.63%

The cost of debt is the YTM of the bonds, so:

P0 = $930 = $37.5(PVIFAR%,30) + $1,000(PVIFR%,30) R = 4.163% YTM = 4.163% × 2 = 8.33%


RD = (1 – .35)(.0833) = .0541 or 5.41%


RP = $6/$91 = .0659 or 6.59%

Now we can calculate the WACC as:

WACC = .0541(.2290) + .1563(.7176) + .0659(.0534) = .1280 or12.80%

17. a. Projects X, Y and Z.

b. Using the CAPM to consider the projects, we need to calculate theexpected return of the project given its level of risk. Thisexpected return should then be compared to the expected return ofthe project. If the return calculated using the CAPM is lower thanthe project expected return, we should accept the project, if not,we reject the project. After considering risk via the CAPM:

E[W]= .05 + .80(.11 – .05) = .0980 < .10, so accept W

B-487 SOLUTIONS

E[X] = .05 + .90(.11 – .05) = .1040 < .12, soaccept XE[Y] = .05 + 1.45(.11 – .05) = .1370 > .13, soreject YE[Z] = .05 + 1.60(.11 – .05) = .1460 < .15, soaccept Z

c. Project W would be incorrectly rejected; Project Y would beincorrectly accepted.

18. a. He should look at the weighted average flotation cost, not justthe debt cost.

CHAPTER 14 B-488

b. The weighted average floatation cost is the weighted average ofthe floatation costs for debt and equity, so:

fT = .05(.75/1.75) + .08(1/1.75) = .0671 or 6.71%

c. The total cost of the equipment including floatation costs is:

Amount raised(1 – .0671) = $20,000,000 Amount raised = $20,000,000/(1 – .0671) = $21,439,510

Even if the specific funds are actually being raised completelyfrom debt, the flotation costs, and hence true investment cost,should be valued as if the firm’s target capital structure isused.

19. We first need to find the weighted average floatation cost. Doing so,we find:

fT = .65(.09) + .05(.06) + .30(.03) = .071 or 7.1%

And the total cost of the equipment including floatation costs is:


Intermediate

20. Using the debt-equity ratio to calculate the WACC, we find:

WACC = (.90/1.90)(.048) + (1/1.90)(.13) = .0912 or 9.12%

Since the project is riskier than the company, we need to adjust theproject discount rate for the additional risk. Using the subjectiverisk factor given, we find:

Project discount rate = 9.12% + 2.00% = 11.12%

We would accept the project if the NPV is positive. The NPV is the PVof the cash outflows plus the PV of the cash inflows. Since we havethe costs, we just need to find the PV of inflows. The cash inflowsare a growing perpetuity. If you remember, the equation for the PV ofa growing perpetuity is the same as the dividend growth equation, so:

B-489 SOLUTIONS

PV of future CF = $2,700,000/(.1112 – .04) = $37,943,787

The project should only be undertaken if its cost is less than$37,943,787 since costs less than this amount will result in apositive NPV.

21. The total cost of the equipment including floatation costs was:

Total costs = $15,000,000 + 850,000 = $15,850,000

CHAPTER 14 B-490

Using the equation to calculate the total cost including floatationcosts, we get:

Amount raised(1 – fT) = Amount needed after floatation costs$15,850,000(1 – fT) = $15,000,000 fT = .0536 or 5.36%

Now, we know the weighted average floatation cost. The equation tocalculate the percentage floatation costs is:

fT = .0536 = .07(E/V) + .03(D/V)

We can solve this equation to find the debt-equity ratio as follows:

.0536(V/E) = .07 + .03(D/E)

We must recognize that the V/E term is the equity multiplier, whichis (1 + D/E), so:

.0536(D/E + 1) = .08 + .03(D/E) D/E = 0.6929

22. To find the aftertax cost of debt for the company, we need to findthe weighted average of the four debt issues. We will begin bycalculating the market value of each debt issue, which is:

MV1 = 1.03($40,000,000) MV1 = $41,200,000

MV2 = 1.08($35,000,000) MV2 = $37,800,000

MV3 = 0.97($55,000,000) MV3 = $53,500,000

MV4 = 1.11($40,000,000) MV4 = $55,500,000

So, the total market value of the company’s debt is:

MVD = $41,200,000 + 37,800,000 + 53,350,000 + 55,500,000 MVD = $187,850,000

B-491 SOLUTIONS

The weight of each debt issue is:

w1 = $41,200,000/$187,850,000 w1 = .2193 or 21.93%

w2 = $37,800,000/$187,850,000 w2 = .2012 or 20.12%

w3 = $53,500,000/$187,850,000 w3 = .2840 or 28.40%

CHAPTER 14 B-492

w4 = $55,500,000/$187,850,000 w4 = .2954 or 29.54%

Next, we need to find the YTM for each bond issue. The YTM for eachissue is:

P1 = $1,030 = $35(PVIFAR%,10) + $1,000(PVIFR%,10) R1 = 2.768% YTM1 = 3.146% × 2 YTM1 = 6.29%

P2 = $1,080 = $42.50(PVIFAR%,16) + $1,000(PVIFR%,16) R2 = 3.584% YTM2 = 3.584% × 2 YTM2 = 7.17%

P3 = $970 = $41(PVIFAR%,31) + $1,000(PVIFR%,31) R3 = 3.654% YTM3 = 4.276% × 2 YTM3 = 8.54%

P4 = $1,110 = $49(PVIFAR%,50) + $1,000(PVIFR%,50) R4 = 4.356% YTM4 = 4.356% × 2 YTM4 = 8.71%

The weighted average YTM of the company’s debt is thus:

YTM = .2193(.0629) + .2012 (.0717) + .2840(.0854) + .2954(.0871)YTM = .0782 or 7.82%


RD = .0782(1 – .034)RD = .0516 or 5.16%

23. a. Using the dividend discount model, the cost of equity is:

RE = [(0.80)(1.05)/$61] + .05 RE = .0638 or 6.38%

b. Using the CAPM, the cost of equity is:

B-493 SOLUTIONS

RE = .055 + 1.50(.1200 – .0550) RE = .1525 or 15.25%

c. When using the dividend growth model or the CAPM, you mustremember that both are estimates for the cost of equity.Additionally, and perhaps more importantly, each method of

estimating the cost of equity depends upon differentassumptions.

CHAPTER 14 B-494

Challenge

24. We can use the debt-equity ratio to calculate the weights of equityand debt. The debt of the company has a weight for long-term debt anda weight for accounts payable. We can use the weight given foraccounts payable to calculate the weight of accounts payable and theweight of long-term debt. The weight of each will be:

Accounts payable weight = .20/1.20 = .17Long-term debt weight = 1/1.20 = .83

Since the accounts payable has the same cost as the overall WACC, wecan write the equation for the WACC as:

WACC = (1/1.7)(.14) + (0.7/1.7)[(.20/1.2)WACC + (1/1.2)(.08)(1– .35)]

Solving for WACC, we find:

WACC = .0824 + .4118[(.20/1.2)WACC + .0433]WACC = .0824 + (.0686)WACC + .0178(.9314)WACC = .1002WACC = .1076 or 10.76%

We will use basically the same equation to calculate the weightedaverage floatation cost, except we will use the floatation cost foreach form of financing. Doing so, we get:

Flotation costs = (1/1.7)(.08) + (0.7/1.7)[(.20/1.2)(0) + (1/1.2)(.04)] = .0608 or 6.08%

The total amount we need to raise to fund the new equipment will be:

Amount raised cost = $45,000,000/(1 – .0608) Amount raised = $47,912,317

Since the cash flows go to perpetuity, we can calculate the presentvalue using the equation for the PV of a perpetuity. The NPV is:

NPV = –$47,912,317 + ($6,200,000/.1076)NPV = $9,719,777

B-495 SOLUTIONS

25. We can use the debt-equity ratio to calculate the weights of equityand debt. The weight of debt in the capital structure is:

wD = 1.20 / 2.20 = .5455 or 54.55%

And the weight of equity is:

wE = 1 – .5455 = .4545 or 45.45%

CHAPTER 14 B-496

Now we can calculate the weighted average floatation costs for thevarious percentages of internally raised equity. To find the portionof equity floatation costs, we can multiply the equity costs by thepercentage of equity raised externally, which is one minus thepercentage raised internally. So, if the company raises all equityexternally, the floatation costs are:

fT = (0.5455)(.08)(1 – 0) + (0.4545)(.035) fT = .0555 or 5.55%

The initial cash outflow for the project needs to be adjusted for thefloatation costs. To account for the floatation costs:

Amount raised(1 – .0555) = $145,000,000 Amount raised = $145,000,000/(1 – .0555) Amount raised = $153,512,993

If the company uses 60 percent internally generated equity, thefloatation cost is:

fT = (0.5455)(.08)(1 – 0.60) + (0.4545)(.035) fT = .0336 or 3.36%

And the initial cash flow will be:


If the company uses 100 percent internally generated equity, thefloatation cost is:

fT = (0.5455)(.08)(1 – 1) + (0.4545)(.035) fT = .0191 or 1.91%

And the initial cash flow will be:


26. The $4 million cost of the land 3 years ago is a sunk cost andirrelevant; the $5.1 million appraised value of the land is an

B-497 SOLUTIONS

opportunity cost and is relevant. The $6 million land value in 5years is a relevant cash flow as well. The fact that the company iskeeping the land rather than selling it is unimportant. The land isan opportunity cost in 5 years and is a relevant cash flow for thisproject. The market value capitalization weights are:

MVD = 240,000($1,000)(0.94) = $225,600,000 MVE = 9,000,000($71) = $639,000,000MVP = 400,000($81) = $32,400,000

The total market value of the company is:

V = $225,600,000 + 639,000,000 + 32,400,000 = $897,000,000

CHAPTER 14 B-498

Next we need to find the cost of funds. We have the informationavailable to calculate the cost of equity using the CAPM, so:

RE = .05 + 1.20(.08) = .1460 or 14.60%

The cost of debt is the YTM of the company’s outstanding bonds, so:

P0 = $940 = $37.50(PVIFAR%,40) + $1,000(PVIFR%,40) R = 4.056%

YTM = 4.056% × 2 = 8.11%


RD = (1 – .35)(.0811) = .0527 or 5.27%


RP = $5.50/$81 = .0679 or 6.79%

a. The weighted average floatation cost is the sum of the weight ofeach source of funds in the capital structure of the company timesthe floatation costs, so:

fT = ($639/$897)(.08) + ($32.4/$897)(.06) + ($225.6/$897)(.04)= .0692 or 6.92%

The initial cash outflow for the project needs to be adjusted forthe floatation costs. To account for the floatation costs:


So the cash flow at time zero will be:

CF0 = –$5,100,000 – 37,602,765 – 1,3000,000 = –$44,002,765

There is an important caveat to this solution. This solutionassumes that the increase in net working capital does not requirethe company to raise outside funds; therefore the floatation costsare not included. However, this is an assumption and the companycould need to raise outside funds for the NWC. If this is true,the initial cash outlay includes these floatation costs, so:

B-499 SOLUTIONS

Total cost of NWC including floatation costs:

$1,300,000/(1 – .0692) = $1,396,674

This would make the total initial cash flow:

CF0 = –$5,100,000 – 37,602,765 – 1,396,674 = –$44,099,439

CHAPTER 14 B-500

b. To find the required return on this project, we first need tocalculate the WACC for the company. The company’s WACC is:

WACC = [($639/$897)(.1460) + ($32.4/$897)(.0679) + ($225.6/$897)(.0527)] = .1197

The company wants to use the subjective approach to this projectbecause it is located overseas. The adjustment factor is 2percent, so the required return on this project is:

Project required return = .1197 + .02 = .1397

c. The annual depreciation for the equipment will be:

$35,000,000/8 = $4,375,000

So, the book value of the equipment at the end of five years willbe:

BV5 = $35,000,000 – 5($4,375,000) = $13,125,000

So, the aftertax salvage value will be:

Aftertax salvage value = $6,000,000 + .35($13,125,000 – 6,000,000)= $8,493,750

d. Using the tax shield approach, the OCF for this project is:

OCF = [(P – v)Q – FC](1 – t) + tCDOCF = [($10,900 – 9,400)(18,000) – 7,000,000](1 – .35)+ .35($35,000,000/8) = $14,531,250

e. The accounting breakeven sales figure for this project is:

QA = (FC + D)/(P – v) = ($7,000,000 + 4,375,000)/($10,900 – 9,400)= 7,583 units

f. We have calculated all cash flows of the project. We just need tomake sure that in Year 5 we add back the aftertax salvage valueand the recovery of the initial NWC. The cash flows for theproject are:

Year Flow Cash

B-501 SOLUTIONS

0 –$44,002,7651 14,531,2502 14,531,2503 14,531,2504 14,531,2505 30,325,000

Using the required return of 13.97 percent, the NPV of the projectis:

NPV = –$44,002,765 + $14,531,250(PVIFA13.97%,4) + $30,325,000/1.13975

NPV = $14,130,713.81

CHAPTER 14 B-502

And the IRR is:

NPV = 0 = –$44,002,765 + $14,531,250(PVIFAIRR%,4) + $30,325,000/(1 +IRR)5

IRR = 25.25%

If the initial NWC is assumed to be financed from outside sources,the cash flows are:

Year Flow Cash0 –$44,099,4391 14,531,2502 14,531,2503 14,531,2504 14,531,2505 30,325,000

With this assumption, and the required return of 13.97 percent,the NPV of the project is:

NPV = –$44,099,439 + $14,531,250(PVIFA13.97%,4) + $30,325,000/1.13975

NPV = $14,034,039.67

And the IRR is:

NPV = 0 = –$44,099,439 + $14,531,250(PVIFAIRR%,4) + $30,325,000/(1 +IRR)5

IRR = 25.15%

CHAPTER 15RAISING CAPITAL


1. A company’s internally generated cash flow provides a source ofequity financing. For a profitable company, outside equity may neverbe needed. Debt issues are larger because large companies have thegreatest access to public debt markets (small companies tend toborrow more from private lenders). Equity issuers are frequentlysmall companies going public; such issues are often quite small.

2. From the previous question, economies of scale are part of theanswer. Beyond this, debt issues are simply easier and less risky tosell from an investment bank’s perspective. The two main reasons arethat very large amounts of debt securities can be sold to arelatively small number of buyers, particularly large institutionalbuyers such as pension funds and insurance companies, and debtsecurities are much easier to price.

3. They are riskier and harder to market from an investment bank’sperspective.

4. Yields on comparable bonds can usually be readily observed, sopricing a bond issue accurately is much less difficult.

5. It is clear that the stock was sold too cheaply, so Eyetech hadreason to be unhappy.

6. No, but, in fairness, pricing the stock in such a situation isextremely difficult.

7. It’s an important factor. Only 6.5 million of the shares wereunderpriced. The other 32 million were, in effect, priced completelycorrectly.

8. The evidence suggests that a non-underwritten rights offering mightbe substantially cheaper than a cash offer. However, such offerings

CHAPTER 15 B-504

are rare, and there may be hidden costs or other factors not yetidentified or well understood by researchers.

9. He could have done worse since his access to the oversubscribed and,presumably, underpriced issues was restricted while the bulk of hisfunds were allocated to stocks from the undersubscribed and, quitepossibly, overpriced issues.

10. a. The price will probably go up because IPOs are generallyunderpriced. This is especially true for smaller issues such asthis one.

b. It is probably safe to assume that they are having trouble movingthe issue, and it is likely that the issue is not substantiallyunderpriced.

B-505 SOLUTIONS



Basic

1. a. The new market value will be the current shares outstanding timesthe stock price plus the rights offered times the rights price,so:

New market value = 500,000($81) + 60,000($70) = $44,700,000

b. The number of rights associated with the old shares is the numberof shares outstanding divided by the rights offered, so:

Number of rights needed = 500,000 old shares/60,000 new shares =8.33 rights per new share

c. The new price of the stock will be the new market value of thecompany divided by the total number of shares outstanding afterthe rights offer, which will be:

PX = $44,700,000/(500,000 + 60,000) = $79.82

d. The value of the right

Value of a right = $81.00 – 79.82 = $1.18

e. A rights offering usually costs less, it protects theproportionate interests of existing share-holders and alsoprotects against underpricing.

2. a. The maximum subscription price is the current stock price, or $53.The minimum price is anything greater than $0.

b. The number of new shares will be the amount raised divided by thesubscription price, so:

Number of new shares = $40,000,000/$48 = 833,333 shares

CHAPTER 15 B-506

And the number of rights needed to buy one share will be thecurrent shares outstanding divided by the number of new sharesoffered, so:

Number of rights needed = 4,100,000 shares outstanding/833,333 newshares = 4.92

B-507 SOLUTIONS

c. A shareholder can buy 4.92 rights on shares for:

4.92($53) = $260.76

The shareholder can exercise these rights for $48, at a total costof:

$260.76 + 48 = $308.76

The investor will then have:

Ex-rights shares = 1 + 4.92Ex-rights shares = 5.92

The ex-rights price per share is:

PX = [4.92($53) + $48]/5.92 = $52.16

So, the value of a right is:

Value of a right = $53 – 52.16 = $0.84

d. Before the offer, a shareholder will have the shares owned at thecurrent market price, or:

Portfolio value = (1,000 shares)($53) = $53,000

After the rights offer, the share price will fall, but theshareholder will also hold the rights, so:

Portfolio value = (1,000 shares)($52.16) + (1,000 rights)($0.84) =$53,000

3. Using the equation we derived in Problem 2, part c to calculate theprice of the stock ex-rights, we can find the number of shares ashareholder will have ex-rights, which is:

PX = $74.80 = [N($81) + $40]/(N + 1) N = 5.613

The number of new shares is the amount raised divided by the per-share subscription price, so:

CHAPTER 15 B-508

Number of new shares = $20,000,000/$40 = 500,000

And the number of old shares is the number of new shares times thenumber of shares ex-rights, so:

Number of old shares = 5.613(500,000) = 2,806,452

B-509 SOLUTIONS

4. If you receive 1,000 shares of each, the profit is:

Profit = 1,000($7) – 1,000($5) = $2,000

Since you will only receive one-half of the shares of theoversubscribed issue, your profit will be:

Expected profit = 500($7) – 1,000($5) = –$1,500

This is an example of the winner’s curse.

5. Using X to stand for the required sale proceeds, the equation tocalculate the total sale proceeds, including floatation costs is:

X(1 – .09) = $60,000,000 X = $65,934,066 required total proceeds from sale.

So the number of shares offered is the total amount raised divided bythe offer price, which is:

Number of shares offered = $65,934,066/$21 = 3,139,717

6. This is basically the same as the previous problem, except we need toinclude the $900,000 of expenses in the amount the company needs toraise, so:

X(1 – .09) = ($60,000,000 + 900,00) X = $66,923,077 required total proceeds from sale.

Number of shares offered = $66,923,077/$21 = 3,186,813

7. We need to calculate the net amount raised and the costs associatedwith the offer. The net amount raised is the number of shares offeredtimes the price received by the company, minus the costs associatedwith the offer, so:

Net amount raised = (10,000,000 shares)($18.20) – 900,000 – 320,000 =$180,780,000

The company received $180,780,000 from the stock offering. Now we cancalculate the direct costs. Part of the direct costs are given in theproblem, but the company also had to pay the underwriters. The stockwas offered at $20 per share, and the company received $18.20 per

CHAPTER 15 B-510

share. The difference, which is the underwriters spread, is also adirect cost. The total direct costs were:

Total direct costs = $900,000 + ($20 – 18.20)(10,000,000 shares) =$18,900,000

We are given part of the indirect costs in the problem. Anotherindirect cost is the immediate price appreciation. The total indirectcosts were:

Total indirect costs = $320,000 + ($25.60 – 20)(10,000,000 shares) =$56,320,000

B-511 SOLUTIONS

This makes the total costs:

Total costs = $18,900,000 + 56,320,000 = $75,220,000

The floatation costs as a percentage of the amount raised is thetotal cost divided by the amount raised, so:

Flotation cost percentage = $75,220,000/$180,780,000 = .4161 or41.61%

8. The number of rights needed per new share is:

Number of rights needed = 120,000 old shares/25,000 new shares = 4.8rights per new share.

Using PRO as the rights-on price, and PS as the subscription price, wecan express the price per share of the stock ex-rights as:

PX = [NPRO + PS]/(N + 1)

a. PX = [4.8($94) + $94]/(4.80 + 1) = $94.00; No change.

b. PX = [4.8($94) + $90]/(4.80 + 1) = $93.31; Price drops by $0.69per share.

c. PX = [4.8($94) + $85]/(4.80 + 1) = $92.45; Price drops by $1.55per share.

Intermediate

9. a. The number of shares outstanding after the stock offer will be thecurrent shares outstanding, plus the amount raised divided by thecurrent stock price, assuming the stock price doesn’t change. So:

Number of shares after offering = 8,000,000 + $35,000,000/$50 =8,700,000

Since the book value per share is $18, the old book value of theshares is the current number of shares outstanding times 18. Fromthe previous solution, we can see the company will sell 700,000shares, and these will have a book value of $50 per share. The sumof these two values will give us the total book value of thecompany. If we divide this by the new number of shares

CHAPTER 15 B-512

outstanding. Doing so, we find the new book value per share willbe:

New book value per share = [8,000,000($18) +700,000($50)]/8,700,000 = $23.53

The current EPS for the company is:

EPS0 = NI0/Shares0 = $17,000,000/8,000,000 shares = $2.13 per share

And the current P/E is:

(P/E)0 = $50/$2.13 = 23.53

B-513 SOLUTIONS

If the net income increases by $1,100,000, the new EPS will be:

EPS1 = NI1/shares1 = $18,100,000/8,700,000 shares = $2.08 per share

Assuming the P/E remains constant, the new share price will be:

P1 = (P/E)0(EPS1) = 23.53($2.08) = $48.95

The current market-to-book ratio is:

Current market-to-book = $50/$18 = 2.778

Using the new share price and book value per share, the newmarket-to-book ratio will be:

New market-to-book = $48.95/$20.57 = 2.379

Accounting dilution has occurred because new shares were issuedwhen the market-to-book ratio was less than one; market valuedilution has occurred because the firm financed a negative NPVproject. The cost of the project is given at $35 million. The NPVof the project is the cost of the new project plus the new marketvalue of the firm minus the current market value of the firm, or:

NPV = –$35,000,000 + [8,700,000($48.95) – 8,000,000($50)] = –$9,117,647

b. For the price to remain unchanged when the P/E ratio is constant,EPS must remain constant. The new net income must be the newnumber of shares outstanding times the current EPS, which gives:

NI1 = (8,700,000 shares)($2.13 per share) = $18,487,500

10. The total equity of the company is total assets minus totalliabilities, or:

Equity = $8,000,000 – 3,400,000Equity = $4,600,000

So, the current ROE of the company is:

ROE0 = NI0/TE0 = $900,000/$4,600,000 = .1957 or 19.57%

CHAPTER 15 B-514

The new net income will be the ROE times the new total equity, or:

NI1 = (ROE0)(TE1) = .1957($4,600,000 + 850,000) = $1,066,304

The company’s current earnings per share are:

EPS0 = NI0/Shares outstanding0 = $900,000/30,000 shares = $30.00

The number of shares the company will offer is the cost of theinvestment divided by the current share price, so:

Number of new shares = $850,000/$84 = 10,119

B-515 SOLUTIONS

The earnings per share after the stock offer will be:

EPS1 = $1,066,304/40,119 shares = $26.58

The current P/E ratio is:

(P/E)0 = $84/$26.58 = 2.800

Assuming the P/E remains constant, the new stock price will be:

P1 = 2.800($26.58) = $74.42

The current book value per share and the new book value per shareare:

BVPS0 = TE0/shares0 = $4,600,000/30,000 shares = $153.33 per share

BVPS1 = TE1/shares1 = ($4,600,000 + 850,000)/40,119 shares = $135.85per share

So the current and new market-to-book ratios are:

Market-to-book0 = $84/$153.33 = 0.5478

Market-to-book1 = $74.42/$135.85 = 0.5478

The NPV of the project is the cost of the project plus the new marketvalue of the firm minus the current market value of the firm, or:

NPV = –$850,000 + [$74.42(40,119) – $84(30,000)] = –$384,348

Accounting dilution takes place here because the market-to-book ratiois less than one. Market value dilution has occurred since the firmis investing in a negative NPV project.

11. Using the P/E ratio to find the necessary EPS after the stock issue,we get:

P1 = $84 = 2.800(EPS1) EPS1 = $30.00

The additional net income level must be the EPS times the new sharesoutstanding, so:

CHAPTER 15 B-516

NI = $30(10,119 shares) = $303,571

And the new ROE is:

ROE1 = $303,571/$850,000 = .3571 or 35.71%

B-517 SOLUTIONS

Next, we need to find the NPV of the project. The NPV of the projectis the cost of the project plus the new market value of the firmminus the current market value of the firm, or:

NPV = –$850,000 + [$84(40,119) – $84(30,000)] = $0

Accounting dilution still takes place, as BVPS still falls from$153.33 to $135.85, but no market dilution takes place because thefirm is investing in a zero NPV project.

12. The number of new shares is the amount raised divided by thesubscription price, so:

Number of new shares = $60,000,000/$PS

And the ex-rights number of shares (N) is equal to:

N = Old shares outstanding/New shares outstanding N = 19,000,000/($60,000,000/$PS) N = 0.03167PS

We know the equation for the ex-rights stock price is:

PX = [NPRO + PS]/(N + 1)

We can substitute in the numbers we are given, and then substitutethe two previous results. Doing so, and solving for the subscriptionprice, we get:

PX = $71 = [N($76) + $PS]/(N + 1) $71 = [$76(0.03167PS) + PS]/(0.03167PS + 1) $71 = $24.0667PS/(1 + 0.03167PS) PS = $27.48

13. Using PRO as the rights-on price, and PS as the subscription price, wecan express the price per share of the stock ex-rights as:

PX = [NPRO + PS]/(N + 1)

And the equation for the value of a right is:

Value of a right = PRO – PX

CHAPTER 15 B-518

Substituting the ex-rights price equation into the equation for thevalue of a right and rearranging, we get:

Value of a right = PRO – {[NPRO + PS]/(N + 1)} Value of a right = [(N + 1)PRO – NPRO – PS]/(N+1) Value of a right = [PRO – PS]/(N + 1)

B-519 SOLUTIONS

14. The net proceeds to the company on a per share basis is thesubscription price times one minus the underwriter spread, so:

Net proceeds to the company = $23(1 – .06) = $21.62 per share

So, to raise the required funds, the company must sell:

New shares offered = $5,600,000/$21.62 = 259,019

The number of rights needed per share is the current number of sharesoutstanding divided by the new shares offered, or:

Number of rights needed = 650,000 old shares/259,019 new sharesNumber of rights needed = 2.51 rights per share

The ex-rights stock price will be:

PX = [NPRO + PS]/(N + 1)PX = [2.51($50) + 23]/(2.51 + 1) = $42.31

So, the value of a right is:

Value of a right = $50 – 42.31 = $7.69

And your proceeds from selling your rights will be:

Proceeds from selling rights = 5,000($7.69) = $38,467.41

15. Using the equation for valuing a stock ex-rights, we find:

PX = [NPRO + PS]/(N + 1)PX = [4($60) + $35]/(4 + 1) = $55

The stock is correctly priced. Calculating the value of a right, wefind:

Value of a right = PRO – PX

Value of a right = $60 – 53 = $7

So, the rights are underpriced. You can create an immediate profit onthe ex-rights day if the stock is selling for $53 and the rights areselling for $3 by executing the following transactions:

CHAPTER 15 B-520

Buy 4 rights in the market for 4($3) = $12. Use these rights topurchase a new share at the subscription price of $35. Immediatelysell this share in the market for $53, creating an instant $6 profit.

CHAPTER 16FINANCIAL LEVERAGE AND CAPITAL STRUCTURE POLICY


1. Business risk is the equity risk arising from the nature of thefirm’s operating activity, and is directly related to the systematicrisk of the firm’s assets. Financial risk is the equity risk that isdue entirely to the firm’s chosen capital structure. As financialleverage, or the use of debt financing, increases, so does financialrisk and, hence, the overall risk of the equity. Thus, Firm B couldhave a higher cost of equity if it uses greater leverage.

2. No, it doesn’t follow. While it is true that the equity and debtcosts are rising, the key thing to remember is that the cost of debtis still less than the cost of equity. Since we are using more andmore debt, the WACC does not necessarily rise.

3. Because many relevant factors such as bankruptcy costs, taxasymmetries, and agency costs cannot easily be identified orquantified, it’s practically impossible to determine the precisedebt-equity ratio that maximizes the value of the firm. However, ifthe firm’s cost of new debt suddenly becomes much more expensive,it’s probably true that the firm is too highly leveraged.

4. The more capital intensive industries, such as airlines, cabletelevision, and electric utilities, tend to use greater financialleverage. Also, industries with less predictable future earnings,such as computers or drugs, tend to use less financial leverage. Suchindustries also have a higher concentration of growth and startupfirms. Overall, the general tendency is for firms with identifiable,tangible assets and relatively more predictable future earnings touse more debt financing. These are typically the firms with thegreatest need for external financing and the greatest likelihood ofbenefiting from the interest tax shelter.

CHAPTER 16 B-522

5. It’s called leverage (or “gearing” in the UK) because it magnifiesgains or losses.

6. Homemade leverage refers to the use of borrowing on the personallevel as opposed to the corporate level.

7. One answer is that the right to file for bankruptcy is a valuableasset, and the financial manager acts in shareholders’ best interestby managing this asset in ways that maximize its value. To the extentthat a bankruptcy filing prevents “a race to the courthouse steps,”it would seem to be a reasonable use of the process.

8. As in the previous question, it could be argued that using bankruptcylaws as a sword may simply be the best use of the asset. Creditorsare aware at the time a loan is made of the possibility ofbankruptcy, and the interest charged incorporates it.

B-523 SOLUTIONS

9. One side is that Continental was going to go bankrupt because itscosts made it uncompetitive. The bankruptcy filing enabledContinental to restructure and keep flying. The other side is thatContinental abused the bankruptcy code. Rather than renegotiate laboragreements, Continental simply abrogated them to the detriment of itsemployees. In this, and the last several, questions, an importantthing to keep in mind is that the bankruptcy code is a creation oflaw, not economics. A strong argument can always be made that makingthe best use of the bankruptcy code is no different from, forexample, minimizing taxes by making best use of the tax code. Indeed,a strong case can be made that it is the financial manager’s duty todo so. As the case of Continental illustrates, the code can bechanged if socially undesirable outcomes are a problem.

10. The basic goal is to minimize the value of non-marketed claims.Solutions to Questions and Problems


Basic

1. a. A table outlining the income statement for the three possiblestates of the economy is shown below. The EPS is the net incomedivided by the 5,000 shares outstanding. The last row shows thepercentage change in EPS the company will experience in arecession or an expansion economy.

Recession Normal ExpansionEBIT $14,000 $28,000 $36,400Interest

0 0 0

NI $14,000 $28,000 $36,400EPS $ 2.80 $ 5.60 $ 7.28%EPS –50 ––– +30

b. If the company undergoes the proposed recapitalization, it willrepurchase:

Share price = Equity / Shares outstandingShare price = $250,000/5,000

CHAPTER 16 B-524

Share price = $50

Shares repurchased = Debt issued / Share priceShares repurchased =$90,000/$50 Shares repurchased = 1,800

B-525 SOLUTIONS

The interest payment each year under all three scenarios will be:

Interest payment = $90,000(.07) = $6,300

The last row shows the percentage change in EPS the company willexperience in a recession or an expansion economy under theproposed recapitalization.


6,300 6,300 6,300

NI $7,700 $21,700 $30,100EPS $2.41 $ 6.78 $9.41%EPS –64.52 ––– +38.71

2. a. A table outlining the income statement with taxes for the threepossible states of the economy is shown below. The share price isstill $50, and there are still 5,000 shares outstanding. The lastrow shows the percentage change in EPS the company will experiencein a recession or an expansion economy.


0 0 0

Taxes 4,900 9,850 12,740NI $9,100 $18,200 $23,600EPS $1.82 $3.64 $4.73%EPS –50 ––– +30

b. A table outlining the income statement with taxes for the threepossible states of the economy and assuming the company undertakesthe proposed capitalization is shown below. The interest paymentand shares repurchased are the same as in part b of Problem 1.


6,300 6,300 6,300

Taxes 2,695 7,595 10,535NI $5,005 $14,105 $19,565EPS $1.56 $4.41 $6.11

CHAPTER 16 B-526

%EPS –64.52 ––– +38.71

Notice that the percentage change in EPS is the same both with andwithout taxes.

B-527 SOLUTIONS

3. a. Since the company has a market-to-book ratio of 1.0, the totalequity of the firm is equal to the market value of equity. Usingthe equation for ROE:

ROE = NI/$250,000

The ROE for each state of the economy under the current capitalstructure and no taxes is:

Recession Normal ExpansionROE .0560 .1120 .1456%ROE –50 ––– +30

The second row shows the percentage change in ROE from the normaleconomy.

b. If the company undertakes the proposed recapitalization, the newequity value will be:

Equity = $250,000 – 90,000 Equity = $160,000

So, the ROE for each state of the economy is:

ROE = NI/$160,000

Recession Normal ExpansionROE .0481 .1356 .1881%ROE –64.52 ––– +38.71

c. If there are corporate taxes and the company maintains its currentcapital structure, the ROE is:

ROE .0364 .0728 .0946%ROE –50 ––– +30

If the company undertakes the proposed recapitalization, and thereare corporate taxes, the ROE for each state of the economy is:

ROE .0313 .0882 .1223%ROE –64.52 ––– +38.71

CHAPTER 16 B-528

Notice that the percentage change in ROE is the same as thepercentage change in EPS. The percentage change in ROE is also thesame with or without taxes.

4. a. Under Plan I, the unlevered company, net income is the same asEBIT with no corporate tax. The EPS under this capitalization willbe:

EPS = $350,000/160,000 shares EPS = $2.19

B-529 SOLUTIONS

Under Plan II, the levered company, EBIT will be reduced by theinterest payment. The interest payment is the amount of debt timesthe interest rate, so:

NI = $500,000 – .08($2,800,000) NI = $126,000

And the EPS will be:

EPS = $126,000/80,000 shares EPS = $1.58

Plan I has the higher EPS when EBIT is $350,000.

b. Under Plan I, the net income is $500,000 and the EPS is:

EPS = $500,000/160,000 shares EPS = $3.13

Under Plan II, the net income is:

NI = $500,000 – .08($2,800,000) NI = $276,000

And the EPS is:

EPS = $276,000/80,000 shares EPS = $3.45

Plan II has the higher EPS when EBIT is $500,000.

c. To find the breakeven EBIT for two different capital structures,we simply set the equations for EPS equal to each other and solvefor EBIT. The breakeven EBIT is:

EBIT/160,000 = [EBIT – .08($2,800,000)]/80,000 EBIT = $448,000

5. We can find the price per share by dividing the amount of debt usedto repurchase shares by the number of shares repurchased. Doing so,we find the share price is:

Share price = $2,800,000/(160,000 – 80,000)

CHAPTER 16 B-530

Share price = $35.00 per share

The value of the company under the all-equity plan is:

V = $35.00(160,000 shares) = $5,600,000

And the value of the company under the levered plan is:

V = $35.00(80,000 shares) + $2,800,000 debt = $5,600,000

B-531 SOLUTIONS

6. a. The income statement for each capitalization plan is:

I II All-equityEBIT $39,000 $39,000 $39,000Interest

16,000 24,000 0

NI $23,000 $15,000 $39,000EPS $ 3.29 $ 3.00 $ 3.55

The all-equity plan; Plan II has the lowest EPS.

b. The breakeven level of EBIT occurs when the capitalization plansresult in the same EPS. The EPS is calculated as:

EPS = (EBIT – RDD)/Shares outstanding

This equation calculates the interest payment (RDD) and subtractsit from the EBIT, which results in the net income. Dividing by theshares outstanding gives us the EPS. For the all-equity capitalstructure, the interest term is zero. To find the breakeven EBITfor two different capital structures, we simply set the equationsequal to each other and solve for EBIT. The breakeven EBIT betweenthe all-equity capital structure and Plan I is:

EBIT/11,000 = [EBIT – .10($160,000)]/7,000 EBIT = $44,000

And the breakeven EBIT between the all-equity capital structureand Plan II is:

EBIT/11,000 = [EBIT – .10($240,000)]/5,000 EBIT = $44,000

The break-even levels of EBIT are the same because of M&MProposition I.

c. Setting the equations for EPS from Plan I and Plan II equal toeach other and solving for EBIT, we get:

[EBIT – .10($160,000)]/7,000 = [EBIT – .10($240,000)]/5,000 EBIT = $44,000

CHAPTER 16 B-532

This break-even level of EBIT is the same as in part b againbecause of M&M Proposition I.

B-533 SOLUTIONS

d. The income statement for each capitalization plan with corporateincome taxes is:

III All-equity

EBIT $39,000 $39,000 $39,000Interest

16,000 24,000 0

Taxes 9,200 6,000 15,600NI $ 13,800 $ 9,000 $ 23,400EPS $ 1.97 $ 1.80 $ 2.13

The all-equity plan still has the highest EPS; Plan II still hasthe lowest EPS.

We can calculate the EPS as:

EPS = [(EBIT – RDD)(1 – tC)]/Shares outstanding

This is similar to the equation we used before, except now we needto account for taxes. Again, the interest expense term is zero inthe all-equity capital structure. So, the breakeven EBIT betweenthe all-equity plan and Plan I is:

EBIT(1 – .40)/11,000 = [EBIT – .10($160,000)](1 – .40)/7,000 EBIT = $44,000

The breakeven EBIT between the all-equity plan and Plan II is:

EBIT(1 – .40)/11,000 = [EBIT – .10($240,000)](1 – .40)/5,000 EBIT = $44,000

And the breakeven between Plan I and Plan II is:

[EBIT – .10($160,000)](1 – .40)/7,000 = [EBIT – .10($240,000)](1 –.40)/5,000 EBIT = $44,000

The break-even levels of EBIT do not change because the additionof taxes reduces the income of all three plans by the samepercentage; therefore, they do not change relative to one another.

CHAPTER 16 B-534

7. To find the value per share of the stock under each capitalizationplan, we can calculate the price as the value of shares repurchaseddivided by the number of shares repurchased. So, under Plan I, thevalue per share is:

P = $160,000/(11,000 – 7,000 shares) P = $40 per share

And under Plan II, the value per share is:

P = $240,000/(11,000 – 5,000 shares) P = $40 per share

This shows that when there are no corporate taxes, the stockholderdoes not care about the capital structure decision of the firm. Thisis M&M Proposition I without taxes.

B-535 SOLUTIONS

8. a. The earnings per share are:

EPS = $32,000/8,000 shares EPS = $4.00

So, the cash flow for the company is:

Cash flow = $4.00(100 shares) Cash flow = $400

b. To determine the cash flow to the shareholder, we need todetermine the EPS of the firm under the proposed capitalstructure. The market value of the firm is:

V = $55(8,000) V = $440,000

Under the proposed capital structure, the firm will raise new debtin the amount of:

D = 0.35($440,000) D = $154,000

in debt. This means the number of shares repurchased will be:

Shares repurchased = $154,000/$55 Shares repurchased = 2,800

Under the new capital structure, the company will have to make aninterest payment on the new debt. The net income with the interestpayment will be:

NI = $32,000 – .08($154,000) NI = $19,680

This means the EPS under the new capital structure will be:

EPS = $19,680/(8,000 – 2,800) shares EPS = $3.7846

Since all earnings are paid as dividends, the shareholder willreceive:

CHAPTER 16 B-536

Shareholder cash flow = $3.7846(100 shares) Shareholder cash flow = $378.46

c. To replicate the proposed capital structure, the shareholdershould sell 35 percent of their shares, or 35 shares, and lend theproceeds at 8 percent. The shareholder will have an interest cashflow of:

Interest cash flow = 35($55)(.08) Interest cash flow = $154

B-537 SOLUTIONS

The shareholder will receive dividend payments on the remaining 65shares, so the dividends received will be:

Dividends received = $3.7846(65 shares) Dividends received = $246

The total cash flow for the shareholder under these assumptionswill be:

Total cash flow = $154 + 246 Total cash flow = $400

This is the same cash flow we calculated in part a.

d. The capital structure is irrelevant because shareholders cancreate their own leverage or unlever the stock to create thepayoff they desire, regardless of the capital structure the firmactually chooses.

9. a. The rate of return earned will be the dividend yield. The companyhas debt, so it must make an interest payment. The net income forthe company is:

NI = $80,000 – .08($300,000) NI = $56,000

The investor will receive dividends in proportion to thepercentage of the company’s share they own. The total dividendsreceived by the shareholder will be:

Dividends received = $56,000($30,000/$300,000) Dividends received = $5,600

So the return the shareholder expects is:

R = $5,600/$30,000 R = .1867 or 18.67%

b. To generate exactly the same cash flows in the other company, theshareholder needs to match the capital structure of ABC. Theshareholder should sell all shares in XYZ. This will net $30,000.The shareholder should then borrow $30,000. This will create aninterest cash flow of:

CHAPTER 16 B-538

Interest cash flow = .08(–$30,000) Interest cash flow = –$2,400

The investor should then use the proceeds of the stock sale andthe loan to buy shares in ABC. The investor will receive dividendsin proportion to the percentage of the company’s share they own.The total dividends received by the shareholder will be:

Dividends received = $80,000($60,000/$600,000) Dividends received = $8,000

B-539 SOLUTIONS

The total cash flow for the shareholder will be:

Total cash flow = $8,000 – 2,400Total cash flow = $5,600

The shareholders return in this case will be:

R = $5,600/$30,000 R = .1867 or 18.67%

c. ABC is an all equity company, so:

RE = RA = $80,000/$600,000 RE = .1333 or 13.33%

To find the cost of equity for XYZ we need to use M&M PropositionII, so:

RE = RA + (RA – RD)(D/E)(1 – tC)RE = .1333 + (.1333 – .08)(1)(1) RE = .1867 or 18.67%

d. To find the WACC for each company we need to use the WACCequation:

WACC = (E/V)RE + (D/V)RD(1 – tC)

So, for ABC, the WACC is:

WACC = (1)(.1333) + (0)(.08) WACC = .1333 or 13.33%

And for XYZ, the WACC is:

WACC = (1/2)(.1867) + (1/2)(.08) WACC = .1333 or 13.33%

When there are no corporate taxes, the cost of capital for thefirm is unaffected by the capital structure; this is M&MProposition II without taxes.

10. With no taxes, the value of an unlevered firm is the interest ratedivided by the unlevered cost of equity, so:

CHAPTER 16 B-540

V = EBIT/WACC $23,000,000 = EBIT/.09EBIT = .09($23,000,000) EBIT = $2,070,000

B-541 SOLUTIONS

11. If there are corporate taxes, the value of an unlevered firm is:

VU = EBIT(1 – tC)/RU

Using this relationship, we can find EBIT as:

$23,000,000 = EBIT(1 – .35)/.09EBIT = $3,184,615.38

The WACC remains at 9 percent. Due to taxes, EBIT for an all-equity firm would have to be higher for the firm to still be worth$23 million.

12. a. With the information provided, we can use the equation forcalculating WACC to find the cost of equity. The equation for WACCis:


The company has a debt-equity ratio of 1.5, which implies theweight of debt is 1.5/2.5, and the weight of equity is 1/2.5, so

WACC = .10 = (1/2.5)RE + (1.5/2.5)(.07)(1 – .35)RE = .1818 or 18.18%

b. To find the unlevered cost of equity we need to use M&MProposition II with taxes, so:

RE = RU + (RU – RD)(D/E)(1 – tC) .1818 = RU + (RU – .07)(1.5)(1 – .35) RU = .1266 or 12.66%

c. To find the cost of equity under different capital structures, wecan again use M&M Proposition II with taxes. With a debt-equityratio of 2, the cost of equity is:

RE = RU + (RU – RD)(D/E)(1 – tC) RE = .1266 + (.1266 – .07)(2)(1 – .35) RE = .2001 or 20.01%

With a debt-equity ratio of 1.0, the cost of equity is:

RE = .1266 + (.1266 – .07)(1)(1 – .35)

CHAPTER 16 B-542

RE = .1634 or 16.34%

And with a debt-equity ratio of 0, the cost of equity is:

RE = .1266 + (.1266 – .07)(0)(1 – .35) RE = RU = .1266 or 12.66%

B-543 SOLUTIONS

13. a. For an all-equity financed company:

WACC = RU = RE = .11 or 11%

b. To find the cost of equity for the company with leverage we needto use M&M Proposition II with taxes, so:

RE = RU + (RU – RD)(D/E)(1 – tC) RE = .11 + (.11 – .082)(.25/.75)(.65) RE = .1161 or 11.61%

c. Using M&M Proposition II with taxes again, we get:

RE = RU + (RU – RD)(D/E)(1 – tC) RE = .11 + (.11 – .082)(.50/.50)(1 – .35) RE = .1282 or 12.82%

d. The WACC with 25 percent debt is:

WACC = (E/V)RE + (D/V)RD(1 – tC) WACC = .75(.1161) + .25(.082)(1 – .35) WACC = .1004 or 10.04%

And the WACC with 50 percent debt is:

WACC = (E/V)RE + (D/V)RD(1 – tC) WACC = .50(.1282) + .50(.082)(1 – .35) WACC = .0908 or 9.08%

14. a. The value of the unlevered firm is:


VU = $92,000(1 – .35)/.15 VU = $398,666.67

b. The value of the levered firm is:

VU = VU + tCD VU = $398,666.67 + .35($60,000) VU = $419,666.67

CHAPTER 16 B-544

15. We can find the cost of equity using M&M Proposition II with taxes.Doing so, we find:

RE = RU + (RU – RD)(D/E)(1 – t) RE = .15 + (.15 – .09)($60,000/$398,667)(1 – .35) RE = .1565 or 15.65%

Using this cost of equity, the WACC for the firm afterrecapitalization is:

WACC = (E/V)RE + (D/V)RD(1 – tC) WACC = .1565($398,667/$419,667) + .09(1 – .35)($60,000/$419,667) WACC = .1425 or 14.25%

When there are corporate taxes, the overall cost of capital for thefirm declines the more highly leveraged is the firm’s capitalstructure. This is M&M Proposition I with taxes.

Intermediate

16. To find the value of the levered firm we first need to find the valueof an unlevered firm. So, the value of the unlevered firm is:

VU = EBIT(1 – tC)/RU VU = ($64,000)(1 – .35)/.15 VU = $277,333.33

Now we can find the value of the levered firm as:

VL = VU + tCDVL = $277,333.33 + .35($95,000) VL = $310,583.33

Applying M&M Proposition I with taxes, the firm has increased itsvalue by issuing debt. As long as M&M Proposition I holds, that is,there are no bankruptcy costs and so forth, then the company shouldcontinue to increase its debt/equity ratio to maximize the value ofthe firm.

17. With no debt, we are finding the value of an unlevered firm, so:


VU = $14,000(1 – .35)/.16

B-545 SOLUTIONS

VU = $56,875

With debt, we simply need to use the equation for the value of alevered firm. With 50 percent debt, one-half of the firm value isdebt, so the value of the levered firm is:

VL = VU + tC(D/V)VU

VL = $56,875 + .35(.50)($56,875) VL = $66,828.13

CHAPTER 16 B-546

And with 100 percent debt, the value of the firm is:

VL = VU + tC(D/V)VU

VL = $56,875 + .35(1.0)($56,875) VL = $76,781.25

18. a. To purchase 5 percent of Knight’s equity, the investor would need:

Knight investment = .05($1,632,000) = $81,600

And to purchase 5 percent of Veblen without borrowing would require:

Veblen investment = .05($2,500,000) = $125,000

In order to compare dollar returns, the initial net cost of bothpositions should be the same. Therefore, the investor will need toborrow the difference between the two amounts, or:

Amount to borrow = $125,000 – 81,600 = $43,400

An investor who owns 5 percent of Knight’s equity will be entitledto 5 percent of the firm’s earnings available to common stockholders at the end of each year. While Knight’s expected operatingincome is $400,000, it must pay $72,000 to debt holders beforedistributing any of its earnings to stockholders. So, the amountavailable to this shareholder will be:

Cash flow from Knight to shareholder = .05($400,000 – 72,000) = $16,400

Veblen will distribute all of its earnings to shareholders, so theshareholder will receive:

Cash flow from Veblen to shareholder = .05($400,000) = $20,000

However, to have the same initial cost, the investor has borrowed$43,400 to invest in Veblen, so interest must be paid on theborrowings. The net cash flow from the investment in Veblen willbe:

Net cash flow from Veblen investment = $20,000 – .06($43,400) = $17,396

B-547 SOLUTIONS

For the same initial cost, the investment in Veblen produces a higher dollar return.

b. Both of the two strategies have the same initial cost. Since thedollar return to the investment in Veblen is higher, all investorswill choose to invest in Veblen over Knight. The process ofinvestors purchasing Veblen’s equity rather than Knight’s willcause the market value of Veblen’s equity to rise and the marketvalue of Knight’s equity to fall. Any differences in the dollarreturns to the two strategies will be eliminated, and the processwill cease when the total market values of the two firms areequal.

CHAPTER 16 B-548

Challenge

19. M&M Proposition II states:

RE = RU + (RU – RD)(D/E)(1 – tC)

And the equation for WACC is:


Substituting the M&M Proposition II equation into the equation forWACC, we get:

WACC = (E/V)[RU + (RU – RD)(D/E)(1 – tC)] + (D/V)RD(1 – tC)

Rearranging and reducing the equation, we get:

WACC = RU[(E/V) + (E/V)(D/E)(1 – tC)] + RD(1 – tC)[(D/V) – (E/V)(D/E)]WACC = RU[(E/V) + (D/V)(1 – tC)] WACC = RU[{(E+D)/V} – tC(D/V)] WACC = RU[1 – tC(D/V)]

20. The return on equity is net income divided by equity. Net income canbe expressed as:

NI = (EBIT – RDD)(1 – tC)

So, ROE is:

RE = (EBIT – RDD)(1 – tC)/E

Now we can rearrange and substitute as follows to arrive at M&MProposition II with taxes:

RE = [EBIT(1 – tC)/E] – [RD(D/E)(1 – tC)]RE = RUVU/E – [RD(D/E)(1 – tC)] RE = RU(VL – tCD)/E – [RD(D/E)(1 – tC)]RE = RU(E + D – tCD)/E – [RD(D/E)(1 – tC)] RE = RU + (RU – RD)(D/E)(1 – tC)

B-549 SOLUTIONS

21. M&M Proposition II, with no taxes is:

RE = RU + (RU – Rf)(D/E)

Note that we use the risk-free rate as the return on debt. This is animportant assumption of M&M Proposition II. The CAPM to calculate thecost of equity is expressed as:

RE = E(RM – Rf) + Rf

We can rewrite the CAPM to express the return on an unlevered companyas:

RA = A(RM – Rf) + Rf

We can now substitute the CAPM for an unlevered company into M&MProposition II. Doing so and rearranging the terms we get:

RE = A(RM – Rf) + Rf + [A(RM – Rf) + Rf – Rf](D/E)RE = A(RM – Rf) + Rf + [A(RM – Rf)](D/E)RE = (1 + D/E)A(RM – Rf) + Rf

Now we set this equation equal to the CAPM equation to calculate thecost of equity and reduce:

E(RM – Rf) + Rf = (1 + D/E)A(RM – Rf) + Rf

E(RM – Rf) = (1 + D/E)A(RM – Rf) E = A(1 + D/E)

22. Using the equation we derived in Problem 21:

E = A(1 + D/E)

The equity beta for the respective asset betas is:

Debt-equityratio

Equity beta

0 1(1 + 0) = 11 1(1 + 1) = 25 1(1 + 5) = 620 1(1 + 20) =

21

CHAPTER 16 B-550

The equity risk to the shareholder is composed of both business andfinancial risk. Even if the assets of the firm are not very risky,the risk to the shareholder can still be large if the financialleverage is high. These higher levels of risk will be reflected inthe shareholder’s required rate of return RE, which will increase withhigher debt/equity ratios.

CHAPTER 17DIVIDENDS AND DIVIDEND POLICY


1. Dividend policy deals with the timing of dividend payments, not theamounts ultimately paid. Dividend policy is irrelevant when thetiming of dividend payments doesn’t affect the present value of allfuture dividends.

2. A stock repurchase reduces equity while leaving debt unchanged. Thedebt ratio rises. A firm could, if desired, use excess cash to reducedebt instead. This is a capital structure decision.

3. Friday, December 29 is the ex-dividend day. Remember not to countJanuary 1 because it is a holiday, and the exchanges are closed.Anyone who buys the stock before December 29 is entitled to thedividend, assuming they do not sell it again before December 29.

4. No, because the money could be better invested in stocks that paydividends in cash which benefit the fundholders directly.

5. The change in price is due to the change in dividends, not due to thechange in dividend policy. Dividend policy can still be irrelevantwithout a contradiction.

6. The stock price dropped because of an expected drop in futuredividends. Since the stock price is the present value of all futuredividend payments, if the expected future dividend payments decrease,then the stock price will decline.

7. The plan will probably have little effect on shareholder wealth. Theshareholders can reinvest on their own, and the shareholders must paythe taxes on the dividends either way. However, the shareholders whotake the option may benefit at the expense of the ones who don’t(because of the discount). Also as a result of the plan, the firmwill be able to raise equity by paying a 10% flotation cost (thediscount), which may be a smaller discount than the market flotationcosts of a new issue for some companies.

CHAPTER 17 B-552

8. If these firms just went public, they probably did so because theywere growing and needed the additional capital. Growth firmstypically pay very small cash dividends, if they pay a dividend atall. This is because they have numerous projects available, and theyreinvest the earnings in the firm instead of paying cash dividends.

9. The stock price drop on the ex-dividend date should be lower. Withtaxes, stock prices should drop by the amount of the dividend, lessthe taxes investors must pay on the dividends. A lower tax ratelowers the investors’ tax liability.

10. With a high tax on dividends and a low tax on capital gains,investors, in general, will prefer capital gains. If the dividend taxrate declines, the attractiveness of dividends increases.

B-553 SOLUTIONS



Basic

1. The aftertax dividend is the pretax dividend times one minus the taxrate, so:

Aftertax dividend = $4.60(1 – .15) = $3.91

The stock price should drop by the aftertax dividend amount, or:

Ex-dividend price = $80.37 – 3.91 = $76.46

2. a. The shares outstanding increases by 10 percent, so:

New shares outstanding = 30,000(1.10) = 33,000

New shares issued = 3,000

Since the par value of the new shares is $1, the capital surplusper share is $29. The total capital surplus is therefore:

Capital surplus on new shares = 3,000($29) = $87,000

Common stock ($1 parvalue)

$33,000

Capital surplus 372,000

Retained earnings 559,180$964,180

b. The shares outstanding increases by 25 percent, so:


New shares issued = 7,500

CHAPTER 17 B-554

Since the par value of the new shares is $1, the capital surplusper share is $29. The total capital surplus is therefore:

Capital surplus on new shares = 7,500($29) = $217,500


$ 37,500

Capital surplus 502,500

Retained earnings 424,180$964,180

B-555 SOLUTIONS

3. a. To find the new shares outstanding, we multiply the current sharesoutstanding times the ratio of new shares to old shares, so:

New shares outstanding = 30,000(4/1) = 120,000

The equity accounts are unchanged except the par value of thestock is changed by the ratio of new shares to old shares, so thenew par value is:

New par value = $1(1/4) = $0.25 per share.

b. To find the new shares outstanding, we multiply the current sharesoutstanding times the ratio of new shares to old shares, so:

New shares outstanding = 30,000(1/5) = 6,000.

The equity accounts are unchanged except the par value of thestock is changed by the ratio of new shares to old shares, so thenew par value is:


4. To find the new stock price, we multiply the current stock price bythe ratio of old shares to new shares, so:

a. $90(3/5) = $54.00

b. $90(1/1.15) = $78.26

c. $90(1/1.425) = $63.16

d. $90(7/4) = $157.50

e. To find the new shares outstanding, we multiply the current sharesoutstanding times the ratio of new shares to old shares, so:

a: 350,000(5/3) = 583,333

b: 350,000(1.15) = 402,500

c: 350,000(1.425) = 498,750

d: 350,000(4/7) = 200,000

CHAPTER 17 B-556

5. The stock price is the total market value of equity divided by theshares outstanding, so:

P0 = $308,500 equity/8,000 shares = $38.56 per share

Ignoring tax effects, the stock price will drop by the amount of thedividend, so:

PX = $38.56 – 1.30 = $37.26

B-557 SOLUTIONS

The total dividends paid will be:

$1.30 per share(8,000 shares) = $10,400

The equity and cash accounts will both decline by $10,400.

6. Repurchasing the shares will reduce cash and shareholders’ equity by$10,400. The shares repurchased will be the total purchase amountdivided by the stock price, so:

Shares bought = $10,400/$38.56 = 269.69

And the new shares outstanding will be:

New shares outstanding = 8,000 – 269.69 = 7,730.31

After repurchase, the new stock price is:

Share price = ($308,500 – 10,400)/7,730.31 shares = $38.56

The repurchase is effectively the same as the cash dividend becauseyou either hold a share worth $38.56, or a share worth $37.26 and$1.30 in cash. Therefore, you participate in the repurchase accordingto the dividend payout percentage; you are unaffected.

7. The stock price is the total market value of equity divided by theshares outstanding, so:

P0 = $471,000 equity/12,000 shares = $39.25 per share

The shares outstanding will increase by 25 percent, so:


The new stock price is the market value of equity divided by the newshares outstanding, so:

PX = $471,000/15,000 shares = $31.40

8. With a stock dividend, the shares outstanding will increase by oneplus the dividend amount, so:


CHAPTER 17 B-558

The capital surplus is the capital paid in excess of par value, whichis $1, so:

Capital surplus for new shares = 60,900($34) = $2,070,600

The new capital surplus will be the old capital surplus plus theadditional capital surplus for the new shares, so:

Capital surplus = $1,340,000 + 2,070,600 = $3,410,600

B-559 SOLUTIONS

The new equity portion of the balance sheet will look like this:


$466,900

Capital surplus 3,410,600Retained earnings 1,295,5

00$5,173,00

0

9. The only equity account that will be affected is the par value of thestock. The par value will change by the ratio of old shares to newshares, so:


The total dividends paid this year will be the dividend amount timesthe number of shares outstanding. The company had 406,000 sharesoutstanding before the split. We must remember to adjust the sharesoutstanding for the stock split, so:

Total dividends paid this year = $0.85(406,000 shares)(4/1 split) =$1,380,400

The dividends increased by 10 percent, so the total dividends paidlast year were:

Last year’s dividends = $1,380,400/1.10 = $1,254,909.09

And to find the dividends per share, we simply divide this amount bythe shares outstanding last year. Doing so, we get:

Dividends per share last year = $1,254,909.09/406,000 shares = $3.09

Intermediate

10. The price of the stock today is the PV of the dividends, so:

P0 = $2.30/1.15 + $53/1.152 = $42.08

To find the equal two year dividends with the same present value asthe price of the stock, we set up the following equation and solve

CHAPTER 17 B-560

for the dividend (Note: The dividend is a two year annuity, so wecould solve with the annuity factor as well):

$42.08 = D/1.15 + D/1.152 D = $25.88

We now know the cash flow per share we want each of the next twoyears. We can find the price of stock in one year, which will be:

P1 = $53/1.15 = $46.09

Since you own 1,000 shares, in one year you want:

Cash flow in Year one = 1,000($25.88) = $25,881.40

B-561 SOLUTIONS

But you’ll only get:

Dividends received in one year = 1,000($2.30) = $2,300

Thus, in one year you will need to sell additional shares in order toincrease your cash flow. The number of shares to sell in year one is:

Shares to sell at time one = ($25,881.40 – 2,300)/$46.09 = 511.67shares

At Year 2, you cash flow will be the dividend payment times thenumber of shares you still own, so the Year 2 cash flow is:

Year 2 cash flow = $53(1,000 – 511.67) = $25,881.40

11. If you only want $750 in Year 1, you will buy:

($2,300 – 750)/$46.09 = 33.63 shares

at time 1. Your dividend payment in Year 2 will be:

Year 2 dividend = (1,000 + 33.63)($53) = $54,782.50

Note, the present value of each cash flow stream is the same. Belowwe show this by finding the present values as:

PV = $750/1.15 + $54,782.50/1.152 = $42,075.61

PV = 1,000($2.30)/1.15 + 1,000($53)/1.152 = $42,075.61

12. a. If the company makes a dividend payment, we can calculate thewealth of a shareholder as:

Dividend per share = $9,000/1,000 shares = $9.00

The stock price after the dividend payment will be:

PX = $64 – 9 = $55 per share

The shareholder will have a stock worth $55 and a $9 dividend fora total wealth of $64. If the company makes a repurchase, thecompany will repurchase:

CHAPTER 17 B-562

Shares repurchased = $9,000/$64 = 140.63 shares

If the shareholder lets their shares be repurchased, they willhave $64 in cash. If the shareholder keeps their shares, they’restill worth $64.

B-563 SOLUTIONS

b. If the company pays dividends, the current EPS is $1.30, and theP/E ratio is:

P/E = $55/$1.30 = 42.31

If the company repurchases stock, the number of shares willdecrease. The total net income is the EPS times the current numberof shares outstanding. Dividing net income by the new number ofshares outstanding, we find the EPS under the repurchase is:

EPS = $1.30(1,000)/(1,000 140.63) = $1.51

The stock price will remain at $64 per share, so the P/E ratio is:

P/E = $64/$1.51 = 42.31

c. A share repurchase would seem to be the preferred course of action.Only those shareholders who wish to sell will do so, giving theshareholder a tax timing option that he or she doesn’t get with adividend payment.

Challenge

13. Assuming no capital gains tax, the aftertax return for the GordonCompany is the capital gains growth rate, plus the dividend yieldtimes one minus the tax rate. Using the constant growth dividendmodel, we get:

Aftertax return = g + D(1 – t) = .15

Solving for g, we get:

.15 = g + .05(1 – .35)g = .1175

The equivalent pretax return for Gordon Company, which pays nodividend, is:

Pretax return = g + D = .1175 + .05 = .1675 or 16.75%

14. Using the equation for the decline in the stock price ex-dividendfor each of the tax rate policies, we get:

CHAPTER 17 B-564

(P0 – PX)/D = (1 – TP)/(1 – TG)

a. P0 – PX = D(1 – 0)/(1 – 0)P0 – PX = D

b. P0 – PX = D(1 – .15)/(1 – 0)P0 – PX = .85D

c. P0 – PX = D(1 – .15)/(1 – .30)P0 – PX = 1.2143D

B-565 SOLUTIONS

d. With this tax policy, we simply need to multiply the personal taxrate times one minus the dividend exemption percentage, so:

P0 – PX = D[1 – (.35)(.30)]/(1 – .35) P0 – PX = 1.377D

e. Since different investors have widely varying tax rates onordinary income and capital gains, dividend payments havedifferent after-tax implications for different investors. Thisdifferential taxation among investors is one aspect of what wehave called the clientele effect.

15. Since the $2,000,000 cash is after corporate tax, the full amountwill be invested. So, the value of each alternative is:

Alternative 1 : The firm invests in T-bills or in preferred stock, and then pays out

as special dividend in 3 years

If the firm invests in T-Bills:

If the firm invests in T-bills, the aftertax yield of the T-bills will be:

Aftertax corporate yield = .05(1 – .35)Aftertax corporate yield = .0325 or 3.25%

So, the future value of the corporate investment in T-bills will be:

FV of investment in T-bills = $2,000,000(1 + .0325)3

FV of investment in T-bills = $2,201,406.16

Since the future value will be paid to shareholders as a dividend, the aftertax cash flow will be:

Aftertax cash flow to shareholders = $2,201,406.16(1 – .15)Aftertax cash flow to shareholders = $1,871,195.23

If the firm invests in preferred stock:

If the firm invests in preferred stock, the assumption would be thatthe dividends received will be reinvested in the same preferredstock. The preferred stock will pay a dividend of:

CHAPTER 17 B-566

Preferred dividend = .08($2,000,000)Preferred dividend = $160,000

Since 70 percent of the dividends are excluded from tax:

Taxable preferred dividends = (1 – .70)($220,000)Taxable preferred dividends = $48,000

And the taxes the company must pay on the preferred dividends will be:

Taxes on preferred dividends = .35($48,000)Taxes on preferred dividends = $16,800

So, the aftertax dividend for the corporation will be:

Aftertax corporate dividend = $160,000 – 16,800Aftertax corporate dividend = $143,200

This means the aftertax corporate dividend yield is:

Aftertax corporate dividend yield = $143,200 / $2,000,000Aftertax corporate dividend yield = .0716 or 7.16%

The future value of the company’s investment in preferred stock will be:

FV of investment in preferred stock = $2,000,000(1 + .0716)3

FV of investment in preferred stock = $2,461,093.48

Since the future value will be paid to shareholders as a dividend, the aftertax cash flow will be:

Aftertax cash flow to shareholders = $2,461,093.48(1 – .15)Aftertax cash flow to shareholders = $2,091,926.46

Alternative 2:

The firm pays out dividend now, and individuals invest on their own.The aftertax cash received by shareholders now will be:

Aftertax cash received today = $2,000,000(1 – .15)

B-567 SOLUTIONS

Aftertax cash received today = $1,700,000

The individuals invest in Treasury bills:

If the shareholders invest the current aftertax dividends in Treasurybills, the aftertax individual yield will be:

Aftertax individual yield on T-bills = .05(1 – .31)Aftertax individual yield on T-bills = .0345 or 3.45%

So, the future value of the individual investment in Treasury bills will be:

FV of investment in T-bills = $1,700,000(1 + .0345)3

FV of investment in T-bills = $1,882,090.08

The individuals invest in preferred stock:

If the individual invests in preferred stock, the assumption would bethat the dividends received will be reinvested in the same preferredstock. The preferred stock will pay a dividend of:

Preferred dividend = .08($1,700,000)Preferred dividend = $136,000

CHAPTER 17 B-568

And the taxes on the preferred dividends will be:

Taxes on preferred dividends = .31($136,000)Taxes on preferred dividends = $42,160

So, the aftertax preferred dividend will be:

Aftertax preferred dividend = $136,000 – 42,160Aftertax preferred dividend = $93,840

This means the aftertax individual dividend yield is:

Aftertax corporate dividend yield = $93,840 / $1,700,000Aftertax corporate dividend yield = .0552 or 5.52%

The future value of the individual investment in preferred stock willbe:

FV of investment in preferred stock = $1,700,000(1 + .0552)3

FV of investment in preferred stock = $1,997,345.84

The aftertax cash flow for the shareholders is maximized when thefirm invests the cash in the preferred stocks and pays a specialdividend later.

16. a. Let x be the ordinary income tax rate. The individual receives anafter-tax dividend of:

Aftertax dividend = $1,000(1 – x)

which she invests in Treasury bonds. The Treasury bond willgenerate aftertax cash flows to the investor of:

Aftertax cash flow from Treasury bonds = $1,000(1 – x)[1 + .06(1 –x)]

If the firm invests the money, its proceeds are:

Firm proceeds = $1,000[1 + .06(1 – .35)]

And the proceeds to the investor when the firm pays a dividend will be:

B-569 SOLUTIONS

Proceeds if firm invests first = (1 – x){$1,000[1 + .06(1 – .35)]}

To be indifferent, the investor’s proceeds must be the samewhether she invests the aftertax dividend or receives the proceedsfrom the firm’s investment and pays taxes on that amount. To findthe rate at which the investor would be indifferent, we can setthe two equations equal, and solve for x. Doing so, we find:

$1,000(1 – x)[1 + .06(1 – x)] = (1 – x){$1,000[1 + .06(1 – .35)]}1 + .06(1 – x) = 1 + .06(1 – .35)x = .35 or 35%

Note that this argument does not depend upon the length of time the investment is held.

CHAPTER 17 B-570

b. Yes, this is a reasonable answer. She is only indifferent if theaftertax proceeds from the $1,000 investment in identicalsecurities are identical. That occurs only when the tax rates areidentical.

c. Since both investors will receive the same pre-tax return, youwould expect the same answer as in part a. Yet, because Carlsonenjoys a tax benefit from investing in stock (70 percent of incomefrom stock is exempt from corporate taxes), the tax rate onordinary income which induces indifference, is much lower. Again,set the two equations equal and solve for x:

$1,000(1 – x)[1 + .09(1 – x)] = (1 – x)($1,000{1 + .09[.70 + (1– .70)(1 – .35)]})1 + .09(1 – x) = 1 + .09[.70 + (1 – .70)(1 – .35)]x = .1050 or 10.50%

d. It is a compelling argument, but there are legal constraints,which deter firms from investing large sums in stock of othercompanies.

CHAPTER 18SHORT-TERM FINANCE AND PLANNING


1. These are firms with relatively long inventory periods and/orrelatively long receivables periods. Thus, such firms tend to keepinventory on hand, and they allow customers to purchase on credit andtake a relatively long time to pay.

2. These are firms that have a relatively long time between the timepurchased inventory is paid for and the time that inventory is soldand payment received. Thus, these are firms that have relativelyshort payables periods and/or relatively long receivable cycles.

3. a. Use: The cash balance declined by $200 to pay the dividend.

b. Source: The cash balance increased by $500, assuming the goodsbought on payables credit were sold for cash.

c. Use: The cash balance declined by $900 to pay for the fixedassets.

d. Use: The cash balance declined by $625 to pay for the higherlevel of inventory.

e. Use: The cash balance declined by $1,200 to pay for theredemption of debt.

4. Carrying costs will decrease because they are not holding goods ininventory. Shortage costs will probably increase depending on howclose the suppliers are and how well they can estimate need. Theoperating cycle will decrease because the inventory period isdecreased.

5. Since the cash cycle equals the operating cycle minus the accountspayable period, it is not possible for the cash cycle to be longerthan the operating cycle if the accounts payable period is positive.Moreover, it is unlikely that the accounts payable period would ever

CHAPTER 18 B-572

be negative since that implies the firm pays its bills before theyare incurred.

6. It lengthened its payables period, thereby shortening its cash cycle.It will have no effect on the operating cycle.

7. The supplier’s receivables period will increase, thereby increasingtheir operating and cash cycles.

8. It is sometimes argued that large firms take advantage of smallerfirms by threatening to take their business elsewhere. However,considering a move to another supplier to get better terms is thenature of competitive free enterprise.

9. They would like to! The payables period is a subject of muchnegotiation, and it is one aspect of the price a firm pays itssuppliers. A firm will generally negotiate the best possiblecombination of payables period and price. Typically, suppliersprovide strong financial incentives for rapid payment. This issue isdiscussed in detail in a later chapter on credit policy.

B-573 SOLUTIONS

10. BlueSky will need less financing because it is essentially borrowingmore from its suppliers. Among other things, BlueSky will likely needless short-term borrowing from other sources, so it will save oninterest expense.



Basic

1. a. No change. A dividend paid for by the sale of debt will not changecash since the cash raised from the debt offer goes immediately toshareholders.

b. No change. The real estate is paid for by the cash raised from thedebt, so this will not change the cash balance.

c. No change. Inventory and accounts payable will increase, butneither will impact the cash account.

d. Decrease. The short-term bank loan is repaid with cash, which willreduce the cash balance.

e. Decrease. The payment of taxes is a cash transaction.

f. Decrease. The preferred stock will be repurchased with cash.

g. No change. Accounts receivable will increase, but cash will notincrease until the sales are paid off.

h. Decrease. The interest is paid with cash, which will reduce thecash balance.

i. Increase. When payments for previous sales, or accountsreceivable, are paid off, the cash balance increases since thepayment must be made in cash.

j. Decrease. The accounts payable are reduced through cash paymentsto suppliers.

CHAPTER 18 B-574

k. Decrease. Here the dividend payments are made with cash, which isgenerally the case. This is different from part a where debt wasraised to make the dividend payment.

l. No change. The short-term note will not change the cash balance.

m. Decrease. The utility bills must be paid in cash.

n. Decrease. A cash payment will reduce cash.

o. Increase. If marketable securities are sold, the company willreceive cash from the sale.

B-575 SOLUTIONS

2. The total liabilities and equity of the company are the net bookworth, or market value of equity, plus current liabilities and long-term debt, so:

Total liabilities and equity = $10,380 + 1,450 + 7,500 Total liabilities and equity = $19,330

This is also equal to the total assets of the company. Since totalassets are the sum of all assets, and cash is an asset, the cashaccount must be equal to total assets minus all other assets, so:

Cash = $19,330 – 15,190 – 2,105 Cash = $2,035

We have NWC other than cash, so the total NWC is:

NWC = $2,105 + 2,035 NWC = $4,140

We can find total current assets by using the NWC equation. NWC isequal to:

NWC = CA – CL$4,140 = CA – $1,450CA = $5,590

3. a. Increase. If receivables go up, the time to collect thereceivables would increase, which increases the operating cycle.

b. Increase. If credit repayment times are increased, customers willtake longer to pay their bills, which will lead to an increase inthe operating cycle.

c. Decrease. If the inventory turnover increases, the inventoryperiod decreases.

d. No change. The accounts payable period is part of the cash cycle,not the operating cycle.

e. Decrease. If the receivables turnover increases, the receivablesperiod decreases.

CHAPTER 18 B-576

f. No change. Payments to suppliers affects the accounts payableperiod, which is part of the cash cycle, not the operating cycle.

4. a. Increase; Increase. If the terms of the cash discount are madeless favorable to customers, the accounts receivable period willlengthen. This will increase both the cash cycle and the operatingcycle.

b. Increase; No change. This will shorten the accounts payableperiod, which will increase the cash cycle. It will have no effecton the operating cycle since the accounts payable period is notpart of the operating cycle.

B-577 SOLUTIONS

c. Decrease; Decrease. If more customers pay in cash, the accountsreceivable period will decrease. This will decrease both the cashcycle and the operating cycle.

d. Decrease; Decrease. Assume the accounts payable period does notchange. Fewer raw materials purchased will reduce the inventoryperiod, which will decrease both the cash cycle and the operatingcycle.

e. Decrease; No change. If more raw materials are purchased oncredit, the accounts payable period will tend to increase, whichwould decrease the cash cycle. We should say that this may not bethe case. The accounts payable period is a decision made by thecompany’s management. The company could increase the accountspayable account and still make the payments in the same number ofdays. This would leave the accounts payable period unchanged,which would leave the cash cycle unchanged. The change in creditpurchases made on credit will not affect the inventory period orthe accounts payable period, so the operating cycle will notchange.

f. Increase; Increase. If more goods are produced for inventory, theinventory period will increase. This will increase both the cashcycle and operating cycle.

5. a. A 45-day collection period implies all receivables outstandingfrom the previous quarter are collected in the current quarter,and:

(90 – 45)/90 = 1/2 of current sales are collected. So:

Q1 Q2 Q3 Q4Beginning receivables $360.00 $395.00 $370.00 $435.00Sales 790.00 740.00 870.00 950.00Cash collections

(755.00)

(765.00)

(805.00)

(910.00)

Ending receivables $395.00 $370.00 $435.00 $475.00

b. A 60-day collection period implies all receivables outstandingfrom previous quarter are collected in the current quarter, and:

CHAPTER 18 B-578

(90-60)/90 = 1/3 of current sales are collected. So:


(623.33)

(773.33)

(783.33)

(896.67)


B-579 SOLUTIONS

c. A 30-day collection period implies all receivables outstandingfrom previous quarter are collected in the current quarter, and:

(90-30)/90 = 2/3 of current sales are collected. So:


(886.67)

(756.67)

(826.67)

(923.33)


6. The operating cycle is the inventory period plus the receivablesperiod. The inventory turnover and inventory period are:

Inventory turnover = COGS/Average inventoryInventory turnover = $56,384/{[$9,780 + 11,380]/2} Inventory turnover = 5.3293 times

Inventory period = 365 days/Inventory turnoverInventory period = 365 days/5.3293 Inventory period = 68.49 days

And the receivables turnover and receivables period are:

Receivables turnover = Credit sales/Average receivablesReceivables turnover = $89,804/{[$4,108 + 4,938]/2} Receivables turnover = 19.8550 times

Receivables period = 365 days/Receivables turnoverReceivables period = 365 days/19.8550 Receivables period = 18.38 days

So, the operating cycle is:

Operating cycle = 68.49 days + 18.38 days Operating cycle = 86.87 days

The cash cycle is the operating cycle minus the payables period. Thepayables turnover and payables period are:

CHAPTER 18 B-580

Payables turnover = COGS/Average payablesPayables turnover = $56,384/{[$7,636 + 7,927]/2} Payables turnover = 7.2459 times

Payables period = 365 days/Payables turnoverPayables period = 365 days/7.2459 Payables period = 50.37 days

B-581 SOLUTIONS

So, the cash cycle is:

Cash cycle = 86.87 days – 50.37 days Cash cycle = 36.50 days

The firm is receiving cash on average 36.50 days after it pays its bills.

7. If we factor immediately, we receive cash on an average of 32 days sooner. The number of periods in a year are:

Number of periods = 365/32 Number of periods = 11.4063

The EAR of this arrangement is:

EAR = (1 + Periodic rate)m – 1EAR = (1 + 1.5/98.5)11.4063 – 1 EAR = .1881 or 18.81%

8. a. The payables period is zero since the company pays immediately.The payment in each period is 30 percent of next period’s sales,so:

Q1 Q2 Q3 Q4

Payment of accounts$258.00 $279.00 $297.00 $282.90

b. Since the payables period is 90 days, the payment in each periodis 30 percent of the last period’s sales, so:

Q1 Q2 Q3 Q4


c. Since the payables period is 60 days, the payment in each periodis 2/3 of last quarter’s orders, plus 1/3 of this quarter’sorders, or:

Quarterly payments = 2/3(.30) times current sales + 1/3(.30) nextperiod sales.

Q1 Q2 Q3 Q4

CHAPTER 18 B-582


B-583 SOLUTIONS

9. Since the payables period is 60 days, the payables in each periodwill be:

Payables each period = 2/3 of last quarter’s orders + 1/3 of thisquarter’s ordersPayables each period = 2/3(.75) times current sales + 1/3(.75) nextperiod sales

Q1 Q2 Q3 Q4

Payment of accounts$722.5

0$732.5

0$847.5

0$897.5

0

Wages, taxes, other expenses 196.00 186.00 214.00 250.00

Long-term financing expenses 90.00 90.00 90.00 90.00

Total$1,008

.50$1,008

.50$1,151

.50$1,237

.50

10. a. The November sales must have been the total uncollected salesminus the uncollected sales from December, divided by thecollection rate two months after the sale, so:

November sales = ($173,000 – 136,000)/0.15 November sales = $246,666.67

b. The December sales are the uncollected sales from December dividedby the collection rate of the previous months’ sales, so:

December sales = $136,000/0.35 December sales = $388,571.43

c. The collections each month for this company are:

Collections = .15(Sales from 2 months ago) + .20(Last month’ssales) + .65 (Current sales)

January collections = .15($246,666.67) + .20($388,571.43)+ .65($235,000) January collections = $267,464.29

February collections = .15($388,571.43) + .20($235,000)+ .65($260,000) February collections = $274,285.71

CHAPTER 18 B-584

March collections = .15($235,000) + .20($260,000) + .65($295,000) March collections = $279,000.00

B-585 SOLUTIONS

11. The sales collections each month will be:

Sales collections = .35(current month sales) + .60(previous monthsales)

Given this collection, the cash budget will be:

April

May

June

Beginning cash balance $140,000 $101,600 $104,100Cash receipts Cash collectionsfrom credit sales 283,500 361,400 371,700 Total cash available $423,500 463,000 475,800Cash disbursements Purchases $168,000 $147,800 $176,300 Wages, taxes, andexpenses 53,800 51,000 78,300 Interest 13,100 13,100 13,100 Equipment purchases 87,000 147,000 0 Total cashdisbursements $321,900 $358,900 $267,700Ending cash balance $101,600 $104,100 $208,100

12. ItemSource/

Use Amount

Cash Source $1,100Accounts receivable Use –$4,300Inventories Use –$3,670Property, plant, andequipment Use –$12,720

Accounts payable Source $2,600Accrued expenses Use –$810Long-term debt Source $3,000Common stock Source $5,000Accumulated retainedearnings Source $1,610

CHAPTER 18 B-586

Intermediate

13. a. If you borrow $50,000,000 for one month, you will pay interest of:

Interest = $50,000,000(.0064) Interest = $320,000

However, with the compensating balance, you will only get the useof:

Amount received = $50,000,000 – 50,000,000(.05) Amount received = $47,500,000

This means the periodic interest rate is:

Periodic interest = $320,000/$47,500,000 Periodic interest = .006737 or 0.674%

B-587 SOLUTIONS

So, the EAR is:

EAR = [1 + ($320,000/$47,500,000)]12 – 1EAR = .0839 or 8.39%

b. To end up with $15,000,000, you must borrow:

Amount to borrow = $15,000,000/(1 – .05) Amount to borrow = $15,789,473.68

The total interest you will pay on the loan is:

Total interest paid = $15,789,473.68(1.0064)6 – 15,789,473.68 Total interest paid = $616,100.02

14. a. The EAR of your investment account is:

EAR = 1.0124 – 1 EAR = .0489 or 4.89%

b. To calculate the EAR of the loan, we can divide the interest onthe loan by the amount of the loan. The interest on the loanincludes the opportunity cost of the compensating balance. Theopportunity cost is the amount of the compensating balance timesthe potential interest rate you could have earned. Thecompensating balance is only on the unused portion of the creditline, so:

Opportunity cost = .04($70,000,000 – 45,000,000)(1.012)4

– .04($70,000,000 – 45,000,000) Opportunity cost = $48,870.93

And the interest you will pay to the bank on the loan is:

Interest cost = $45,000,000(1.023)4 – 45,000,000 Interest cost = $4,285,032.65

So, the EAR of the loan in the amount of $45 million is:

EAR = ($4,285,032.65 + 48,870.93)/$45,000,000 EAR = .0963 or 9.63%

CHAPTER 18 B-588

c. The compensating balance is only applied to the unused portion ofthe credit line, so the EAR of a loan on the full credit line is:

EAR = 1.0234 – 1 EAR = .0952 or 9.52%

B-589 SOLUTIONS

15. a. A 45-day collection period means sales collections each quarterare:

Collections = 1/2 current sales + 1/2 old sales

A 36-day payables period means payables each quarter are:

Payables = 3/5 current orders + 2/5 old orders

So, the cash inflows and disbursements each quarter are:

Q1 Q2 Q3 Q4 Beginning receivables $68.00 $105.00 $90.00 $122.50 Sales 210.00 180.00 245.00 280.00 Collection of accounts 173.00 195.00 212.50 262.50 Ending receivables $105.00 $90.00 $122.50 $140.00 Payment of accounts $86.40 $98.55 $119.70 $115.20

Wages, taxes, and expenses 52.50 45.00 61.25 70.00

Capital expenditures 80.00 Interest & dividends 12.00 12.00 12.00 12.00

Total cash disbursements $150.90 $235.55 $192.95 $197.20

Total cash collections $173.00 $195.00 $212.50 $262.50

Total cash disbursements 150.90 235.55 192.95 197.20

Net cash inflow $22.10 $(40.55) $19.55 $65.30

The company’s cash budget will be:

WILDCAT, INC.Cash Budget(in millions)

Q1 Q2 Q3 Q4Beginning cash balance $64.00 $86.10 $45.55 $65.10Net cash inflow 22.10 –40.55 19.55 65.30Ending cash balance $86.10 $45.55 $65.10 $130.40Minimum cash balance –30.00 –30.00 –30.00 –30.00Cumulative surplus $56.10 $15.55 $35.10 $100.40

CHAPTER 18 B-590

(deficit)

B-591 SOLUTIONS

With a $30 million minimum cash balance, the short-term financialplan will be:

WILDCAT, INC.Short-Term Financial Plan

(in millions) b. Q1 Q2 Q3 Q4

Beginning cash balance $30.00 $30.00 $30.00 $30.00Net cash inflow 22.10 –40.55 19.55 65.30New short-term investments –22.78 0 –19.90 –65.70Income on short-term investments 0.68 1.14 0.35 0.40Short-term investments sold 0 39.41 0 0New short-term borrowing 0 0 0 0Interest on short-term borrowing 0 0 0 0Short-term borrowing repaid 0 0 0 0Ending cash balance $30.00 $30.00 $30.00 $30.00Minimum cash balance –30.00 –30.00 –30.00 –30.00Cumulative surplus (deficit) $0 $0 $0 $0

Beginning short-term investments $34.00 $56.78 $0 $17.37Ending short-term investments $56.78 $17.37 $19.90 $83.46Beginning short-term debt $0 $0 $0 $0Ending short-term debt $0 $0 $0 $0

Below you will find the interest paid (or received) for eachquarter:

Q1: excess funds at start of quarter of $34 invested for 1 quarterearns .02($34) = $0.68 income

CHAPTER 18 B-592

Q2: excess funds of $56.78 invested for 1 quarterearns .02($56.78) = $1.14 in income



Net cash cost = $0.68 + 1.14 + 0.35 + 0.40 = $2.56

B-593 SOLUTIONS

16. a. With a minimum cash balance of $50 million, the short-term financial plan will be:


(in millions)

Q1 Q2 Q3 Q4Beginning cash balance $50.00 $50.00 $50.00 $50.00Net cash inflow 22.10 –40.55 19.55 65.30New short-terminvestments –22.38 0 –16.00 –65.62Income on short-terminvestments 0.28 0.73 0 0.32Short-term investmentssold 0 36.38 0 0New short-term borrowing 0 3.44 0 0Interest on short-termborrowing 0 0 –0.10 0Short-term borrowingrepaid 0 0 –3.44 0Ending cash balance $50.00 $50.00 $50.00 $50.00Minimum cash balance –50.00 –50.00 –50.00 –50.00Cumulative surplus(deficit) $0 $0 $0 $0

Beginning short-terminvestments $14.00 $36.38 $0 $0Ending short-terminvestments $36.38 $0 $16.00 $65.94Beginning short-termdebt $0 $0 $3.44 $0Ending short-term debt $0 $3.44 $0 $0




CHAPTER 18 B-594

Q3: shortage of funds of $3.44 borrowed for 1 quartercosts .03($3.44) = $0.10 in interest


Net cash cost = $0.28 + 0.73 – 0.10 + 0.32 = $1.22

B-595 SOLUTIONS

b. And with a minimum cash balance of $10 million, the short-termfinancial plan will be:


(in millions)

Q1 Q2 Q3 Q4Beginning cash balance $10.00 $10.00 $10.00 $10.00Net cash inflow 22.10 –40.55 19.55 65.30New short-terminvestments –23.18 0 –20.31 –65.71Income on short-terminvestments 1.08 1.54 0.76 0.41Short-term investmentssold 0 39.01 0 0New short-term borrowing 0 0 0 0Interest on short-termborrowing 0 0 0 0Short-term borrowingrepaid 0 0 0 0Ending cash balance $10.00 $10.00 $10.00 $10.00Minimum cash balance –10.00 –10.00 –10.00 –10.00Cumulative surplus(deficit) 0 0 0 0

Beginning short-terminvestments $54.00 $77.18 0 $38.17Ending short-terminvestments $77.18 $38.17 $20.31 $104.29Beginning short-termdebt 0 0 0 0Ending short-term debt 0 0 0 0




CHAPTER 18 B-596

Q3: excess funds of $38.17 invested for 1 quarterearns .02($38.17) = $0.76 in interest


Net cash cost = $1.08 + 1.54 + 0.76 + 0.41 = $3.79

Since cash has an opportunity cost, the firm can boost its profit ifit keeps its minimum cash balance low and invests the cash instead.However, the tradeoff is that in the event of unforeseencircumstances, the firm may not be able to meet its short-runobligations if enough cash is not available.

B-597 SOLUTIONS

Challenge

17. a. For every dollar borrowed, you pay interest of:

Interest = $1(.019) = $0.019

You also must maintain a compensating balance of 4 percent of thefunds borrowed, so for each dollar borrowed, you will onlyreceive:

Amount received = $1(1 – .04) = $0.96

We can adjust the EAR equation we have been using to account forthe compensating balance by dividing the EAR by one minus thecompensating balance, so:

EAR = [(1.019)4 – 1]/(1 – .04) EAR = .08145 or 8.15%

Another way to calculate the EAR is using the FVIF (or PVIF). Foreach dollar borrowed, we must repay:

Amount owed = $1(1.019)4

Amount owed = $1.0782

At the end of the year the compensating will be returned, so yournet cash flow at the end of the year will be:

End of year cash flow = $1.0782 – .04End of year cash flow = $1.0382

The present value of the end of year cash flow is the amount youreceive at the beginning of the year, so the EAR is:

FV = PV(1 + R)$1.0382 = $0.96(1 + R)R = $1.0382/$0.96 – 1EAR = .08145 or 8.15%

b. The EAR is the amount of interest paid on the loan divided by theamount received when the loan is originated. The amount ofinterest you will pay on the loan is the amount of the loan timesthe effective annual interest rate, so:

CHAPTER 18 B-598

Interest = $130,000,000[(1.019)4 – 1]Interest = $10,165,163.62

For whatever loan amount you take, you will only receive 96percent of that amount since you must maintain a 4 percentcompensating balance on the portion of the credit line used. Thecredit line also has a fee of .140 percent, so you will only getto use:

Amount received = .96($130,000,000) – .00140($400,000,000)Amount received = $124,240,000

B-599 SOLUTIONS

So, the EAR of the loan is:

EAR = $10,165,163.62/$124,240,000EAR = .08182 or 8.18%

18. You will pay interest of:

Interest = $25,000,000(.10) = $2,500,000

Additionally, the compensating balance on the loan is:

Compensating balance = $25,000,000(.05) = $1,250,000

Since this is a discount loan, you will receive the loan amount minusthe interest payment. You will also not get to use the compensatingbalance. So, the amount of money you will actually receive on a $25million loan is:

Cash received = $25,000,000 – 2,500,000 – 1,250,000 = $21,250,000

The EAR is the interest amount divided by the loan amount, so:

EAR = $2,500,000/$21,250,000 EAR = .1176 or 11.76%

We can also use the FVIF (or PVIF) here to calculate the EAR. Yourcash flow at the beginning of the year is $21,250,000. At the end ofthe year, your cash flow loan repayment, but you will also receiveyour compensating balance back, so:

End of year cash flow = $25,000,000 – 1,250,000End of year cash flow = $23,750,000

So, using the time value of money, the EAR is:

$23,750,000 = $21,250,000(1 + R)R = $23,750,000/$21,250,000 – 1EAR = .1176 or 11.76%

CHAPTER 19CASH AND LIQUIDITY MANAGEMENT


1. Yes. Once a firm has more cash than it needs for operations andplanned expenditures, the excess cash has an opportunity cost. Itcould be invested (by shareholders) in potentially more profitableways. Question 10 discusses another reason.

2. If it has too much cash it can simply pay a dividend, or, more likelyin the current financial environment, buy back stock. It can alsoreduce debt. If it has insufficient cash, then it must either borrow,sell stock, or improve profitability.

3. Probably not. Creditors would probably want substantially more.

4. In the case of Ford, the company generally argued that it held cashto guard against future economic downturns. For Goldman Sachs,investment banks traditionally hold large cash positions, althoughthe amount of cash in early 2008 was an all-time high.

5. Cash management is associated more with the collection anddisbursement of cash. Liquidity management is broader and concernsthe optimal level of liquid assets needed by a firm. Thus, forexample, a company’s stockpiling of cash is liquidity management;whereas, evaluating a lockbox system is cash management.

6. Such instruments go by a variety of names, but the key feature isthat the dividend adjusts, keeping the price relatively stable. Thisprice stability, along with the dividend tax exemption, makes so-called adjustable rate preferred stock very attractive relative tointerest-bearing instruments.

7. Net disbursement float is more desirable because the bank thinks thefirm has more money than it actually does, and the firm is,therefore, receiving interest on funds it has already spent.

B-601 SOLUTIONS

8. The firm has a net disbursement float of $500,000. If this is anongoing situation, the firm may be tempted to write checks for morethan it actually has in its account.

9. a. About the only disadvantage to holding T-bills are the generallylower yields compared to alternative money market investments.

b. Some ordinary preferred stock issues pose both credit and pricerisks that are not consistent with most short-term cash managementplans.

c. The primary disadvantage of NCDs is the normally largetransactions sizes, which may not be feasible for the short-terminvestment plans of many smaller to medium-sized corporations.

d. The primary disadvantages of the commercial paper market are thehigher default risk characteristics of the security and the lackof an active secondary market which may excessively restrict theflexibility of corporations to meet their liquidity adjustmentneeds.

CHAPTER 19 B-602

e. The primary disadvantages of RANs is that some possess non-triviallevels of default risk, and also, corporations are somewhatrestricted in the type and amount of these tax-exempts that theycan hold in their portfolios.

f. The primary disadvantage of the repo market is the generally veryshort maturities available.

10. The concern is that excess cash on hand can lead to poorly thought-out management decisions. The thought is that keeping cash levelsrelatively low forces management to pay careful attention to cashflow and capital spending.

11. A potential advantage is that the quicker payment often means abetter price. The disadvantage is that doing so increases the firm’scash cycle.

12. This is really a capital structure decision. If the firm has anoptimal capital structure, paying off debt moves it to an under-leveraged position. However, a combination of debt reduction andstock buy-backs could be structured to leave capital structureunchanged.

13. It is unethical because you have essentially tricked the grocerystore into making you an interest-free loan, and the grocery store isharmed because it could have earned interest on the money instead ofloaning it to you.



Basic

1. The average daily float is the average amount of checks received perday times the average number of days delay, divided by the number ofdays in a month. Assuming 30 days in a month, the average daily floatis:

Average daily float = 4($156,000)/30 Average daily float = $20,800

B-603 SOLUTIONS

2. a. The disbursement float is the average monthly checks written timesthe average number of days for the checks to clear, so:

Disbursement float = 4($14,000) Disbursement float = $56,000

The collection float is the average monthly checks received timesthe average number of days for the checks to clear, so:

Collection float = 2(–$26,000) Collection float = –$52,000

CHAPTER 19 B-604

The net float is the disbursement float plus the collection float,so:

Net float = $56,000 – 52,000 Net float = $4,000

b. The new collection float will be:

Collection float = 1(–$26,000) Collection float = –$26,000

And the new net float will be:

Net float = $56,000 – 26,000 Net float = $30,000

3. a. The collection float is the average daily checks received timesthe average number of days for the checks to clear, so:

Collection float = 3($19,000) Collection float = $57,000

b. The firm should pay no more than the amount of the float, or$57,000, to eliminate the float.

c. The maximum daily charge the firm should be willing to pay is thecollection float times the daily interest rate, so:

Maximum daily charge = $57,000(.00019) Maximum daily charge = $10.83

4. a. Total float = 4($17,000) + 5($6,000) Total float = $98,000

b. The average daily float is the total float divided by the numberof days in a month. Assuming 30 days in a month, the average dailyfloat is:

Average daily float = $98,000/30 Average daily float = $3,266.67

c. The average daily receipts are the average daily checks receiveddivided by the number of days in a month. Assuming a 30 day month:

B-605 SOLUTIONS

Average daily receipts = ($17,000 + 6,000)/30 Average daily receipts = $766.67

The weighted average delay is the sum of the days to clear acheck, times the amount of the check divided by the average dailyreceipts, so:

Weighted average delay = 4($16,000/$23,000) + 5($6,000/$23,000) Weighted average delay = 4.26 days

CHAPTER 19 B-606

5. The average daily collections are the number of checks received timesthe average value of a check, so:

Average daily collections = $108(8,500) Average daily collections = $918,000

The present value of the lockbox service is the average dailyreceipts times the number of days the collection is reduced, so:

PV = (2 day reduction)($918,000) PV = $1,836,000

The daily cost is a perpetuity. The present value of the cost is thedaily cost divided by the daily interest rate. So:

PV of cost = $225/.00016 PV of cost = $1,406,250

The firm should take the lockbox service. The NPV of the lockbox isthe cost plus the present value of the reduction in collection time,so:

NPV = –$1,406,250 + 1,836,000NPV = $429,750

The annual savings excluding the cost would be the future value ofthe savings minus the costs, so:

Annual savings = $1,836,000(1.00016)365 – 1,836,000 Annual savings = $110,406.05

And the annual cost would be the future value of the daily cost,which is an annuity, so:

Annual cost = $225(FVIFA365,.016%) Annual cost = $84,563.46

So, the annual net savings would be:

Annual net savings = $110,406.05 – 84,563.46 Annual net savings = $25,842.59

B-607 SOLUTIONS

6. a. The average daily float is the sum of the percentage each checkamount is of the total checks received times the number of checksreceived times the amount of the check times the number of daysuntil the check clears, divided by the number of days in a month.Assuming a 30 day month, we get:

Average daily float = [.60(5,300)($55)(2) + .40(5,300)($80)(3)]/30Average daily float = $28,620

On average, there is $28,620 that is uncollected and not availableto the firm.

CHAPTER 19 B-608

b. The total collections are the sum of the percentage of each checkamount received times the total checks received times the amountof the check, so:

Total collections = .60(5,300)($55) + .40(5,300)($80) Total collections = $344,500

The weighted average delay is the sum of the average number ofdays a check of a specific amount is delayed, times the percentagethat check amount makes up of the total checks received, so:

Weighted average delay = 2[.60(5,300)($55)/$344,500] +3[.40(5,300)($80) /$344,500] Weighted average delay = 2.49 days

The average daily float is the weighted average delay times theaverage checks received per day. Assuming a 30 day month, we get:

Average daily float = 2.49($344,500/30 days) Average daily float = $28,620

c. The most the firm should pay is the total amount of the averagefloat, or $28,620.

d. The average daily interest rate is:

1.07 = (1 + R)365 R = .01854% per day

The daily cost of float is the average daily float times the dailyinterest rate, so:

Daily cost of the float = $28,620(.0001854) Daily cost of the float = $5.31

e. The most the firm should pay is still the average daily float.Under the reduced collection time assumption, we get:

New average daily float = 1.5($344,500/30) New average daily float = $17,225

7. a. The present value of adopting the system is the number of dayscollections are reduced times the average daily collections, so:

B-609 SOLUTIONS

PV = 3(385)($1,105) PV = $1,276,275

b. The NPV of adopting the system is the present value of the savingsminus the cost of adopting the system. The cost of adopting thesystem is the present value of the fee per transaction times thenumber of transactions. This is a perpetuity, so:

NPV = $1,276,275 – [$0.50(385)/.0002] NPV = $313,775

CHAPTER 19 B-610

c. The net cash flows is the present value of the average dailycollections times the daily interest rate, minus the transactioncost per day, so:

Net cash flow per day = $1,276,275(.0002) – $0.50(385) Net cash flow per day = $62.76

The net cash flow per check is the net cash flow per day dividedby the number of checks received per day, or:

Net cash flow per check = $62.76/385Net cash flow per check = $0.16

Alternatively, we could find the net cash flow per check as thenumber of days the system reduces collection time times theaverage check amount times the daily interest rate, minus thetransaction cost per check. Doing so, we confirm our previousanswer as:

Net cash flow per check = 3($1,105)(.0002) – $0.50 Net cash flow per check = $0.16 per check

8. a. The reduction in cash balance from adopting the lockbox is thenumber of days the system reduces collection time times theaverage daily collections, so:

Cash balance reduction = 3($145,000) Cash balance reduction = $435,000

b. The dollar return that can be earned is the average daily interestrate times the cash balance reduction. The average daily interestrate is:

Average daily rate = 1.091/365 – 1 Average daily rate = .0236% per day

The daily dollar return that can be earned from the reduction indays to clear the checks is:

Daily dollar return = $435,000(.000236) Daily dollar return = $102.72

B-611 SOLUTIONS

c. If the company takes the lockbox, it will receive three paymentsearly, with the first payment occurring today. We can use thedaily interest rate from part b, so the savings are:

Savings = $145,000 + $145,000(PVIFA.0236%,2) Savings = $434,897.32

If the lockbox payments occur at the end of the month, we need theeffective monthly interest rate, which is:

Monthly interest rate = 1.091/12 – 1 Monthly interest rate = 0.7207%

CHAPTER 19 B-612

Assuming the lockbox payments occur at the end of the month, thelockbox payments, which are a perpetuity, will be:

PV = C/R$434,897.32 = C / .007207 C = $3,134.35

It could also be assumed that the lockbox payments occur at thebeginning of the month. If so, we would need to use the PV of aperpetuity due, which is:

PV = C + C / R

Solving for C:

C = (PV × R) / (1 + R)C = (434,897.32 × .007207) / (1 + .007207)C = $3,112.02

9. The interest that the company could earn will be the amount of thechecks times the number of days it will delay payment times thenumber of weeks that checks will be disbursed times the dailyinterest rate, so:

Interest = $93,000(7)(52/2)(.00015) Interest = $2,538.90

10. The benefit of the new arrangement is the $4 million in acceleratedcollections since the new system will speed up collections by oneday. The cost is the new compensating balance, but the company willrecover the existing compensating balance, so:

NPV = $4,000,000 – ($400,000 – 500,000) NPV = $3,900,000

The company should proceed with the new system. The savings are theNPV times the annual interest rate, so:

Net savings = $3,900,000(.05) Net savings = $195,000

Intermediate

B-613 SOLUTIONS

11. To find the NPV of taking the lockbox, we first need to calculate thepresent value of the savings. The present value of the savings willbe the reduction in collection time times the average dailycollections, so:

PV = 2(750)($980) PV = $1,470,000

And the daily interest rate is:

Daily interest rate = 1.0701/365 – 1 Daily interest rate = .00019 or .019% per day

CHAPTER 19 B-614

The transaction costs are a perpetuity. The cost per day is the costper transaction times the number of transactions per day, so the NPVof taking the lockbox is:

NPV = $1,470,000 – [$0.35(980)/.00019] NPV = $54,015.17

Without the fee, the lockbox system should be accepted. To calculatethe NPV of the lockbox with the annual fee, we can simply use the NPVof the lockbox without the annual fee and subtract the addition cost.The annual fee is a perpetuity, so, with the fee, the NPV of takingthe lockbox is:

NPV = $54,015.17 – [$5,000/.07] NPV = –$17,413.40

With the fee, the lockbox system should not be accepted.

12. To find the minimum number of payments per day needed to make thelockbox system feasible is the number of checks that makes the NPV ofthe decision equal to zero. The average daily interest rate is:

Daily interest rate = 1.051/365 – 1 Daily interest rate = .0134% per day

The present value of the savings is the average payment amount timesthe days the collection period is reduced times the number ofcustomers. The costs are the transaction fee and the annual fee. Bothare perpetuities. The total transaction costs are the transactioncosts per check times the number of checks. The equation for the NPVof the project, where N is the number of checks transacted per day,is:

NPV = 0 = ($5,300)(1)N – $0.10(N)/.000134 – $20,000/.05$400,000 = $5,300N – $748.05N$4,551.95N = $400,000N = 87.87 88 customers per day

B-615 SOLUTIONS

APPENDIX 19A

1. a. Decrease. This will lower the trading costs, which will cause adecrease in the target cash balance.

b. Decrease. This will increase the holding cost, which will cause adecrease in the target cash balance.

c. Increase. This will increase the amount of cash that the firm hasto hold in non-interest bearing accounts, so they will have toraise the target cash balance to meet this requirement.

d. Decrease. If the credit rating improves, then the firm can borrowmore easily, allowing it to lower the target cash balance andborrow if a cash shortfall occurs.

e. Increase. If the cost of borrowing increases, the firm will needto hold more cash to protect against cash shortfalls as itsborrowing costs become more prohibitive.

f. Either. This depends somewhat on what the fees apply to, but ifdirect fees are established, then the compensating balance may belowered, thus lowering the target cash balance. If, on the otherhand, fees are charged on the number of transactions, then thefirm may wish to hold a higher cash balance so they are nottransferring money into the account as often.

2. The target cash balance using the BAT model is:

C* = [(2T × F)/R]1/2 C* = [2($8,500)($25)/.06]1/2 C* = $2,661.45

The initial balance should be $2,661.45, and whenever the balancedrops to $0, another $2,661.45 should be transferred in.

3. The holding cost is the average daily cash balance times the interestrate, so:

Holding cost = ($1,300)(.05) Holding cost = $65.00

CHAPTER 19 B-616

The trading costs are the total cash needed times the replenishingcosts, divided by the average daily balance times two, so:

Trading cost = [($43,000)($8)]/[($1,300)(2)] Trading cost = $132.31

The total cost is the sum of the holding cost and the trading cost,so:

Total cost = $65.00 + 132.31 Total cost = $197.31

B-617 SOLUTIONS

The target cash balance using the BAT model is:

C* = [(2T × F)/R]1/2 C* = [2($43,000)($8)/.05]1/2 C* = $3,709.45

They should increase their average daily cash balance to:

New average cash balance = $3,709.45/2 New average cash balance = $1,854.72

This would minimize the costs. The new total cost would be:

New total cost = ($1,845.72)(.05) + [($43,000)($8)]/[2($1,854.72)] New total cost = $185.47

4. a. The opportunity costs are the amount transferred times theinterest rate, divided by two, so:

Opportunity cost = ($1,500)(.05)/2 Opportunity cost = $37.50

The trading costs are the total cash balance times the tradingcost per transaction, divided by the amount transferred, so:

Trading cost = ($16,000)($25)/$1,500 Trading cost = $266.67

The firm keeps too little in cash because the trading costs aremuch higher than the opportunity costs.

b. The target cash balance using the BAT model is:

C* = [(2T × F)/R]1/2 C* = [2($16,000)($25)/.05]1/2 C* = $4,000

5. The total cash needed is the cash shortage per month times twelvemonths, so:

Total cash = 12($140,000) Total cash = $1,680,000

CHAPTER 19 B-618

The target cash balance using the BAT model is:

C* = [(2T × F)/R]1/2 C* = [2($1,680,000)($500)/.057]1/2 C* = $171,679.02

B-619 SOLUTIONS

The company should invest:

Invest = $690,000 – 171,679.02 Invest = $518,320.98

of its current cash holdings in marketable securities to bring thecash balance down to the optimal level. Over the rest of the year,sell securities:

Sell securities = $1,680,000/$171,679.02 Sell securities = 9.79 10 times.

6. The lower limit is the minimum balance allowed in the account, andthe upper limit is the maximum balance allowed in the account. Whenthe account balance drops to the lower limit:

Securities sold = $80,000 – 43,000 Securities sold = $37,000

in marketable securities will be sold, and the proceeds deposited inthe account. This moves the account balance back to the target cashlevel. When the account balance rises to the upper limit, then:

Securities purchased = $125,000 – 80,000 Securities purchased = $45,000

of marketable securities will be purchased. This expenditure bringsthe cash level back down to the target balance of $80,000.

7. The target cash balance using the Miller-Orr model is:

C* = L + (3/4 × F × 2 / R]1/3 C* = $1,500 + [3/4($40)($70)2/.00021]1/3 C* = $2,387.90

The upper limit is:

U* = 3 × C* – 2 × LU* = 3($2,387.90) – 2($1,500) U* = $4,163.71

CHAPTER 19 B-620

When the balance in the cash account drops to $1,500, the firm sells:

Sell = $2,387.90 – 1,500 Sell = $887.90

of marketable securities. The proceeds from the sale are used toreplenish the account back to the optimal target level of C*.Conversely, when the upper limit is reached, the firm buys:

Buy = $4,163.71 – 2,387.90 Buy = $1,775.81

of marketable securities. This expenditure lowers the cash level backdown to the optimal level of $2,387.90.

8. As variance increases, the upper limit and the spread will increase,while the lower limit remains unchanged. The lower limit does notchange because it is an exogenous variable set by management. As thevariance increases, however, the amount of uncertainty increases.When this happens, the target cash balance, and therefore the upperlimit and the spread, will need to be higher. If the variance dropsto zero, then the lower limit, the target balance, and the upperlimit will all be the same.

9. The average daily interest rate is:

Daily rate = 1.071/365 – 1 Daily rate = .000185 or .0185% per day

The target cash balance using the Miller-Orr model is:

C* = L + (3/4 × F × 2 / R]1/3 C* = $160,000 + [3/4($300)($890,000)/.000185]1/3 C* = $170,260.47

The upper limit is:

U* = 3 × C* – 2 × LU* = 3($170,260.47) – 2($160,000) U* = $190,781.41

10. Using the BAT model and solving for R, we get:

B-621 SOLUTIONS

C* = [(2T × F)/R]1/2

$2,700 = [2($28,000)($10)/R]1/2 R = [2($28,000)($10)]/$2,7002

R = .0768 or 7.68%

CHAPTER 20CREDIT AND INVENTORY MANAGEMENTAnswers to Concepts Review and Critical Thinking Questions

1. a. A sight draft is a commercial draft that is payable immediately.b. A time draft is a commercial draft that does not require immediate

payment.c. A bankers acceptance is when a bank guarantees the future payment

of a commercial draft.d. A promissory note is an IOU that the customer signs.e. A trade acceptance is when the buyer accepts the commercial draft

and promises to pay it in the future.

2. Trade credit is usually granted on open account. The invoice is thecredit instrument.

3. Credit costs: cost of debt, probability of default, and the cashdiscountNo-credit costs: lost salesThe sum of these are the carrying costs.

4. 1. Character: determines if a customer is willing to pay his orher debts.

2. Capacity: determines if a customer is able to pay debts out ofoperating cash flow.

3. Capital: determines the customer’s financial reserves in caseproblems occur with operating cash flow.

4. Collateral: assets that can be liquidated to pay off the loanin case of default.

5. Conditions: customer’s ability to weather an economic downturnand whether such a downturn is likely.

5. 1. Perishability and collateral value2. Consumer demand3. Cost, profitability, and standardization4. Credit risk5. The size of the account6. Competition7. Customer type

B-623 SOLUTIONS

If the credit period exceeds a customer’s operating cycle, then thefirm is financing the receivables and other aspects of the customer’sbusiness that go beyond the purchase of the selling firm’smerchandise.

6. a. B: A is likely to sell for cash only, unless the product reallyworks. If it does, then they might grant longer credit periodsto entice buyers.

b. A: Landlords have significantly greater collateral, and thatcollateral is not mobile.

c. A: Since A’s customers turn over inventory less frequently, theyhave a longer inventory period, and thus, will most likelyhave a longer credit period as well.

d. B: Since A’s merchandise is perishable and B’s is not, B willprobably have a longer credit period.

CHAPTER 20 B-624

e. A: Rugs are fairly standardized and they are transportable, whilecarpets are custom fit and are not particularly transportable.

7. The three main categories of inventory are: raw material (initialinputs to the firm’s production process), work-in-progress (partiallycompleted products), and finished goods (products ready for sale).From the firm’s perspective, the demand for finished goods isindependent from the demand for the other types of inventory. Thedemand for raw material and work-in-progress is derived from, ordependent on, the firm’s needs for these inventory types in order toachieve the desired levels of finished goods.

8. JIT systems reduce inventory amounts. Assuming no adverse effects onsales, inventory turnover will increase. Since assets will decrease,total asset turnover will also increase. Recalling the DuPontequation, an increase in total asset turnover, all else being equal,has a positive effect on ROE.

9. Carrying costs should be equal to order costs. Since the carryingcosts are low relative to the order costs, the firm should increasethe inventory level.

10. Since the price of components can decline quickly, Dell does not haveinventory which is purchased and then declines quickly in valuebefore it is sold. If this happens, the inventory may be sold at aloss. While this approach is valuable, it is difficult to implement.For example, Dell manufacturing plants will often have areas setaside that are for the suppliers. When parts are needed, it is amatter of going across the floor to get new parts. In fact, m0stcomputer manufacturers are trying to implement similar inventorysystems.



Basic

1. a. There are 30 days until account is overdue. If you take the fullperiod, you must remit:

B-625 SOLUTIONS

Remittance = 400($125) Remittance = $50,000

b. There is a 1 percent discount offered, with a 10 day discountperiod. If you take the discount, you will only have to remit:

Remittance = (1 – .01)($50,000) Remittance = $49,500

c. The implicit interest is the difference between the two remittanceamounts, or:

Implicit interest = $50,000 – 49,500 Implicit interest = $500

CHAPTER 20 B-626

The number of days’ credit offered is:

Days’ credit = 30 – 10 Days’ credit = 20 days

2. The receivables turnover is:

Receivables turnover = 365/Average collection periodReceivables turnover = 365/36 Receivables turnover = 10.139 times

And the average receivables are:

Average receivables = Sales/Receivables periodAverage receivables = $47,000,000 / 10.139 Average receivables = $4,635,616

3. a. The average collection period is the percentage of accounts takingthe discount times the discount period, plus the percentage ofaccounts not taking the discount times the days’ until fullpayment is required, so:

Average collection period = .65(10 days) + .35(30 days) Average collection period = 17 days

b. And the average daily balance is:

Average balance = 1,300($1,700)(17)(12/365) Average balance = $1,235,178.08

4. The daily sales are:

Daily sales = $19,400 / 7 Daily sales = $2,771.43

Since the average collection period is 34 days, the average accountsreceivable is:

Average accounts receivable = $2,771.43(34) Average accounts receivable = $94,228.57

5. The interest rate for the term of the discount is:

B-627 SOLUTIONS

Interest rate = .01/.99 Interest rate = .0101 or 1.01%

And the interest is for:

35 – 10 = 25 days

CHAPTER 20 B-628

So, using the EAR equation, the effective annual interest rate is:

EAR = (1 + Periodic rate)m – 1EAR = (1.0101)365/25 – 1 EAR = .1580 or 15.80%

a. The periodic interest rate is:

Interest rate = .02/.98 Interest rate = .0204 or 2.04%

And the EAR is:

EAR = (1.0204)365/25 – 1 EAR = .3431 or 34.31%

b. The EAR is:

EAR = (1.0101)365/50 – 1 EAR = .0761 or = 7.61%

c. The EAR is:

EAR = (1.0101)365/20 – 1 EAR = .2013 or 20.13%

6. The receivables turnover is:

Receivables turnover = 365/Average collection period Receivables turnover = 365/39 Receivables turnover = 9.3590 times

And the annual credit sales are:

Annual credit sales = Receivables turnover × Average dailyreceivablesAnnual credit sales = 9.3590($47,500) Annual credit sales = $444,551.28

7. The total sales of the firm are equal to the total credit sales sinceall sales are on credit, so:

Total credit sales = 5,600($425)

B-629 SOLUTIONS

Total credit sales = $2,380,000

The average collection period is the percentage of accounts takingthe discount times the discount period, plus the percentage ofaccounts not taking the discount times the days’ until full paymentis required, so:

Average collection period = .60(10) + .40(40) Average collection period = 22 days

CHAPTER 20 B-630

The receivables turnover is 365 divided by the average collectionperiod, so:

Receivables turnover = 365/22 Receivables turnover = 16.591 times

And the average receivables are the credit sales divided by thereceivables turnover so:

Average receivables = $2,380,000/16.591 Average receivables = $143,452.05

If the firm increases the cash discount, more people will pay sooner,thus lowering the average collection period. If the ACP declines, thereceivables turnover increases, which will lead to a decrease in theaverage receivables.

8. The average collection period is the net credit terms plus the daysoverdue, so:

Average collection period = 30 + 8 Average collection period = 38 days

The receivables turnover is 365 divided by the average collectionperiod, so:

Receivables turnover = 365/38 Receivables turnover = 9.6053 times

And the average receivables are the credit sales divided by thereceivables turnover so:

Average receivables = $8,400,000 / 9.6053 Average receivables = $874,520.55

9. a. The cash outlay for the credit decision is the variable cost ofthe engine. If this is a one-time order, the cash inflow is thepresent value of the sales price of the engine times one minus thedefault probability. So, the NPV per unit is:

NPV = –$1,600,000 + (1 – .005)($1,870,000)/1.029 NPV = $208,211.86 per unit

B-631 SOLUTIONS

The company should fill the order.

b. To find the breakeven probability of default, , we simply use theNPV equation from part a, set it equal to zero, and solve for .Doing so, we get:

NPV = 0 = –$1,600,000 + (1 – )($1,870,000)/1.029 = .1196 or 11.96%

We would not accept the order if the default probability washigher than 11.96 percent.

CHAPTER 20 B-632

c. If the customer will become a repeat customer, the cash inflowchanges. The cash inflow is now one minus the default probability,times the sales price minus the variable cost. We need to use thesales price minus the variable cost since we will have to buildanother engine for the customer in one period. Additionally, thiscash inflow is now a perpetuity, so the NPV under theseassumptions is:

NPV = –$1,600,000 + (1 – .005)($1,870,000 – 1,600,000)/.029 NPV = $7,663,793.10 per unit

The company should fill the order. The breakeven defaultprobability under these assumptions is:

NPV = 0 = –$1,600,000 + (1 – )($1,870,000 – 1,600,000)/.029 = .8281 or 82.81%

We would not accept the order if the default probability washigher than 82.81 percent. This default probability is much higherthan in part b because the customer may become a repeat customer.

d. It is assumed that if a person has paid his or her bills in thepast, they will pay their bills in the future. This implies thatif someone doesn’t default when credit is first granted, then theywill be a good customer far into the future, and the possiblegains from the future business outweigh the possible losses fromgranting credit the first time.

10. The cost of switching is the lost sales from the existing policy plusthe incremental variable costs under the new policy, so:

Cost of switching = $720(1,305) + $495(1,380 – 1,305) Cost of switching = $976,725

The benefit of switching is the new sales price minus the variablecosts per unit, times the incremental units sold, so:

Benefit of switching = ($720 – 495)(1,380 – 1,305) Benefit of switching = $16,875

The benefit of switching is a perpetuity, so the NPV of the decisionto switch is:

B-633 SOLUTIONS

NPV = –$976,275 + $16,875/.015 NPV = $148,275.00

The firm will have to bear the cost of sales for one month beforethey receive any revenue from credit sales, which is why the initialcost is for one month. Receivables will grow over the one monthcredit period and will then remain stable with payments and new salesoffsetting one another.

11. The carrying costs are the average inventory times the cost ofcarrying an individual unit, so:

Carrying costs = (2,500/2)($9) = $11,250

CHAPTER 20 B-634

The order costs are the number of orders times the cost of an order,so:

Order costs = (52)($1,700) = $88,400

The economic order quantity is:

EOQ = [(2T × F)/CC]1/2 EOQ = [2(52)(2,500)($1,700)/$9]1/2 EOQ = 7,007.93

The firm’s policy is not optimal, since the carrying costs and theorder costs are not equal. The company should increase the order sizeand decrease the number of orders.

12. The carrying costs are the average inventory times the cost ofcarrying an individual unit, so:

Carrying costs = (300/2)($41) = $6,150

The order costs are the number of orders times the cost of an order,so:

Restocking costs = 52($95) = $4,940

The economic order quantity is:

EOQ = [(2T × F)/CC]1/2 EOQ = [2(52)(300)($95)/$41]1/2 EOQ = 268.87

The number of orders per year will be the total units sold per yeardivided by the EOQ, so:

Number of orders per year = 52(300)/268.87 Number of orders per year = 58.02

The firm’s policy is not optimal, since the carrying costs and theorder costs are not equal. The company should decrease the order sizeand increase the number of orders.

Intermediate

B-635 SOLUTIONS

13. The total carrying costs are:

Carrying costs = (Q/2) CC

where CC is the carrying cost per unit. The restocking costs are:

Restocking costs = F (T/Q)

Setting these equations equal to each other and solving for Q, wefind:

CC (Q/2) = F (T/Q)Q2 = 2 F T /CCQ = [2F T /CC]1/2 = EOQ

CHAPTER 20 B-636

14. The cash flow from either policy is:

Cash flow = (P – v)Q

So, the cash flows from the old policy are:

Cash flow from old policy = ($91 – 47)(3,850) Cash flow from old policy = $169,400

And the cash flow from the new policy would be:

Cash flow from new policy = ($94 – 47)(3,940) Cash flow from new policy = $185,180

So, the incremental cash flow would be:

Incremental cash flow = $185,180 – 169,400 Incremental cash flow = $15,780

The incremental cash flow is a perpetuity. The cost of initiating thenew policy is:

Cost of new policy = –[PQ + v(Q – Q)]

So, the NPV of the decision to change credit policies is:

NPV = –[($94)(3,850) + ($47)(3,940 – 3,850)] + $15,780/.025 NPV = $276,620

15. The cash flow from the old policy is:

Cash flow from old policy = ($290 – 230)(1,105) Cash flow from old policy = $66,300

And the cash flow from the new policy will be:

Cash flow from new policy = ($295 – 234)(1,125) Cash flow from new policy = $68,625

The incremental cash flow, which is a perpetuity, is the differencebetween the old policy cash flows and the new policy cash flows, so:

Incremental cash flow = $66,300 – 68,625

B-637 SOLUTIONS

Incremental cash flow = $2,325

CHAPTER 20 B-638

The cost of switching credit policies is:

Cost of new policy = –[PQ + Q(v – v) + v(Q – Q)]

In this cost equation, we need to account for the increased variablecost for all units produced. This includes the units we already sell,plus the increased variable costs for the incremental units. So, theNPV of switching credit policies is:

NPV = –[($290)(1,105) + (1,105)($234 – 230) + ($234)(1,125 – 1,120)]+ ($2,325/.0095) NPV = –$84,813.16

16. If the cost of subscribing to the credit agency is less than thesavings from collection of the bad debts, the company shouldsubscribe. The cost of the subscription is:

Cost of the subscription = $490 + $5(500) Cost of the subscription = $2,950

And the savings from having no bad debts will be:

Savings from not selling to bad credit risks = ($490)(500)(0.04) Savings from not selling to bad credit risks = $9,800

So, the company’s net savings will be:

Net savings = $9,800 – 2,950Net savings = $6,850

The company should subscribe to the credit agency.

Challenge

17. The cost of switching credit policies is:


And the cash flow from switching, which is a perpetuity, is:

Cash flow from new policy = [Q(P – v) – Q(P – v)]

B-639 SOLUTIONS

To find the breakeven quantity sold for switching credit policies, weset the NPV equal to zero and solve for Q. Doing so, we find:

NPV = 0 = –[($91)(3,850) + ($47)(Q – 3,850)] + [(Q)($94 – 47) –(3,850)($91 – 47)]/.0250 = –$350,350 – $47Q + $180,950 + $1,880Q – $6,776,000$1,833Q = $6,945,400 Q = 3,789.09

CHAPTER 20 B-640

18. We can use the equation for the NPV we constructed in Problem 17.Using the sales figure of 4,100 units and solving for P, we get:

NPV = 0 = [–($91)(3,850) – ($47)(4,100 – 3,850)] + [(P – 47)(4,100) –($91 – 47)(3,850)]/.0250 = –$350,350 – 11,750 + $164,000P – 7,708,000 – 6,776,000 $164,000P = $14,846,100 P = $90.53

19. From Problem 15, the incremental cash flow from the new credit policywill be:

Incremental cash flow = Q(P – v) – Q(P – v)

And the cost of the new policy is:


Setting the NPV equal to zero and solving for P, we get:

NPV = 0 = –[($290)(1,105) + ($234 – 230)(1,105) + ($234)(1,125 –1,105)] + [(1,125)(P – 234) –

(1,105)($290 – 230)]/.00950 = –[$320,450 + 4,420 + 4,680] + $118,421.05P – 27,710,526.32 –6,978,947.37$118,421.05P = $35,019,023.68 P = $295.72

20. Since the company sells 700 suits per week, and there are 52 weeksper year, the total number of suits sold is:

Total suits sold = 700 × 52 = 36,400

And, the EOQ is 500 suits, so the number of orders per year is:

Orders per year = 36,400 / 500 = 72.80

To determine the day when the next order is placed, we need to determine when the last order was placed. Since the suits arrived on Monday and there is a 3 day delay from the time the order was placed until the suits arrive, the last order was placed Friday. Since thereare five days between the orders, the next order will be placed on Wednesday

B-641 SOLUTIONS

Alternatively, we could consider that the store sells 100 suits perday (700 per week / 7 days). This implies that the store will be atthe safety stock of 100 suits on Saturday when it opens. Since thesuits must arrive before the store opens on Saturday, they should beordered 3 days prior to account for the delivery time, which againmeans the suits should be ordered in Wednesday.

CHAPTER 20 B-642

APPENDIX 20A

1. The cash flow from the old policy is the quantity sold times theprice, so:

Cash flow from old policy = 40,000($510) Cash flow from old policy = $20,400,000

The cash flow from the new policy is the quantity sold times the newprice, all times one minus the default rate, so:

Cash flow from new policy = 40,000($537)(1 – .03)Cash flow from new policy = $20,835,600

The incremental cash flow is the difference in the two cash flows,so:

Incremental cash flow = $20,835,600 – 20,400,000 Incremental cash flow = $435,600

The cash flows from the new policy are a perpetuity. The cost is theold cash flow, so the NPV of the decision to switch is:

NPV = –$20,400,000 + $435,600/.025 NPV = –$2,976,000

2. a. The old price as a percentage of the new price is:

$90/$91.84 = .98

So the discount is:

Discount = 1 – .98 = .02 or 2%

The credit terms will be:

Credit terms: 2/15, net 30

b. We are unable to determine for certain since no information isgiven concerning the percentage of customers who will take thediscount. However, the maximum receivables would occur if allcustomers took the credit, so:

B-643 SOLUTIONS

Receivables = 3,300($90) Receivables = $297,000 (at a maximum)

c. Since the quantity sold does not change, variable cost is the sameunder either plan.

CHAPTER 20 B-644

d. No, because:

d – = .02 – .11 d – = –.09 or –9%

Therefore the NPV will be negative. The NPV is:

NPV = –3,300($90) + (3,300)($91.84)(.02 – .11)/(.01) NPV = –$3,023,592

The breakeven credit price is:

P(1 + r)/(1 – ) = $90(1.01)/(.89) P = $102.13

This implies that the breakeven discount is:

Breakeven discount = 1 – ($90/$102.13) Breakeven discount = .1188 or 11.88%

The NPV at this discount rate is:

NPV = –3,300($90) + (3,300)($102.13)(.1188 – .11)/(.01) NPV 0

3. a. The cost of the credit policy switch is the quantity sold timesthe variable cost. The cash inflow is the price times the quantitysold, times one minus the default rate. This is a one-time, lumpsum, so we need to discount this value one period. Doing so, wefind the NPV is:

NPV = –15($760) + (1 – .2)(15)($1,140)/1.02 NPV = $2,011.76

The order should be taken since the NPV is positive.

b. To find the breakeven default rate, , we just need to set the NPVequal to zero and solve for the breakeven default rate. Doing so,we get:

NPV = 0 = –15($760) + (1 – )(12)($1,140)/1.02 = .3200 or 32.00%

B-645 SOLUTIONS

c. Effectively, the cash discount is:

Cash discount = ($1,140 – 1,090)/$1,140 Cash discount = .0439 or 4.39%

Since the discount rate is less than the default rate, creditshould not be granted. The firm would be better off taking the$1,090 up-front than taking an 80% chance of making $1,140.

CHAPTER 20 B-646

4. a. The cash discount is:

Cash discount = ($75 – 71)/$75 Cash discount = .0533 or 5.33%

The default probability is one minus the probability of payment,or:

Default probability = 1 – .90 Default probability = .10

Since the default probability is greater than the cash discount,credit should not be granted; the NPV of doing so is negative.

b. Due to the increase in both quantity sold and credit price whencredit is granted, an additional incremental cost is incurred of:

Additional cost = (6,200)($33 – 32) + (6,900 – 6,200)($33) Additional cost = $29,300

The breakeven price under these assumptions is:

NPV = 0 = –$29,300 – (6,200)($71) + {6,900[(1 – .10)P – $33] – 6,200($71 – 32)}/(1.00753 – 1)NPV = –$34,100 – 440,200 + 273,940.31P – 10,044,478.08 –10,666,468.16$21,185,246.24 = $273,940.31P P = $77.32

c. The credit report is an additional cost, so we have to include itin our analysis. The NPV when using the credit reports is:

NPV = 6,200(32) – .90(6,900)33 – 6,200(71) – 6,900($1.50) +{6,900[0.90(75 – 33) – 1.50]

– 6,200(71 – 32)}/(1.00753 – 1)NPV = $198,400 – 204,930 – 440,200 – 10,350 + 384,457.73NPV = –$74,622.27

The reports should not be purchased and credit should not begranted.

B-647 SOLUTIONS

5. We can express the old cash flow as:

Old cash flow = (P – v)Q

And the new cash flow will be:

New cash flow = (P – v)(1 – )Q + Q [(1 – )P – v]

So, the incremental cash flow is

Incremental cash flow = –(P – v)Q + (P – v)(1 – )Q + Q [(1 – )P –v]Incremental cash flow = (P – v)(Q – Q) + Q [(1 – )P – P]

Thus:

NPV = (P – v)(Q – Q) – PQ + [(P - v )(Q' - Q )+αQ' {(1 - π)P' - P}R ]

CHAPTER 21INTERNATIONAL CORPORATE FINANCEAnswers to Concepts Review and Critical Thinking Questions

1. a. The dollar is selling at a premium because it is more expensive inthe forward market than in the spot market (SFr 1.53 versus SFr1.50).

b. The franc is expected to depreciate relative to the dollar becauseit will take more francs to buy one dollar in the future than itdoes today.

c. Inflation in Switzerland is higher than in the United States, asare interest rates.

2. The exchange rate will increase, as it will take progressively morepesos to purchase a dollar. This is the relative PPP relationship.

3. a. The Australian dollar is expected to weaken relative to thedollar, because it will take more A$ in the future to buy onedollar than it does today.

b. The inflation rate in Australia is higher.

c. Nominal interest rates in Australia are higher; relative realrates in the two countries are the same.

4. A Yankee bond is most accurately described by d.

5. It depends. For example, if a country’s currency strengthens, importsbecome cheaper (good), but its exports become more expensive forothers to buy (bad). The reverse is true for currency depreciation.

6. Additional advantages include being closer to the final consumer and,thereby, saving on transportation, significantly lower wages, andless exposure to exchange rate risk. Disadvantages include politicalrisk and costs of supervising distant operations.

B-649 SOLUTIONS

7. One key thing to remember is that dividend payments are made in thehome currency. More generally, it may be that the owners of themultinational are primarily domestic and are ultimately concernedabout their wealth denominated in their home currency because, unlikea multinational, they are not internationally diversified.

8. a. False. If prices are rising faster in Great Britain, it will takemore pounds to buy the same amount of goods that one dollar canbuy; the pound will depreciate relative to the dollar.

b. False. The forward market would already reflect the projecteddeterioration of the euro relative to the dollar. Only if you feelthat there might be additional, unanticipated weakening of theeuro that isn’t reflected in forward rates today will the forwardhedge protect you against additional declines.

CHAPTER 21 B-650

c. True. The market would only be correct on average, while you wouldbe correct all the time.

9. a. American exporters: their situation in general improves because asale of the exported goods for a fixed number of euros will beworth more dollars.American importers: their situation in general worsens because thepurchase of the imported goods for a fixed number of euros willcost more in dollars.

b. American exporters: they would generally be better off if theBritish government’s intentions result in a strengthened pound.American importers: they would generally be worse off if the poundstrengthens.

c. American exporters: would generally be much worse off, because anextreme case of fiscal expansion like this one will make Americangoods prohibitively expensive to buy, or else Brazilian sales, iffixed in reais, would become worth an unacceptably low number ofdollars.American importers: would generally be much better off, becauseBrazilian goods will become much cheaper to purchase in dollars.

10. IRP is the most likely to hold because it presents the easiest andleast costly means to exploit any arbitrage opportunities. RelativePPP is least likely to hold since it depends on the absence of marketimperfections and frictions in order to hold strictly.



Basic

1. Using the quotes from the table, we get:

a. $100(€0.6509/$1) = €65.09

b. $1.5363

B-651 SOLUTIONS

c. €5M($1.5363/€) = $7,681,672

d. New Zealand dollar

e. Mexican peso

f. (P10.3584/$1)($1.5363/€1) = P15.9140/€

This is a cross rate.

g. Most valuable: Kuwait dinar = $3.7595Least valuable: Vietnam dong = $0.00006020

CHAPTER 21 B-652

2. a. You would prefer £100, since:

(£100)(£1/$0.5135) = $194.74

b. You would still prefer £100. Using the $/£ exchange rate and theSF/£ exchange rate to find the amount of Swiss francs £100 willbuy, we get:

(£100)($1.9474/£1)($/SF 0.9531) = SF 240.3227

c. Using the quotes in the book to find the SF/£ cross rate, we find:

(SF 0.9531/$)($1.9474/£1) = SF 2.0432/£1

The £/SF exchange rate is the inverse of the SF/£ exchange rate,so:

£1/SF 2.0432 = £0.4894/SF 1

3. a. F180 = ¥106.96 (per $). The yen is selling at a premium because itis more expensive in the forward market than in the spot market($0.009241 versus $0.0093493).

b. F90 = $0.9706/C$1. The dollar is selling at a discount because itis less expensive in the forward market than in the spot market($0.9716 versus $0.9706).

c. The value of the dollar will fall relative to the yen, since ittakes more dollars to buy one yen in the future than it doestoday. The value of the dollar will rise relative to the Canadiandollar, because it will take fewer dollars to buy one Canadiandollar in the future than it does today.

4. a. The U.S. dollar, since one Canadian dollar will buy:

(Can$1)/(Can$1.06/$1) = $0.9434

b. The cost in U.S. dollars is:

(Can$2.50)/(Can$1.06/$1) = $2.36

B-653 SOLUTIONS

Among the reasons that absolute PPP doesn’t hold are tariffs andother barriers to trade, transactions costs, taxes, and differenttastes.

c. The U.S. dollar is selling at a premium, because it is moreexpensive in the forward market than in the spot market (Can$1.11versus Can$1.06).

d. The Canadian dollar is expected to depreciate in value relative tothe dollar, because it takes more Canadian dollars to buy one U.S.dollar in the future than it does today.

e. Interest rates in the United States are probably lower than theyare in Canada.

5. a. The cross rate in ¥/£ terms is:

(¥112/$1)($1.93/£1) = ¥216.16/£1

CHAPTER 21 B-654

b. The yen is quoted too low relative to the pound. Take out a loanfor $1 and buy ¥112. Use the ¥112 to purchase pounds at the cross-rate, which will give you:

¥112(£1/¥209) = £0.53589

Use the pounds to buy back dollars and repay the loan. The cost torepay the loan will be:

£0.53589($1.93/£1) = $1.0343

You arbitrage profit is $0.0343 per dollar used.

6. We can rearrange the interest rate parity condition to answer thisquestion. The equation we will use is:

RFC = (Ft – S0)/S0 + RUS

Using this relationship, we find:

Great Britain: RFC = (£0.5204 – £0.5135)/£0.5135 + .022 = .0354 or3.54%

Japan: RFC = (¥106.96 – ¥108.21)/¥108.21 + .022 = .0104 or 1.04%

Switzerland: RFC = (SFr 1.0478 – SFr 1.0492)/SFr 1.0492 + .022 = .0207or 2.07%

7. If we invest in the U.S. for the next three months, we will have:

$30,000,000(1.0037)3 = $30,334,233.62

If we invest in Great Britain, we must exchange the dollars today forpounds, and exchange the pounds for dollars in three months. Aftermaking these transactions, the dollar amount we would have in threemonths would be:

($30,000,000)(£0.55/$1)(1.0051)3/(£0.56/$1) = $29,917,392.29

The company should invest in the U.S.

8. Using the relative purchasing power parity equation:

B-655 SOLUTIONS

Ft = S0 × [1 + (hFC – hUS)]t

We find:

Z2.26 = Z2.17[1 + (hFC – hUS)]3 hFC – hUS = (Z2.26/Z2.17)1/3 – 1 hFC – hUS = .0136 or 1.36%

Inflation in Poland is expected to exceed that in the U.S. by 1.36%over this period.

CHAPTER 21 B-656

9. The profit will be the quantity sold, times the sales price minus thecost of production. The production cost is in Singapore dollars, sowe must convert this to U.S. dollars. Doing so, we find that if theexchange rates stay the same, the profit will be:

Profit = 30,000[$150 – {(S$204.70)/(S$1.3803/$1)}] Profit = $50,967.18

If the exchange rate rises, we must adjust the cost by the increasedexchange rate, so:

Profit = 30,000[$150 – {(S$204.70)/(1.1(S$1.3803/$1))}] Profit = $455,424.71

If the exchange rate falls, we must adjust the cost by the decreasedexchange rate, so:

Profit = 30,000[$150 – {(S$204.70)/(0.9(S$1.3803/$1))}] Profit = –$443,369.80

To calculate the breakeven change in the exchange rate, we need tofind the exchange rate that make the cost in Singapore dollars equalto the selling price in U.S. dollars, so:

$150 = S$204.70/ST ST = S$1.3647/$1 ST = –0.0113 or –1.13% decline

10. a. If IRP holds, then:

F180 = (Kr 5.15)[1 + (.057 – .038)]1/2 F180 = Kr 5.1987

Since given F180 is Kr5.22, an arbitrage opportunity exists; theforward premium is too high. Borrow Kr1 today at 5.7% interest.Agree to a 180-day forward contract at Kr 5.22. Convert the loanproceeds into dollars:

Kr 1 ($1/Kr 5.15) = $0.19417

Invest these dollars at 3.8%, ending up with $0.19778. Convert thedollars back into krone as

B-657 SOLUTIONS

$0.19778(Kr 5.22/$1) = Kr 1.03241

Repay the Kr 1 loan, ending with a profit of:

Kr1.03241 – Kr1.02771 = Kr 0.0469

b. To find the forward rate that eliminates arbitrage, we use theinterest rate parity condition, so:

F180 = (Kr 5.15)[1 + (.057 – .038)]1/2 F180 = Kr 5.1987

CHAPTER 21 B-658

11. The international Fisher effect states that the real interest rateacross countries is equal. We can rearrange the international Fishereffect as follows to answer this question:

RUS – hUS = RFC – hFC hFC = RFC + hUS – RUS

a. hAUS = .04 + .039 – .058 hAUS = .021 or 2.1%

b. hCAN = .07 + .039 – .058 hCAN = .051 or 5.1%

c. hTAI = .09 + .039 – .058 hTAI = .071 or 7.1%

12. a. The yen is expected to get weaker, since it will take more yen tobuy one dollar in the future than it does today.

b. hUS – hJAP (¥116.03 – ¥114.32)/¥114.32 hUS – hJAP = 0.0150 or 1.50%

(1 + .0150)4 – 1 = 0.0612 or 6.12%

The approximate inflation differential between the U.S. and Japanis 6.12% annually.

13. We need to find the change in the exchange rate over time so we needto use the interest rate parity relationship:

Ft = S0 × [1 + (RFC – RUS)]t

Using this relationship, we find the exchange rate in one year shouldbe:

F1 = 152.93[1 + (.086 – .049)]1 F1 = HUF 158.59

The exchange rate in two years should be:

F2 = 152.93[1 + (.086 – .049)]2 F2 = HUF 164.46

B-659 SOLUTIONS

And the exchange rate in five years should be:

F5 = 152.93[1 + (.086 – .049)]5 F5 = HUF 183.39

CHAPTER 21 B-660

Intermediate

14. First, we need to forecast the future spot rate for each of the nextthree years. From interest rate and purchasing power parity, theexpected exchange rate is:

E(ST) = [(1 + RUS) / (1 + RFC)]T S0

So:

E(S1) = (1.0480 / 1.0410)1 $1.28/€ = $1.2886/€E(S2) = (1.0480 / 1.0410)2 $1.28/€ = $1.2973/€E(S3) = (1.0480 / 1.0410)3 $1.28/€ = $1.3060/€

Now we can use these future spot rates to find the dollar cash flows.The dollar cash flow each year will be:

Year 0 cash flow = –€$14,000,000($1.28/€) = –$17,920,000.00Year 1 cash flow = €$2,100,000($1.2886/€) = $2,706,074.93Year 2 cash flow = €$3,400,000($1.2973/€) = $4,410,725.12Year 3 cash flow = (€4,300,000 + 9,600,000)($1.3060/€) =

$18,153,335.30

And the NPV of the project will be:

NPV = –$17,920,000 + $2,706,074.93/1.13 + $4,410,725.12/1.132 + $18,153,335.30/1.133

NPV = $510,173.30

15. a. Implicitly, it is assumed that interest rates won’t change overthe life of the project, but the exchange rate is projected todecline because the Euroswiss rate is lower than the Eurodollarrate.

b. We can use relative purchasing power parity to calculate thedollar cash flows at each time. The equation is:

E[St] = (SFr 1.09)[1 + (.07 – .08)]t E[St] = 1.09(.99)t

So, the cash flows each year in U.S. dollar terms will be:

B-661 SOLUTIONS

t SFr E[St] US$0 –24.0M 1.0900 –$22,018,348.631 +6.6M 1.0791 $6,116,207.952 +6.6M 1.0683 $6,177,987.833 +6.6M 1.0576 $6,240,391.754 +6.6M 1.0470 $6,303,426.015 +6.6M 1.0366 $6,367,096.98

CHAPTER 21 B-662

And the NPV is:

NPV = –$22,018,348.62 + $6,116,207.95/1.12 + $6,177,987.83/1.122 +$6,240,391.75 /1.123 +

$6,303,426.01/1.124 + $6,367,096.98/1.125

NPV = $428,195.99

c. Rearranging the relative purchasing power parity equation to findthe required return in Swiss francs, we get:

RSFr = 1.12[1 + (.07 – .08)] – 1 RSFr = 10.88%

So the NPV in Swiss francs is:

NPV = –SFr 24.0M + SFr 6.6M(PVIFA10.88%,5)NPV = SFr 466,733.63

Converting the NPV to dollars at the spot rate, we get the NPV inU.S. dollars as:

NPV = (SFr 466,733.63)($1/SFr 1.09) NPV = $428,195.99

16. a. To construct the balance sheet in dollars, we need to convert theaccount balances to dollars. At the current exchange rate, we get:

Assets = solaris 23,000 / ($ / solaris 1.20) = $19,166.67Debt = solaris 9,000 / ($ / solaris 1.20) = $7,500.00Equity = solaris 14,000 / ($ / solaris 1.20) = $19,166.67

b. In one year, if the exchange rate is solaris 1.40/$, the accountswill be:


b. If the exchange rate is solaris 1.12/$, the accounts will be:


B-663 SOLUTIONS

CHAPTER 21 B-664

Challenge

17. First, we need to construct the end of year balance sheet in solaris.Since the company has retained earnings, the equity account willincrease, which necessarily implies the assets will also increase bythe same amount. So, the balance sheet at the end of the year insolaris will be:

Balance Sheet (solaris) Liabilities $9,000.00 Equity 15,250.00

Assets $24,250.00 Total liabilities &equity $24,250.00

Now we need to convert the balance sheet accounts to dollars, whichgives us:


18. a. The domestic Fisher effect is:

1 + RUS = (1 + rUS)(1 + hUS)1 + rUS = (1 + RUS)/(1 + hUS)

This relationship must hold for any country, that is:

1 + rFC = (1 + RFC)/(1 + hFC)

The international Fisher effect states that real rates are equalacross countries, so:

1 + rUS = (1 + RUS)/(1 + hUS) = (1 + RFC)/(1 + hFC) = 1 + rFC

b. The exact form of unbiased interest rate parity is:

E[St] = Ft = S0 [(1 + RFC)/(1 + RUS)]t

c. The exact form for relative PPP is:

E[St] = S0 [(1 + hFC)/(1 + hUS)]t

B-665 SOLUTIONS

d. For the home currency approach, we calculate the expected currencyspot rate at time t as:

E[St] = (€0.5)[1.07/1.05]t = (€0.5)(1.019)t

We then convert the euro cash flows using this equation at everytime, and find the present value. Doing so, we find:

NPV = – [€2M/(€0.5)] + {€0.9M/[1.019(€0.5)]}/1.1 +{€0.9M/[1.0192(€0.5)]}/1.12 +

{€0.9M/[1.0193(€0.5/$1)]}/1.13 NPV = $316,230.72

For the foreign currency approach we first find the return in theeuros as:

RFC = 1.10(1.07/1.05) – 1 = 0.121

Next, we find the NPV in euros as:

NPV = – €2M + (€0.9M)/1.121 + (€0.9M)/1.1212 + (€0.9M)/1.1213 =€158,115.36

And finally, we convert the euros to dollars at the currentexchange rate, which is:

NPV ($) = €158,115.36 /(€0.5/$1) = $316,230.72

CHAPTER 22BEHAVIORAL FINANCE: IMPLICATIONSFOR FINANCIAL MANAGEMENT


1. The least likely limit to arbitrage is firm-specific risk. Forexample, in the 3Com/Palm case, the stocks are perfect substitutesafter accounting for the exchange ratio. An investor could investin a risk neutral portfolio by purchasing the underpriced asset andselling the overpriced asset. When the prices of the assets revertto an equilibrium, the positions could be closed.

2. Overconfidence is the belief that one’s abilities are greater thanthey are. An overconfident financial manager could believe thatthey are correct in the face of evidence to the contrary. Forexample, the financial manager could believe that a new productwill be a great success (or failure) even though market researchpoints to the contrary. This could mean that the company invests orinvests too much in the new product or misses out on the newinvestment, an opportunity cost. In each case, shareholder value isnot maximized.

3. Frame dependence is the argument that an investor’s choice isdependent on the way the question is posed. An investor can frame adecision problem in broad terms (like wealth) or in narrow terms(like gains and losses). Broad and narrow frames often lead theinvestor to make different choices. While it is human nature to usea narrow frame (like gains and losses), doing so can lead toirrational decisions. Using broad frames, like overall wealth,results in better investment decisions.

4. A noise trader is someone whose trades are not based on informationor financially meaningful analysis. Noise traders could, inprinciple, act together to worsen a mispricing in the short-run.Noise trader risk is important because the worsening of a

B-667 SOLUTIONS

mispricing could force the arbitrageur to liquidate early andsustain steep losses.

5. As long as it is a fair coin the probability in both cases is 50percent as coins have no memory. Although many believe theprobability of flipping a tail would be greater given the long runof heads, this is an example of the gambler’s fallacy.

6. Taken at face value, this fact suggests that markets have becomemore efficient. The increasing ease with which information isavailable over the Internet lends strength to this conclusion. Onthe other hand, during this particular period, large-capitalizationgrowth stocks were the top performers. Value-weighted indexes suchas the S&P 500 are naturally concentrated in such stocks, thusmaking them especially hard to beat during this period. So, it maybe that the dismal record compiled by the pros is just a matter ofbad luck or benchmark error.

CHAPTER 22 B-668

7. The statement is false because every investor has a different riskpreference. Although the expected return from every well-diversified portfolio is the same after adjusting for risk,investors still need to choose funds that are consistent with theirparticular risk level.

9. Behavioral finance attempts to explain both the 1987 stock marketcrash and the Internet bubble by changes in investor sentiment andpsychology. These changes can lead to non-random price behavior.

9. Behavioral finance states that the market is not efficient. Adherents argue that: 1) Investors are not rational. 2) Deviations from rationality are similar across investors. 3) Arbitrage, being costly, will not eliminate inefficiencies.

10. Frame dependence means that the decision made is affected by theway in which the question is asked. In this example, consider thatthe $78 is a sunk cost. You will not get this money back whether ornot you accept the deal. In this case, the values from the deal area gain of $78 with 20 percent probability or a loss of $22 with an80 percent probability. The expected value of the deal is $78(.20)– $22(.80) = –$2. Notice this is the same as the difference betweenthe loss of $78 and the expected loss of $80 which we calculatedusing no net loss and a loss of $100.

CHAPTER 23RISK MANAGEMENT: AN INTRODUCTION TO FINANCIAL ENGINEERING


1. Since the firm is selling futures, it wants to be able to deliver thelumber; therefore, it is a supplier. Since a decline in lumber priceswould reduce the income of a lumber supplier, it has hedged its pricerisk by selling lumber futures. Losses in the spot market due to afall in lumber prices are offset by gains on the short position inlumber futures.

2. Buying call options gives the firm the right to purchase porkbellies; therefore, it must be a consumer of pork bellies. While arise in pork belly prices is bad for the consumer, this risk isoffset by the gain on the call options; if pork belly prices actuallydecline, the consumer enjoys lower costs, while the call optionexpires worthless.

3. Forward contracts are usually designed by the parties involved fortheir specific needs and are rarely sold in the secondary market;forwards are somewhat customized financial contracts. All gains andlosses on the forward position are settled at the maturity date.Futures contracts are standardized to facilitate their liquidity andto allow them to be effectively traded on organized futuresexchanges. Gains and losses on futures are marked-to-market daily.The default risk is greatly reduced with futures, since the exchangeacts as an intermediary between the two parties, guaranteeingperformance; default risk is also reduced because the dailysettlement procedure keeps large loss positions from accumulating.You might prefer to use forwards instead of futures if your hedgingneeds were different from the standard contract size and maturitydates offered by the futures contract.

4. The firm is hurt by declining oil prices, so it should sell oilfutures contracts. The firm may not be able to create a perfect hedge

CHAPTER 23 B-670

because the quantity of oil it needs to hedge doesn’t match thestandard contract size on crude oil futures, or perhaps the exactsettlement date the company requires isn’t available on these futures(exposing the firm to basis risk), or maybe the firm produces adifferent grade of crude oil than that specified for delivery in thefutures contract.

5. The firm is directly exposed to fluctuations in the price of naturalgas, since it is a natural gas user. In addition, the firm isindirectly exposed to fluctuations in the price of oil. If oilbecomes less expensive relative to natural gas, its competitors willenjoy a cost advantage relative to the firm.

6. Buying the call options is a form of insurance policy for the firm.If cotton prices rise, the firm is protected by the call, while ifprices actually decline, they can just allow the call to expireworthless. However, options hedges are costly because of the initialpremium that must be paid. The futures contract can be entered intoat no initial cost, with the disadvantage that the firm is locking inone price for cotton; it can’t profit from cotton price declines.

7. The put option on the bond gives the owner the right to sell the bondat the option’s strike price. If bond prices decline, the owner ofthe put option profits. However, since bond prices and interest ratesmove in opposite directions, if the put owner profits from a declinein bond prices, he would also profit from a rise in interest rates.Hence, a call option on interest rates is conceptually the same thingas a put option on bond prices.

8. The company would like to lock in the current low rates, or at leastbe protected from a rise in rates, allowing for the possibility ofbenefit if rates actually fall. The former hedge could be implementedby selling bond futures; the latter could be implemented by buyingput options on bond prices or buying call options on interest rates.

9. A swap contract is an agreement between parties to exchange assetsover several time intervals in the future. The swap contract isusually an exchange of cash flows, but not necessarily so. Since aforward contract is also an agreement between parties to exchangeassets in the future, but at a single point in time, a swap can beviewed as a series of forward contracts with different settlementdates. The firm participating in the swap agreement is exposed to thedefault risk of the dealer, in that the dealer may not make the cash

B-671 SOLUTIONS

flow payments called for in the contract. The dealer faces the samerisk from the contracting party, but can more easily hedge itsdefault risk by entering into an offsetting swap agreement withanother party.

10. The firm will borrow at a fixed rate of interest, receive fixed ratepayments from the dealer as part of the swap agreement, and makefloating rate payments back to the dealer; the net position of thefirm is that it has effectively borrowed at floating rates.

11. Transactions exposure is the short-term exposure due to uncertainprices in the near future. Economic exposure is the long-termexposure due to changes in overall economic conditions. There are avariety of instruments available to hedge transaction exposure, butvery few long-term hedging instruments exist. It is much moredifficult to hedge against economic exposure, since fundamentalchanges in the business generally must be made to offset long-runchanges in the economic environment.

12. The risk is that the dollar will strengthen relative to the yen,since the fixed yen payments in the future will be worth fewerdollars. Since this implies a decline in the $/¥ exchange rate, thefirm should sell yen futures.

13. a. Buy oil and natural gas futures contracts, since these areprobably your primary resource costs. If it is a coal-fired plant,a cross-hedge might be implemented by selling natural gas futures,since coal and natural gas prices are somewhat negatively relatedin the market; coal and natural gas are somewhat substitutable.

b. Buy sugar and cocoa futures, since these are probably your primarycommodity inputs.

c. Sell corn futures, since a record harvest implies low corn prices.d. Buy silver and platinum futures, since these are primary commodity

inputs required in the manufacture of photographic film.e. Sell natural gas futures, since excess supply in the market

implies low prices.f. Assuming the bank doesn’t resell its mortgage portfolio in the

secondary market, buy bond futures.g. Sell stock index futures, using an index most closely associated

with the stocks in your fund, such as the S&P 100 or the MajorMarket Index for large blue-chip stocks.

CHAPTER 23 B-672

h. Buy Swiss franc futures, since the risk is that the dollar willweaken relative to the franc over the next six month, whichimplies a rise in the $/SFr exchange rate.

i. Sell euro futures, since the risk is that the dollar willstrengthen relative to the euro over the next three months, whichimplies a decline in the $/€ exchange rate.

14. Sysco must have felt that the combination of fixed plus swap wouldresult in an overall better rate. In other words, variable rateavailable via a swap may have been more attractive than the rateavailable from issuing a floating-rate bond.



Basic

1. The initial price is $3,122 per metric ton and each contract is for10 metric tons, so the initial contract value is:

Initial contract value = ($3,122 per ton)(10 tons per contract) =$31,220

And the final contract value is:

Final contract value = ($3,081 per ton)(10 tons per contract) =$30,810

You will have a loss on this futures position of:

Loss on futures contract = $31,220 – 30,810 = $410

2. The price quote is $16.607 per ounce and each contract is for 5,000ounces, so the initial contract value is:

Initial contract value = ($16.607 per oz.)(5,000 oz. per contract) =$83,035

At a final price of $16.81 per ounce, the value of the position is:

B-673 SOLUTIONS

Final contract value = ($16.81 per oz.)(5,000 oz. per contract) =$84,050

Since this is a short position, there is a net loss of:

$84,050 – 83,035 = $1,015 per contract

Since you sold five contracts, the net loss is:

Net loss = 5($1,015) = $5,075

CHAPTER 23 B-674

At a final price of $16.32 per ounce, the value of the position is:

Final contract value = ($16.32 per oz.)(5,000 oz. per contract) =$81,600

Since this is a short position, there is a net gain of $83,035 –81,600 = $1,435

Since you sold five contracts, the net gain is:

Net gain = 5($1,435) = $7,175

With a short position, you make a profit when the price falls, andincur a loss when the price rises.

3. The price quote is $0.0835 per pound and each contract is for 15,000pounds, so the cost per contract is:

Cost = ($0.0835 per pound)(15,000 pounds per contract) = $1,252.50

If the price of orange juice at expiration is $1.29 per pound, thecall is out of the money since the strike price is above the spotprice. The contracts will expire worthless, so your loss will be theinitial investment of $1,253.

If orange juice prices at contract expiration are $1.67 per pound,the call is in the money since the price per pound is above thestrike price. The payoff on your position is the current price minusthe strike price, times the 15,000 pounds per contract, or:

Payoff = ($1.67 – 1.40)(15,000) = $4,050

And the profit is the payoff minus the initial cost of the contract,or:

Profit = $4,050 – 1,252.50 = $2,797.50

4. The call options give the manager the right to purchase oil futurescontracts at a futures price of $140 per barrel. The manager willexercise the option if the price rises above $140. Selling putoptions obligates the manager to buy oil futures contracts at afutures price of $140 per barrel. The put holder will exercise theoption if the price falls below $140. The payoffs per barrel are:

B-675 SOLUTIONS

Oil futures price: $135 $137 $140 $143 $145Value of call option position: 0 0 0 3 5Value of put option position: –5 –3 0 0 0 Total value: –$5 –$3 $0 $3 $5

The payoff profile is identical to that of a forward contract with a$140 strike price.

5. The price quote is $0.1880 per pound and each contract is for 15,000pounds, so the cost per contract is:

Cost = ($0.1880 per pound)(15,000 pounds per contract) = $2,820.00

CHAPTER 23 B-676

If the price of orange juice at expiration is $1.14 per pound, theput is in the money since the strike price is greater than the spotprice. The payoff on your position is the strike price minus thecurrent price, times the 15,000 pounds per contract, or:

Payoff = ($1.35 – 1.14)(15,000) = $3,150

And the profit is the payoff minus the initial cost of the contract,or:

Profit = $3,150 – 2,820 = $330

If the price of orange juice at expiration is $1.47 per pound, theput is out of the money since the strike price is less than the spotprice. The contracts will expire worthless, so your loss will be theinitial investment of $2,820.

Intermediate

6. a. You’re concerned about a rise in corn prices, so you would buyDecember contracts. Since each contract is for 5,000 bushels, thenumber of contracts you would need to buy is:

Number of contracts to buy = 105,000/5,000 = 21

By doing so, you’re effectively locking in the settle price inDecember, 2008 of $7.65 per bushel of corn, or:

Total price for 105,000 bushels = 21($7.65)(5,000) = $803,250

b. If the price of corn at expiration is $7.41 per bushel, the valueof you futures position is:

Value of future position = 21($7.41)(5,000) = $778,050

Ignoring any transaction costs, your loss on the futures positionwill be:

Loss = $803,250 – 778,050 = $25,200

While the price of the corn your firm needs has become $25,200less expensive since June, your loss from the futures position hasnetted out this lower cost.

B-677 SOLUTIONS

7. a. XYZ has a comparative advantage relative to ABC in borrowing atfixed interest rates, while ABC has a comparative advantagerelative to XYZ in borrowing at floating interest rates. Since thespread between ABC and XYZ’s fixed rate costs is only 1%, whiletheir differential is 2% in floating rate markets, there is anopportunity for a 3% total gain by entering into a fixed forfloating rate swap agreement.

CHAPTER 23 B-678

b. If the swap dealer must capture 2% of the available gain, there is1% left for ABC and XYZ. Any division of that gain is feasible; inan actual swap deal, the divisions would probably be negotiated bythe dealer. One possible combination is ½% for ABC and ½% for XYZ:

Challenge

8. The financial engineer can replicate the payoffs of owning a putoption by selling a forward contract and buying a call. For example,suppose the forward contract has a settle price of $50 and theexercise price of the call is also $50. The payoffs below show thatthe position is the same as owning a put with an exercise price of$50:

Price of coal: $40 $45 $50 $55 $60Value of call option position: 0 0 0 5 10Value of forward position: 10 5 0 –5 – 10Total value: $10 $5 $0 $0 $0

Value of put position: $10 $5 $0 $0 $0

The payoffs for the combined position are exactly the same as thoseof owning a put. This means that, in general, the relationshipbetween puts, calls, and forwards must be such that the cost of thetwo strategies will be the same, or an arbitrage opportunity exists.In general, given any two of the instruments, the third can besynthesized.

CHAPTER 24OPTIONS AND CORPORATE FINANCE


1. A call option confers the right, without the obligation, to buy anasset at a given price on or before a given date. A put optionconfers the right, without the obligation, to sell an asset at agiven price on or before a given date. You would buy a call option ifyou expect the price of the asset to increase. You would buy a putoption if you expect the price of the asset to decrease. A calloption has unlimited potential profit, while a put option has limitedpotential profit; the underlying asset’s price cannot be less thanzero.

2. a. The buyer of a call option pays money for the right to buy....b. The buyer of a put option pays money for the right to sell....c. The seller of a call option receives money for the obligation to

sell....d. The seller of a put option receives money for the obligation to

buy....

3. The intrinsic value of a call option is Max [S – E,0]. It is thevalue of the option at expiration.

4. The value of a put option at expiration is Max[E – S,0]. Bydefinition, the intrinsic value of an option is its value atexpiration, so Max[E – S,0] is the intrinsic value of a put option.

5. The call is selling for less than its intrinsic value; an arbitrageopportunity exists. Buy the call for $10, exercise the call by paying$35 in return for a share of stock, and sell the stock for $50.You’ve made a riskless $5 profit.

6. The prices of both the call and the put option should increase. Thehigher level of downside risk still results in an option price ofzero, but the upside potential is greater since there is a higherprobability that the asset will finish in the money.

CHAPTER 24 B-680

7. False. The value of a call option depends on the total variance ofthe underlying asset, not just the systematic variance.

8. The call option will sell for more since it provides an unlimitedprofit opportunity, while the potential profit from the put islimited (the stock price cannot fall below zero).

9. The value of a call option will increase, and the value of a putoption will decrease.

10. The reason they don’t show up is that the U.S. government uses cashaccounting; i.e., only actual cash inflows and outflows are counted,not contingent cash flows. From a political perspective, debtguarantees would make the deficit larger, so that is another reasonnot to count them! Whether they should be included depends onwhether we feel cash accounting is appropriate or not, but thesecontingent liabilities should be measured and reported. Theycurrently are not, at least not in a systematic fashion.

11. The option to abandon reflects our ability to shut down a project ifit is losing money. Since this option acts to limit losses, we willunderestimate NPV if we ignore it.

B-681 SOLUTIONS

12. The option to expand reflects our ability to increase production ifthe new product sells more than we initially expected. Since thisoption increases the potential future cash flows beyond our initialestimate, we will underestimate NPV if we ignore it.

13. This is a good example of the option to expand.

14. With oil, for example, we can simply stop pumping if prices drop toofar, and we can do so quickly. The oil itself is not affected; itjust sits in the ground until prices rise to a point where pumping isprofitable. Given the volatility of natural resource prices, theoption to suspend output is very valuable.

15. There are two possible benefits. First, awarding employee stockoptions may better align the interests of the employees with theinterests of the stockholders, lowering agency costs. Secondly, ifthe company has little cash available to pay top employees, employeestock options may help attract qualified employees for less pay.



Basic

1. a. The value of the call is the stock price minus the present valueof the exercise price, so:

C0 = $66 – [$55/1.056] = $13.92

The intrinsic value is the amount by which the stock price exceedsthe exercise price of the call, so the intrinsic value is $11.

b. The value of the call is the stock price minus the present valueof the exercise price, so:

C0 = $66 – [$45/1.056] = $23.39

The intrinsic value is the amount by which the stock price exceedsthe exercise price of the call, so the intrinsic value is $21.

CHAPTER 24 B-682

c. The value of the put option is $0 since there is no possibilitythat the put will finish in the money. The intrinsic value is also$0.

2. a. The calls are in the money. The intrinsic value of the calls is$5.

b. The puts are out of the money. The intrinsic value of the puts is$0.

B-683 SOLUTIONS

c. The Mar call and the Oct put are mispriced. The call is mispricedbecause it is selling for less than its intrinsic value. If theoption expired today, the arbitrage strategy would be to buy thecall for $2.80, exercise it and pay $80 for a share of stock, andsell the stock for $85. A riskless profit of $2.20 results. TheOctober put is mispriced because it sells for less than the Julyput. To take advantage of this, sell the July put for $3.90 andbuy the October put for $3.65, for a cash inflow of $0.25. Theexposure of the short position is completely covered by the longposition in the October put, with a positive cash inflow today.

3. a. Each contract is for 100 shares, so the total cost is:

Cost = 10(100 shares/contract)($3.23) Cost = $3,230

b. If the stock price at expiration is $134, the payoff is:

Payoff = 10(100)($134 – 120) Payoff = $14,000

If the stock price at expiration is $126, the payoff is:

Payoff = 10(100)($126 – 120) Payoff = $6,000

c. Remembering that each contract is for 100 shares of stock, thecost is:

Cost = 10(100)($9.10) Cost = $9,100

The maximum gain on the put option would occur if the stock pricegoes to $0. We also need to subtract the initial cost, so:

Maximum gain = 10(100)($120) – $9,100 Maximum gain = $110,900

If the stock price at expiration is $109, the position will beworth:

Position value = 10(100)($120 – 109) Position value = $11,000

CHAPTER 24 B-684

And your profit will be:

Profit = $11,000 – 9,100Profit = $1,900

d. At a stock price of $108 the put is in the money. As the writeryou will lose:

Net gain(loss) = $9,100 – 10(100)($120 – 108) Net gain(loss) = –$2,900

B-685 SOLUTIONS

At a stock price of $132 the put is out of the money, so thewriter will make the initial cost:

Net gain = $9,100

At the breakeven, you would recover the initial cost of $9,100,so:

$9,100 = 10(100)($120 – ST) ST = $110.90

For terminal stock prices above $110.90, the writer of the putoption makes a net profit (ignoring transaction costs and theeffects of the time value of money).


C0 = $70 – 65/1.05 C0 = $8.10

b. Using the equation presented in the text to prevent arbitrage, wefind the value of the call is:

$70 = [($86 – 62)/($86 – 75)]C0 + $62/1.05 C0 = $5.02


C0 = $85 – $65/1.06 C0 = $23.68

b. Using the equation presented in the text to prevent arbitrage, wefind the value of the call is:

$85 = [($95 – 75)/($95 – 70)]C0 + $75/1.06C0 = $17.81

6. Each option contract is for 100 shares of stock, so the price of acall on one share is:

C0 = $1,300/100 shares per contract

CHAPTER 24 B-686

C0 = $13

Using the no arbitrage model, we find that the price of the stock is:

S0 = $13[($67 – 48)/($67 – 60)] + $48/1.08 S0 = $78.73

7. a. The equity can be valued as a call option on the firm with anexercise price equal to the value of the debt, so:

E0 = $1,050 – [$1,000/1.07] E0 = $115.42

B-687 SOLUTIONS

b. The current value of debt is the value of the firm’s assets minusthe value of the equity, so:

D0 = $1,050 – 115.42 D0 = $934.58

We can use the face value of the debt and the current market valueof the debt to find the interest rate, so:

Interest rate = [$1,000/$934.58] – 1 Interest rate = .07 or 7%

c. The value of the equity will increase. The debt then requires ahigher return; therefore the present value of the debt is lesswhile the value of the firm does not change.

8. a. Using the no arbitrage valuation model, we can use the currentmarket value of the firm as the stock price, and the par value ofthe bond as the strike price to value the equity. Doing so, weget:

$1,140 = [($1,430 – 920)/($1,430 – 1,000)]E0 + [$920/1.06] E0 = $229.40

The current value of the debt is the value of the firm’s assetsminus the value of the equity, so:

D0 = $1,140 – 229.40 D0 = $910.60

b. Using the no arbitrage model as in part a, we get:

$1,140 = [($1,600 – 800)/($1,600 – 1,000)]E0 + [$800/1.06] E0 = $288.96

The stockholders will prefer the new asset structure because theirpotential gain increases while their maximum potential lossremains unchanged.

9. The conversion ratio is the par value divided by the conversionprice, so:

Conversion ratio = $1,000/$35

CHAPTER 24 B-688

Conversion ratio = 28.57

The conversion value is the conversion ratio times the stock price,so:

Conversion value = 28.57($46) Conversion value = $1,314.29

B-689 SOLUTIONS

10. a. The minimum bond price is the greater of the straight bond valueor the conversion price. The straight bond value is:

Straight bond value = $26(PVIFA3.5%,60) + $1,000/1.03560 Straight bond value = $775.50

The conversion ratio is the par value divided by the conversionprice, so:

Conversion ratio = $1,000/$55 Conversion ratio = 18.18

The conversion value is the conversion ratio times the stockprice, so:

Conversion value = 18.18($41) Conversion value = $745.45

The minimum value for this bond is the straight bond value of$775.50.

b. The option embedded in the bond adds the extra value.

11. a. The minimum bond price is the greater of the straight bond valueor the conversion value. The straight bond value is:


The conversion ratio is the par value divided by the conversionprice, so:

Conversion ratio = $1,000/$45 Conversion ratio = 22.22

The conversion price is the conversion ratio times the stockprice, so:

Conversion value = 22.22($39) Conversion value = $866.67

The minimum value for this bond is the convertible floor value of$866.67.

CHAPTER 24 B-690

b. The conversion premium is the difference between the current stockprice and conversion price, divided by the current stock price,so:

Conversion premium = ($45 – 39)/$39 = .1538 or 15.38%

B-691 SOLUTIONS

12. The value of the bond without warrants is:

Straight bond value = $45(PVIFA7%,15) + $1,000/1.0715 Straight bond value = $772.30

The value of the warrants is the selling price of the bond minus thevalue of the bond without warrants, so:

Total warrant value = $1,000 – 772.30 Total warrant value = $227.70

Since the bond has 25 warrants attached, the price of each warrantis:

Price of one warrant = $227.70/25 Price of one warrant = $9.11

13. If we purchase the machine today, the NPV is the cost plus the presentvalue of the increased cash flows, so:

NPV0 = –$1,800,000 + $320,000(PVIFA12%,10) NPV0 = $8,071.37

We should not necessarily purchase the machine today, but rather wewould want to purchase the machine when the NPV is the highest. So, weneed to calculate the NPV each year. The NPV each year will be thecost plus the present value of the increased cash savings. We must becareful however. In order to make the correct decision, the NPV foreach year must be taken to a common date. We will discount all of theNPVs to today. Doing so, we get:

Year 1: NPV1 = [–$1,680,000 + $320,000(PVIFA12%,9)] / 1.12 NPV1 = $22,357.08



Year 4: NPV4 = [–$1,320,000 + $320,000(PVIFA12%,6)] / 1.124 NPV4 = –$2,764.29

CHAPTER 24 B-692



The company should purchase the machine 2 years from now when the NPVis the highest.

B-693 SOLUTIONS

Intermediate

14. a. The base-case NPV is:

NPV = –$2,300,000 + $510,000(PVIFA14%,10) NPV = $360,218.98

b. We would abandon the project if the cash flow from selling theequipment is greater than the present value of the future cashflows. We need to find the sale quantity where the two are equal,so:

$1,500,000 = ($68)Q(PVIFA14%,9) Q = $1,500,000/[$68(4.9464)] Q = 4,460

Abandon the project if Q < 4,460 units, because the NPV ofabandoning the project is greater than the NPV of the future cashflows.

c. The $1,500,000 is the market value of the project. If you continuewith the project in one year, you forego the $1,500,000 that couldhave been used for something else.

15. a. If the project is a success, present value of the future cashflows will be:

PV future CFs = $68(9,500)(PVIFA14%,9) PV future CFs = $3,195,356.21

From the previous question, if the quantity sold is 4,000, wewould abandon the project, and the cash flow would be $1,500,000.Since the project has an equal likelihood of success or failure inone year, the expected value of the project in one year is theaverage of the success and failure cash flows, plus the cash flowin one year, so:

Expected value of project at year 1 = [($3,195,356.21 +$1,500,000)/2] + $510,000 Expected value of project at year 1 = $2,857,678.10

The NPV is the present value of the expected value in one yearplus the cost of the equipment, so:

CHAPTER 24 B-694

NPV = –$2,300,000 + ($2,857,678.10)/1.14 NPV = $206,735.18

b. If we couldn’t abandon the project, the present value of thefuture cash flows when the quantity is 4,000 will be:


The gain from the option to abandon is the abandonment value minusthe present value of the cash flows if we cannot abandon theproject, so:

Gain from option to abandon = $1,500,000 – 1,345,413.14 Gain from option to abandon = $154,586.86

B-695 SOLUTIONS

We need to find the value of the option to abandon times thelikelihood of abandonment. So, the value of the option to abandontoday is:

Option value = (.50)($154,586.86)/1.14 Option value = $67,801.25

16. If the project is a success, present value of the future cash flowswill be:


If the sales are only 4,000 units, from Problem #14, we know we willabandon the project, with a value of $1,500,000. Since the projecthas an equal likelihood of success or failure in one year, theexpected value of the project in one year is the average of thesuccess and failure cash flows, plus the cash flow in one year, so:

Expected value of project at year 1 = [($6,390,712.41 +$1,500,000)/2] + $510,000 Expected value of project at year 1 = $4,455,356.21

The NPV is the present value of the expected value in one year plusthe cost of the equipment, so:

NPV = –$2,300,000 + $4,455,356.21/1.14 NPV = $1,608,207.20

The gain from the option to expand is the present value of the cashflows from the additional units sold, so:

Gain from option to expand = $68(9,500)(PVIFA14%,9) Gain from option to expand = $3,195,356.21

We need to find the value of the option to expand times thelikelihood of expansion. We also need to find the value of the optionto expand today, so:

Option value = (.50)($3,195,356.21)/1.14 Option value = $1,401,472.02

CHAPTER 24 B-696

17. a. The value of the call is the maximum of the stock price minus thepresent value of the exercise price, or zero, so:

C0 = Max[$65 – ($75/1.05),0] C0 = $0

The option isn’t worth anything.

b. The stock price is too low for the option to finish in the money.The minimum return on the stock required to get the option in themoney is:

Minimum stock return = ($75 – 65)/$65 Minimum stock return = .1538 or 15.38%

which is much higher than the risk-free rate of interest.

B-697 SOLUTIONS

18. B is the more typical case; A presents an arbitrage opportunity. Youcould buy the bond for $800 and immediately convert it into stockthat can be sold for $1,000. A riskless $200 profit results.

19. a. The conversion ratio is given at 22. The conversion price is thepar value divided by the conversion ratio, so:

Conversion price = $1,000/22 Conversion price = $45.45

The conversion premium is the percent increase in stock price thatresults in no profit when the bond is converted, so:

Conversion premium = ($45.45 – 35)/$35 Conversion premium = .2987 or 29.87%

b. The straight bond value is:


And the conversion value is the conversion ratio times the stockprice, so:

Conversion value = 22($35) Conversion value = $770.00

c. We simply need to set the straight bond value equal to theconversion ratio times the stock price, and solve for the stockprice, so:

$723.98 = 22S S = $32.91

d. There are actually two option values to consider with aconvertible bond. The conversion option value, defined as themarket value less the floor value, and the speculative optionvalue, defined as the floor value less the straight bond value.When the conversion value is less than the straight-bond value,the speculative option is worth zero.

Conversion option value = $960 – 770 = $190

CHAPTER 24 B-698

Speculative option value = $770 – 723.98 = $46.02

Total option value = $190.00 + 46.02 = $236.02

20. a. The NPV of the project is the sum of the present value of the cashflows generated by the project. The cash flows from this projectare an annuity, so the NPV is:

NPV = –$75,000,000 + $18,000,000(PVIFA14%,8)NPV = $8,499,550.09

B-699 SOLUTIONS

b. The company should abandon the project if the PV of the revisedcash flows for the next 7 years is less than the project’saftertax salvage value. Since the option to abandon the projectoccurs in year 1, discount the revised cash flows to year 1 aswell. To determine the level of expected cash flows below whichthe company should abandon the project, calculate the equivalentannual cash flows the project must earn to equal the aftertaxsalvage value. We will solve for C2, the revised cash flowbeginning in year 2. So, the revised annual cash flow below whichit makes sense to abandon the project is:

Aftertax salvage value = C2(PVIFA14%,7)$30,000,000 = C2(PVIFA14%,9)C2 = $30,000,000 / PVIFA14%,7

C2 = $6,995,771.32

Challenge

21. The straight bond value today is:

Straight bond value = $54(PVIFA9%,25) + $1,000/1.0925 Straight bond value = $646.39

And the conversion value of the bond today is:

Conversion value = $41.40($1,000/$150) Conversion value = $276.00

We expect the bond to be called when the conversion value increasesto $1,300, so we need to find the number of periods it will take forthe current conversion value to reach the expected value at which thebond will be converted. Doing so, we find:

$276.00(1.11)t = $1,300 t = 14.85 years

The bond will be called in 14.85 years.

The bond value is the present value of the expected cash flows. Thecash flows will be the annual coupon payments plus the conversionprice. The present value of these cash flows is:

Bond value = $54(PVIFA9%,14.85) + $1,300/1.0914.85 = $794.68

CHAPTER 24 B-700

B-701 SOLUTIONS

22. We will use the bottom up approach to calculate the operating cashflow. Assuming we operate the project for all four years, the cashflows are:

Year 0 1 2 3 4

Sales$9,100,00

0$9,100,00

0$9,100,00

0$9,100,00

0Operating costs 3,700,000 3,700,000 3,700,000 3,700,000Depreciation 3,000,000 3,000,000 3,000,000 3,000,000

EBT$2,400,00

0$2,400,00

0$2,400,00

0$2,400,00

0Tax 912,000 912,000 912,000 912,000

Net income$1,488,00

0$1,488,00

0$1,488,00

0$1,488,00

0+Depreciation 3,000,000 3,000,000 3,000,000 3,000,000

Operating CF$4,488,00

0$4,488,00

0$4,488,00

0$4,488,00

0

Change in NWC –$900,000 0 0 0 $900,000

Capital spending

–$12,000,0

00 0 0 0 0

Total cash flow

–$12,900,0

00$4,488,00

0$4,488,00

0$4,488,00

0$5,388,00

0

There is no salvage value for the equipment. The NPV is:

NPV = –$12,900,000 + $4,488,000(PVIFA13%,3) + $5,388,000/1.134

NPV = $1,001,414.16

b. The cash flows if we abandon the project after one year are:

Year 0 1

Sales$9,100,00

0Operating costs 3,700,000Depreciation 3,000,000

EBT$2,400,00

0

CHAPTER 24 B-702

Tax 912,000

Net income$1,488,00

0+Depreciation 3,000,000


0

Change in NWC –$900,000 $900,000

Capital spending

–$12,000,0

00$8,504,00

0

Total cash flow

–$12,900,0

00$13,892,0

00

The book value of the equipment is:

Book value = $12,000,000 – (1)($12,000,000/4) Book value = $9,000,000

B-703 SOLUTIONS

So, the taxes on the salvage value will be:

Taxes = ($9,000,000 – 8,200,000)(.38)Taxes = $304,000

This makes the aftertax salvage value:

Aftertax salvage value = $8,200,000 + 304,000 Aftertax salvage value = $8,504,000

The NPV if we abandon the project after one year is:

NPV = –$12,900,000 + $13,892,000/1.13NPV = –$606,194.69

If we abandon the project after two years, the cash flows are:

Year 0 1 2

Sales$9,100,00

0$9,100,00

0Operating costs 3,700,000 3,700,000Depreciation 3,000,000 3,000,000

EBT$2,400,00

0$2,400,00

0Tax 912,000 912,000

Net income$1,488,00

0$1,488,00

0+Depreciation 3,000,000 3,000,000


0$4,488,00

0

Change in NWC –$900,000 0 $900,000

Capital spending

–$12,000,0

00 0$6,062,00

0

Total cash flow

–$12,900,0

00$4,488,00

0$11,450,0

00


CHAPTER 24 B-704


So the taxes on the salvage value will be:

Taxes = ($6,000,000 – 6,100,000)(.38)Taxes = –$38,000


Aftertax salvage value = $6,100,000 – 38,000 Aftertax salvage value = $6,062,000

B-705 SOLUTIONS

The NPV if we abandon the project after two years is:

NPV = –$12,900,000 + $4,488,000/1.13 + $11,450,000/1.132

NPV = $38,710.94

If we abandon the project after three years, the cash flows are:

Year 0 1 2 3

Sales$9,100,00

0$9,100,00

0$9,100,00

0Operating costs 3,700,000 3,700,000 3,700,000Depreciation 3,000,000 3,000,000 3,000,000

EBT$2,400,00

0$2,400,00

0$2,400,00

0Tax 912,000 912,000 912,000

Net income$1,488,00

0$1,488,00

0$1,488,00

0+Depreciation 3,000,000 3,000,000 3,000,000


0$4,488,00

0$4,488,00

0

Change in NWC –$900,000 0 0 $900,000

Capital spending

–$12,000,0

00 0 0$4,054,00

0

Total cash flow

–$12,900,0

00$4,488,00

0$4,488,00

0$9,442,00

0



So the taxes on the salvage value will be:

Taxes = ($3,000,000 – 4,700,000)(.38)Taxes = –$646,000


CHAPTER 24 B-706

Aftertax salvage value = $4,700,000 – 646,000 Aftertax salvage value = $4,054,000

The NPV if we abandon the project after three years is:

NPV = –$12,900,000 + $4,488,000(PVIFA13%,2) + $9,442,000/1.133

NPV = $1,130,223.36

We should abandon the equipment after three years since the NPV ofabandoning the project after three years has the highest NPV.

CHAPTER 25OPTION VALUATION


1. Increasing the time to expiration increases the value of an option.The reason is that the option gives the holder the right to buy orsell. The longer the holder has that right, the more time there isfor the option to increase (or decrease in the case of a put) invalue. For example, imagine an out-of-the-money option that is aboutto expire. Because the option is essentially worthless, increasingthe time to expiration would obviously increase its value.

2. An increase in volatility acts to increase both call and put valuesbecause the greater volatility increases the possibility of favorablein-the-money payoffs.

3. Interest rate increases are good for calls and bad for puts. Thereason is that if a call is exercised in the future, we have to pay afixed amount at that time. The higher the interest rate, the lowerthe present value of that fixed amount. The reverse is true for putsin that we receive a fixed amount.

4. If you buy a put option on a stock that you already own you guaranteethat you can sell the stock for the exercise price of the put. Thus,you have effectively insured yourself against a stock price declinebelow this point. This is the protective put strategy.

5. The intrinsic value of a call is Max[S – E, 0]. The intrinsic valueof a put is Max[E – S, 0]. The intrinsic value of an option is thevalue at expiration.

6. The time value of both a call option and a put option is thedifference between the price of the option and the intrinsic value.For both types of options, as maturity increases, the time valueincreases since you have a longer time to realize a price increase(decrease). A call option is more sensitive to the maturity of thecontract.

CHAPTER 25 B-708

7. Since you have a large number of stock options in the company, youhave an incentive to accept the second project, which will increasethe overall risk of the company and reduce the value of the firm’sdebt. However, accepting the risky project will increase your wealth,as the options are more valuable when the risk of the firm increases.

8. Rearranging the put-call parity formula, we get: S – PV(E) = C – P.Since we know that the stock price and exercise price are the same,assuming a positive interest rate, the left hand side of the equationmust be greater than zero. This implies the price of the call must behigher than the price of the put in this situation.

9. Rearranging the put-call parity formula, we get: S – PV(E) = C – P.If the call and the put have the same price, we know C – P = 0. Thismust mean the stock price is equal to the present value of theexercise price, so the put is in-the-money.

10. A stock can be replicated using a long call (to capture the upsidegains), a short put (to reflect the downside losses) and a T-bill (toreflect the time value component – the “wait” factor).

B-709 SOLUTIONS



Basic

1. With continuous compounding, the FV is:

FV = $1,000 e.11(9) = $2,691.23

2. With continuous compounding, the PV is:

PV = $15,000 e–.09(8) = $7,301.28

3. Using put-call parity and solving for the put price, we get:

$62 + P = $60e–(.026)(.25) + $4.10 P = $1.71

4. Using put-call parity and solving for the call price we get:

$47 + $5.08 = $50e–(.048)(.5) + C C = $3.27

5. Using put-call parity and solving for the stock price we get:

S + $3.10 = $70e–(.048)(3/12) + $4.35S = $70.42

6. Using put-call parity, we can solve for the risk-free rate as follows:

$69.38 + $1.05 = $65e–R(4/12) + $6.27$64.16 = $65e–R(4/12) 0.9871 = e–R(4/12)

ln(0.9871) = ln(e–R(4/12))–0.0130 = –R(4/12)Rf = .0390 or 3.90%

CHAPTER 25 B-710

7. Using put-call parity, we can solve for the risk-free rate as follows:

$66.81 + $8.10 = $70e–R(5/12) + $6.12$68.79 = $70e–R(5/12) 0.9827 = e–R(5/12)

ln(0.9827) = ln(e–R(5/12))–0.0174 = –R(5/12)Rf = .0418 or 4.18%

B-711 SOLUTIONS

8. Using the Black-Scholes option pricing model to find the price of thecall option, we find:

d1 = [ln($69/$70) + (.06 + .412/2) (3/12)] / (.41 √3/12 ) = .1055

d2 = .1055 – (.41 √3/12 ) = –.0995

N(d1) = .5420

N(d2) = .4604

Putting these values into the Black-Scholes model, we find the call price is:

C = $69(.5420) – ($70e–.06(.25))(.4604) = $5.65

Using put-call parity, the put price is:

Put = $70e–.06(.25) + 5.65 – 69 = $5.61


d1 = [ln($86/$90) + (.055 + .622/2) (4/12)] / (.62 √4/12 ) = .1032

d2 = .1032 – (.62 √4/12 ) = –.2548

N(d1) = .5411

N(d2) = .3995


C = $86(.5411) – ($90e–.055(4/12))(.3955) = $11.24

Using put-call parity, the put price is:

Put = $90e–.055(4/12) + 11.24 – 86 = $13.60

10. The delta of a call option is N(d1), so:

CHAPTER 25 B-712

d1 = [ln($89/$85) + (.05 + .392/2) .75] / (.39 √.75 ) = .4161

N(d1) = .6613

For a call option the delta is .6613. For a put option, the delta is:

Put delta = .6613 – 1 = –.3387

The delta tells us the change in the price of an option for a $1change in the price of the underlying asset.

B-713 SOLUTIONS

11. Using the Black-Scholes option pricing model, with a ‘stock’ price is$1,900,000 and an exercise price is $2,050,000, the price you shouldreceive is:

d1 = [ln($1,900,000/$2,050,000) + (.05 + .202/2) (12/12)] / (.20 √12/12 ) = –.0299

d2 = –.0299 – (.20 √12/12 ) = –.2299

N(d1) = .4881

N(d2) = .4091


C = $1,900,000(.4881) – ($2,050,000e–.05(1))(.4091) = $129,615.91

12. Using the call price we found in the previous problem and put-call parity, you would need to pay:

Put = $2,050,000e–.05(1) + 129,615.91 – 1,900,000 = $179,636.23

You would have to pay $179,636.23 in order to guarantee the right to sell the land for $2,050,000.


d1 = [ln($84/$80) + (.06 + .532/2) (6/12)] / (.53 √(6 /12 ) ) = .3976

d2 = .3976 – (.53 √6/12 ) = .0229

N(d1) = .6545

N(d2) = .5091


C = $84(.6545) – ($80e–.06(.50))(.5091) = $15.46

CHAPTER 25 B-714

Using put-call parity, we find the put price is:

Put = $80e–.06(.50) + 15.46 – 84 = $9.09

a. The intrinsic value of each option is:

Call intrinsic value = Max[S – E, 0] = $4

Put intrinsic value = Max[E – S, 0] = $0

B-715 SOLUTIONS

b. Option value consists of time value and intrinsic value, so:

Call option value = Intrinsic value + Time value$15.46 = $4 + TVTV = $11.46

Put option value = Intrinsic value + Time value$9.09 = $0 + TV TV = $9.09

c. The time premium (theta) is more important for a call option than aput option, therefore, the time

premium is, in general, larger for a call option.

14. Using put-call parity, the price of the put option is:

$27.05 + P = $30e–.05(1/3) + $3.81P = $6.26

Intermediate

15. If the exercise price is equal to zero, the call price will equal thestock price, which is $85.

16. If the standard deviation is zero, d1 and d2 go to +8, so N(d1) andN(d2) go to 1. This is the no risk call option formula we discussed inan earlier chapter, so:

C = S – Ee–rt C = $74 – $70e–.05(6/12) = $5.73

17. If the standard deviation is infinite, d1 goes to positive infinity soN(d1) goes to 1, and d2 goes to negative infinity so N(d2) goes to 0.In this case, the call price is equal to the stock price, which is$42.

18. We can use the Black-Scholes model to value the equity of a firm.Using the asset value of $16,100 as the stock price, and the facevalue of debt of $15,000 as the exercise price, the value of thefirm’s equity is:

d1 = [ln($16,100/$15,000) + (.06 + .322/2) 1] / (.32 √1 ) = .5687

CHAPTER 25 B-716

d2 = .5687 – (.32 √1 ) = .2487

N(d1) = .7152

N(d2) = .5982

Putting these values into the Black-Scholes model, we find the equityvalue is:

Equity = $16,100(.7152) – ($15,000e–.06(1))(.5982) = $3,064.54

B-717 SOLUTIONS

The value of the debt is the firm value minus the value of theequity, so:

D = $16,100 – 3,064.54 = $13,035.46

19. a. We can use the Black-Scholes model to value the equity of a firm.Using the asset value of $17,600 as the stock price, and the facevalue of debt of $15,000 as the exercise price, the value of thefirm if it accepts project A is:

d1 = [ln($17,600/$15,000) + (.06 + .462/2) 1] / (.46 √1 )= .7079

d2 = .7079 – (.46 √1 ) = .2479

N(d1) = .7605

N(d2) = .5979

Putting these values into the Black-Scholes model, we find the equity value is:

EA = $17,600(.7605) – ($15,000e–.06(1))(.5979) = $4,938.60


DA = $17,600 – 4,938.60 = $12,661.40

And the value of the firm if it accepts Project B is:

d1 = [ln($18,200/$15,000) + (.06 + .242/2) 1] / (.24 √1 ) =1.1757

d2 = 1.1757 – (.24 √1 ) = .9357

N(d1) = .8801

N(d2) = .8253

Putting these values into the Black-Scholes model, we find theequity value is:

CHAPTER 25 B-718

EB = $18,200(.8801) – ($15,000e–.06(1))(.8253) = $4,360.22


DB = $18,200 – 4,360.22 = $13,839.78

b. Although the NPV of project B is higher, the equity value withproject A is higher. While NPV represents the increase in the valueof the assets of the firm, in this case, the increase in the valueof the firm’s assets resulting from project B is mostly allocatedto the debtholders, resulting in a smaller increase in the value ofthe equity. Stockholders would, therefore, prefer project A eventhough it has a lower NPV.

B-719 SOLUTIONS

c. Yes. If the same group of investors have equal stakes in the firmas bondholders and stockholders, then total firm value matters andproject B should be chosen, since it increases the value of thefirm to $18,200 instead of $17,600.

d. Stockholders may have an incentive to take on more risky, lessprofitable projects if the firm is leveraged; the higher the firm’sdebt load, all else the same, the greater is this incentive.

20. We can use the Black-Scholes model to value the equity of a firm.Using the asset value of $27,300 as the stock price, and the facevalue of debt of $25,000 as the exercise price, the value of thefirm’s equity is:

d1 = [ln($27,300/$25,000) + (.06 + .432/2) 1] / (.43 √1 ) = .5592

d2 = .5592 – (.43 √1 ) = .1292

N(d1) = .7120

N(d2) = .5514

Putting these values into the Black-Scholes model, we find the equityvalue is:

Equity = $27,300(.7120) – ($25,000e–.06(1))(.5514) = $6,455.02


D = $27,300 – 6,455.02 = $20,844.98

The return on the company’s debt is:

$20,844.98 = $25,000e–R(1) .8338 = e–R RD = –ln(.8338) = .1818 or 18.18%

21. a. The combined value of equity and debt of the two firms is:

Equity = $3,064.54 + 6,455.02 = $9,519.56

CHAPTER 25 B-720

Debt = $13,035.46 + 20,844.98 = $33,880.44

b. For the new firm, the combined market value of assets is $43,400,and the combined face value of debt is $40,000. Using Black-Scholesto find the value of equity for the new firm, we find:

d1 = [ln($43,400/$40,000) + (.06 + .192/2) 1] / (.19 √1 )= .8402

d2 = .8402 – (.19 √1 ) = .6502

N(d1) = .7996

N(d2) = .7422

B-721 SOLUTIONS

Putting these values into the Black-Scholes model, we find theequity value is:

E = $43,400(.7996) – ($40,000e–.06(1))(.7422) = $6,742.92


D = $43,400 – 6,742.92 = $36,657.08

c. The change in the value of the firm’s equity is:

Equity value change = $6,742.92 – 9,519.56 = –$2,776.65

The change in the value of the firm’s debt is:

Debt = $36,657.08 – 33,880.44 = $2,776.65

d. In a purely financial merger, when the standard deviation of theassets declines, the value of the

equity declines as well. The shareholders will lose exactly theamount the bondholders gain. The bondholders gain as a result ofthe coinsurance effect. That is, it is less likely that the newcompany will default on the debt.

22. a. Using Black-Scholes model to value the equity, we get:

d1 = [ln($16,000,000/$25,000,000) + (.06 + .412/2) 10] / (.41 √10 ) = .7668

d2 = .7668 – (.41 √10 ) = –.5297

N(d1) = .7784

N(d2) = .2982

Putting these values into Black-Scholes:

E = $16,000,000(.7784) – ($25,000,000e–.06(10))(.2982) = $8,363,715.92

b. The value of the debt is the firm value minus the value of theequity, so:

CHAPTER 25 B-722

D = $16,000,000 – 8,363,715.92 = $7,636,284.08

c. Using the equation for the PV of a continuously compounded lumpsum, we get:

$7,636,284.08 = $25,000,000e–R(10) .3055 = e–R(10) RD = –(1/10)ln(.3055) = .1186 or 11.86%

B-723 SOLUTIONS

d. Using Black-Scholes model to value the equity, we get:

d1 = [ln($16,750,000/$25,000,000) + (.06 + .412/2) 10] / (.41 √10 ) = .8022

d2 = .8022 – (.33 √10 ) = –.4944

N(d1) = .7888

N(d2) = .3105


E = $16,750,000(.7888) – ($25,000,000e–.06(10))(.3105) = $8,951,454.33

e.The value of the debt is the firm value minus the value of theequity, so:

D = $16,750,000 – 8,951,454.33 = $7,798,545.67

Using the equation for the PV of a continuously compounded lumpsum, we get:

$7,798,545.67 = $25,000,000e–R(10) .3119 = e–R10 RD = –(1/10)ln(.3119) = .1165 or 11.65%

When the firm accepts the new project, part of the NPV accrues tobondholders. This increases the present value of the bond, thusreducing the return on the bond. Additionally, the new projectmakes the firm safer in the sense it increases the value of assets,thus increasing the probability the call will end in-the-money andthe bondholders will receive their payment.

Challenge

23. a. Using the equation for the PV of a continuouslycompounded lump sum, we get:

PV = $40,000 e–.07(2) = $34,774.33

b. Using Black-Scholes model to value the equity, we get:

CHAPTER 25 B-724

d1 = [ln($17,000/$40,000) + (.07 + .602/2) 2] / (.60 √2 ) =–.4192

d2 = –.4192 – (.60 √2 ) = –1.2677

N(d1) = .3376

N(d2) = .1025


E = $17,000(.3376) – ($40,000e–.07(2))(.1025) = $2,175.55

B-725 SOLUTIONS

And using put-call parity, the price of the put option is:

Put = $40,000e–.07(10) + 2,175.55 – 17,000 = $19,949.88

c. The value of a risky bond is the value of a risk-free bond minusthe value of a put option on the firm’s equity, so:

Value of risky bond = $34,774.33 – 19,949.88 = $14,824.45

Using the equation for the PV of a continuously compounded lump sumto find the return on debt, we get:

$14,824.45 = $40,000e–R(2) .3706 = e–R2 RD = –(1/2)ln(.3706) = .4963 or 49.63%

d. The value of the debt with five years to maturity at the risk-freerate is:

PV = $40,000 e–.07(5) = $28,187.52

Using Black-Scholes model to value the equity, we get:

d1 = [ln($17,000/$40,000) + (.07 + .602/2) 5] / (.60 √5 )= .2939

d2 = .2939 – (.60 √5 ) = –1.0477

N(d1) = .6156

N(d2) = .1474


E = $17,000(.6156) – ($40,000e–.07(5))(.1474) = $6,310.66


Put = $40,000e–.07(5) + 6,310.66 – 17,000 = $17,498.18

The value of a risky bond is the value of a risk-free bond minusthe value of a put option on the firm’s equity, so:

CHAPTER 25 B-726



$10,689.34 = $40,000e–R(5) .2672 = e–R5 RD = –(1/5)ln(.2672) = .2639 or 26.39%

B-727 SOLUTIONS

The value of the debt declines because of the time value of money,i.e., it will be longer until shareholders receive their payment.However, the required return on the debt declines. Under thecurrent situation, it is not likely the company will have theassets to pay off bondholders. Under the new plan where the companyoperates for five more years, the probability of increasing thevalue of assets to meet or exceed the face value of debt is higherthan if the company only operates for two more years.

24. a. Using the equation for the PV of a continuouslycompounded lump sum, we get:

PV = $50,000 e–.07(5) = $35,234.40

b. Using Black-Scholes model to value the equity, we get:

d1 = [ln($46,000/$50,000) + (.07 + .442/2) 5] / (.44 √5 )= .7629

d2 = .7629 – (.44 √5 ) = –.2209

N(d1) = .7772

N(d2) = .4126


E = $46,000(.7772) – ($50,000e–.07(5))(.4126) = $21,216.74


Put = $50,000e–.07(5) + 21,216.74 – 46,000 = $10,451.14

c. The value of a risky bond is the value of a risk-free bond minusthe value of a put option on the firm’s equity, so:



$24,783.26 = $50,000e–R(5)

CHAPTER 25 B-728

.4957 = e–R(5) RD = –(1/5)ln(.4957) = .1404 or 14.04%

B-729 SOLUTIONS

d. Using the equation for the PV of a continuously compounded lumpsum, we get:

PV = $50,000 e–.07(5) = $35,234.40

Using Black-Scholes model to value the equity, we get:

d1 = [ln($46,000/$50,000) + (.07 + .552/2) 5] / (.55 √5 )= .8317

d2 = .8317 – (.55 √5 ) = –.3981

N(d1) = .7972

N(d2) = .3453


E = $46,000(.7972) – ($50,000e–.07(5))(.3453) = $24,506.51


Put = $50,000e–.07(5) + 24,506.51 – 46,000 = $13,740.92

The value of a risky bond is the value of a risk-free bond minusthe value of a put option on the firm’s equity, so:



$21,493.49 = $50,000e–R(5) .4299 = e–R(5) RD = –(1/5)ln(.4299) = .1689 or 16.89%

The value of the debt declines. Since the standard deviation of thecompany’s assets increases, the value of the put option on the facevalue of the bond increases which decreases the bond’s currentvalue.

e. From c and d, bondholders lose: $21,493.49 – 30,132.15 = –$3,289.77

CHAPTER 25 B-730

From c and d, stockholders gain: $24,506.51 – 21,216.74 = $3,289.77

This is an agency problem for bondholders. Management, acting toincrease shareholder wealth in this manner, will reduce bondholderwealth by the exact amount that shareholder wealth is increased.

B-731 SOLUTIONS

25. a. Going back to the chapter on dividends, the price of thestock will decline by the amount of the dividend (less any taxeffects). Therefore, we would expect the price of the stock to dropwhen a dividend is paid, reducing the upside potential of the callby the amount of the dividend. The price of a call option willdecrease when the dividend yield increases.

b. Using the Black-Scholes model with dividends, we get:

d1 = [ln($114/$105) + (.05 – .03 + .502/2) .5] / (.50 √.5 )= .4377

d2 = .4377 – (.50 √.5 ) = .0841

N(d1) = .6692

N(d2) = .5335

C = $114e–(.03)(.5)(.6692) – ($105e–.05(.5))(.5335) = $20.52 26. a.Going back to the chapter on dividends, the price of the stock will

decline by the amount of the dividend (less any tax effects).Therefore, we would expect the price of the stock to drop when adividend is paid. The price of put option will increase when thedividend yield increases.

b. Using put-call parity to find the price of the put option, we get:

$114e–.03(.5) + P = $105e–.05(.5) + 20.52P = $10.62

27. N(d1) is the probability that “z” is less than or equal to N(d1), so 1– N(d1) is the probability that “z” is greater than N(d1). Because ofthe symmetry of the normal distribution, this is the same thing asthe probability that “z” is less than N(–d1). So:

N(d1) – 1 = N(–d1).

28. From put-call parity:

P = E × e-Rt + C – S

CHAPTER 25 B-732

Substituting the Black-Scholes call option formula for C and using the result in the previous question produces the put option formula:

P = E × e-Rt + C – SP = E × e-Rt + S ×N(d1) – E × e-Rt ×N(d2) – SP = S ×(N(d1) – 1) + E × e-Rt ×(1 – N(d2))P = E × e-Rt ×N(–d2) – S × N(–d1)

B-733 SOLUTIONS

29. Based on Black-Scholes, the call option is worth $50! The reason isthat present value of the exercise price is zero, so the second termdisappears. Also, d1 is infinite, so N(d1) is equal to one. Theproblem is that the call option is European with an infiniteexpiration, so why would you pay anything for it since you can neverexercise it? The paradox can be resolved by examining the price ofthe stock. Remember that the call option formula only applies to anon-dividend paying stock. If the stock will never pay a dividend, it(and a call option to buy it at any price) must be worthless.

30. The delta of the call option is N(d1) and the delta of the put optionis N(d1) – 1. Since you are selling a put option, the delta of theportfolio is N(d1) – [N(d1) – 1]. This leaves the overall delta ofyour position as 1. This position will change dollar for dollar invalue with the underlying asset. This position replicates the dollar“action” on the underlying asset.

CHAPTER 26MERGERS AND ACQUISITIONS


1. In the purchase method, assets are recorded at market value, andgoodwill is created to account for the excess of the purchase priceover this recorded value. In the pooling of interests method, thebalance sheets of the two firms are simply combined; no goodwill iscreated. The choice of accounting method has no direct impact onthe cash flows of the firms. EPS will probably be lower under thepurchase method because reported income is usually lower due to therequired amortization of the goodwill created in the purchase.

2. a. Greenmail refers to the practice of paying unwanted suitors whohold an equity stake in the firm a premium over the market valueof their shares to eliminate the potential takeover threat.

b. A white knight refers to an outside bidder that a target firmbrings in to acquire it, rescuing the firm from a takeover bysome other unwanted hostile bidder.

c. A golden parachute refers to lucrative compensation andtermination packages granted to management in the event the firmis acquired.

d. The crown jewels usually refer to the most valuable orprestigious assets of the firm, which in the event of a hostiletakeover attempt, the target sometimes threatens to sell.

e. Shark repellent generally refers to any defensive tacticemployed by the firm to resist hostile takeover attempts.

f. A corporate raider usually refers to a person or firm thatspecializes in the hostile takeover of other firms.

g. A poison pill is an amendment to the corporate charter grantingthe shareholders the right to purchase shares at little or nocost in the event of a hostile takeover, thus making theacquisition prohibitively expensive for the hostile bidder.

h. A tender offer is the legal mechanism required by the SEC when abidding firm goes directly to the shareholders of the targetfirm in an effort to purchase their shares.

B-735 SOLUTIONS

i. A leveraged buyout refers to the purchase of the shares of apublicly-held company and its subsequent conversion into aprivately-held company, financed primarily with debt.

3. Diversification doesn’t create value in and of itself becausediversification reduces unsystematic, not systematic, risk. Asdiscussed in the chapter on options, there is a more subtle issueas well. Reducing unsystematic risk benefits bondholders by makingdefault less likely. However, if a merger is done purely todiversify (i.e., no operating synergy), then the NPV of the mergeris zero. If the NPV is zero, and the bondholders are better off,then stockholders must be worse off.

4. A firm might choose to split up because the newer, smaller firmsmay be better able to focus on their particular markets. Thus,reverse synergy is a possibility. An added advantage is thatperformance evaluation becomes much easier once the split is madebecause the new firm’s financial results (and stock prices) are nolonger commingled.

5. It depends on how they are used. If they are used to protectmanagement, then they are not good for stockholders. If they areused by management to negotiate the best possible terms of amerger, then they are good for stockholders.

6. One of the primary advantages of a taxable merger is the write-upin the basis of the target firm’s assets, while one of the primarydisadvantages is the capital gains tax that is payable. Thesituation is the reverse for a tax-free merger.

The basic determinant of tax status is whether or not the oldstockholders will continue to participate in the new company, whichis usually determined by whether they get any shares in the biddingfirm. An LBO is usually taxable because the acquiring group paysoff the current stockholders in full, usually in cash.

7. Economies of scale occur when average cost declines as outputlevels increase. A merger in this particular case might make sensebecause Eastern and Western may need less total capital investmentto handle the peak power needs, thereby reducing average generationcosts.

CHAPTER 26 B-736

8. Among the defensive tactics often employed by management areseeking white knights, threatening to sell the crown jewels,appealing to regulatory agencies and the courts (if possible), andtargeted share repurchases. Frequently, antitakeover charteramendments are available as well, such as poison pills, poisonputs, golden parachutes, lockup agreements, and supermajorityamendments, but these require shareholder approval, so they can’tbe immediately used if time is short. While target firmshareholders may benefit from management actively fightingacquisition bids, in that it encourages higher bidding and maysolicit bids from other parties as well, there is also the dangerthat such defensive tactics will discourage potential bidders fromseeking the firm in the first place, which harms the shareholders.

9. In a cash offer, it almost surely does not make sense. In a stockoffer, management may feel that one suitor is a better long-runinvestment than the other, but this is only valid if the market isnot efficient. In general, the highest offer is the best one.

10. Various reasons include: (1) Anticipated gains may be smaller thanthought; (2) Bidding firms are typically much larger, so any gainsare spread thinly across shares; (3) Management may not be actingin the shareholders’ best interest with many acquisitions; (4)Competition in the market for takeovers may force prices for targetfirms up to the zero NPV level; and (5) Market participants mayhave already discounted the gains from the merger before it isannounced.



Basic

1. For the merger to make economic sense, the acquirer must feel theacquisition will increase value by at least the amount of thepremium over the market value, so:

Minimum economic value = $538,000,000 – 495,000,000 = $43,000,000

B-737 SOLUTIONS

CHAPTER 26 B-738

2. a) Since neither company has any debt, using the pooling method,the asset value of the combined must equal the value of theequity, so:

Assets = Equity = 30,000($17) + 20,000($6) = $630,000

b) With the purchase method, the assets of the combined firm willbe the book value of Firm X, the acquiring company, plus themarket value of Firm Y, the target company, so:

Assets from X = 30,000($17) = $510,000 (book value)Assets from Y = 20,000($16) = $320,000 (market value)

The purchase price of Firm Y is the number of shares outstandingtimes the sum of the current stock price per share plus thepremium per share, so:

Purchase price of Y = 20,000($16 + 5) = $420,000

The goodwill created will be:

Goodwill = $420,000 – 320,000 = $100,000

And the total asset of the combined company will be:

Total assets XY = Total equity XY = $510,000 + 320,000 + 100,000= $930,000

3. In the pooling method, all accounts of both companies are addedtogether to total the accounts in the new company, so the post-merger balance sheet will be:

Meat Co., post-merger

Current assets $12,600 Current liabilities$ 7,600Fixed assets 38,500 Long-term debt 11,400

Equity 32,100Total $51,100 $51,100

4. Since the acquisition is funded by long-term debt, the post-mergerbalance sheet will have long-term debt equal to the original long-

B-739 SOLUTIONS

term debt of Meat’s balance sheet, plus the original long-term debton Loaf’s balance sheet, plus the new long-term debt issue, so:

Post-merger long-term debt = $9,300 + 2,100 + 19,000 = $30,400

Goodwill will be created since the acquisition price is greaterthan the book value. The goodwill amount is equal to the purchaseprice minus the market value of assets, plus the market value ofthe acquired company’s debt. Generally, the market value of currentassets is equal to the book value, so:

Goodwill = $19,000 – ($12,500 market value FA) – ($3,600 market value CA) + ($1,800 + 2,100)Goodwill = $6,800

CHAPTER 26 B-740

Equity will remain the same as the pre-merger balance sheet of theacquiring firm. Current assets and debt accounts will be the sum ofthe two firm’s pre-merger balance sheet accounts, and the fixedassets will be the sum of the pre-merger fixed assets of theacquirer and the market value of fixed assets of the target firm.The post-merger balance sheet will be:

Meat Co., post-mergerCurrent assets $12,600

Current liabilities $ 7,600

Net fixed assets 44,500

Long-term debt 30,400

Goodwill 6,800 Equity 25,900 Total $63,900 $63,900

5. In the pooling method, all accounts of both companies are addedtogether to total the accounts in the new company, so the post-merger balance sheet will be:

Silver Enterprises, post-mergerCurrent assets $ 4,600


Other assets 1,380Long-term debt 6,500

Net fixed assets 18,400 Equity 14,480 Total $24,380 $24,380

6. Since the acquisition is funded by long-term debt, the post-mergerbalance sheet will have long-term debt equal to the original long-term debt of Silver’s balance sheet plus the new long-term debtissue, so:

Post-merger long-term debt = $6,500 + 11,000 = $17,500

Equity will remain the same as the pre-merger balance sheet of theacquiring firm. Current assets, current liabilities, and otherassets will be the sum of the two firm’s pre-merger balance sheetaccounts, and the fixed assets will be the sum of the pre-mergerfixed assets of the acquirer and the market value of fixed assetsof the target firm. We can calculate the goodwill as the plug

B-741 SOLUTIONS

variable which makes the balance sheet balance. The post-mergerbalance sheet will be:

Silver Enterprises, post-mergerCurrent assets $ 4,600


Other assets 1,380Long-term debt 17,500

Net fixed assets 20,000 Equity 7,600Goodwill 2,520 Total $28,500 $28,500

CHAPTER 26 B-742

7. a. The cash cost is the amount of cash offered, so the cash cost is$73 million.

To calculate the cost of the stock offer, we first need tocalculate the value of the target to the acquirer. The value ofthe target firm to the acquiring firm will be the market valueof the target plus the PV of the incremental cash flowsgenerated by the target firm. The cash flows are a perpetuity,so

V* = $58,000,000 + $2,400,000/.10 = $82,000,000

The cost of the stock offer is the percentage of the acquiringfirm given up times the sum of the market value of the acquiringfirm and the value of the target firm to the acquiring firm. So,the equity cost will be:

Equity cost = .40($107,000,000 + 82,000,000) = $75,600,000

b. The NPV of each offer is the value of the target firm to theacquiring firm minus the cost of acquisition, so:

NPV cash = $82,000,000 – 73,000,000 = $9,000,000

NPV stock = $82,000,000 – 75,600,000 = $6,400,000

c. Since the NPV is greater with the cash offer the acquisitionshould be in cash.

8. a. The EPS of the combined company will be the sum of the earningsof both companies divided by the shares in the combined company.Since the stock offer is one share of the acquiring firm forthree shares of the target firm, new shares in the acquiringfirm will increase by one-third. So, the new EPS will be:

EPS = ($245,000 + 730,000)/[194,000 + (1/3)(92,337)] = $4.338

The market price of Pitt will remain unchanged if it is a zeroNPV acquisition. Using the PE ratio, we find the current marketprice of Pitt stock, which is:

P = 22($730,000)/194,000 = $82.78

B-743 SOLUTIONS

If the acquisition has a zero NPV, the stock price should remainunchanged. Therefore, the new PE will be:

P/E = $82.78/$4.338 = 19.09

b. The value of Jolie to Pitt must be the market value of thecompany since the NPV of the acquisition is zero. Therefore, thevalue is:

V* = $245,000(10.4) = $2,548,000

The cost of the acquisition is the number of shares offeredtimes the share price, so the cost is:

Cost = (1/3)(92,337)($82.78) = $2,548,000

CHAPTER 26 B-744

So, the NPV of the acquisition is:

NPV = 0 = V* + V – Cost = $2,548,000 + V – 2,548,000 V = $0

Although there is no economic value to the takeover, it ispossible that Pitt is motivated to purchase Jolie for other thanfinancial reasons.

9. a. The NPV of the merger is the market value of the target firm,plus the value of the synergy, minus the acquisition costs, so:

NPV = 1,500($18) + $6,000 – 1,500($20.50) = $2,250

b. Since the NPV goes directly to stockholders, the share price ofthe merged firm will be the market value of the acquiring firmplus the NPV of the acquisition, divided by the number of sharesoutstanding, so:

Share price = [3,400($43) + $2,250]/3,400 = $43.66

c. The merger premium is the premium per share times the number ofshares of the target firm outstanding, so the merger premium is:

Merger premium = 1,500($20.50 – 18) = $3,750

d. The number of new shares will be the number of shares of thetarget times the exchange ratio, so:

New shares created = 1,500(1/2) = 750 new shares

The value of the merged firm will be the market value of theacquirer plus the market value of the target plus the synergybenefits, so:

VBT = 3,400($43) + 1,500($18) + 6,000 = $179,200

The price per share of the merged firm will be the value of themerged firm divided by the total shares of the new firm, whichis:

P = $179,200/(3,400 + 750) = $43.18

B-745 SOLUTIONS

e. The NPV of the acquisition using a share exchange is the marketvalue of the target firm plus synergy benefits, minus the cost.The cost is the value per share of the merged firm times thenumber of shares offered to the target firm shareholders, so:

NPV = 1,500($18) + $6,000 – 750($43.18) = $614.46

CHAPTER 26 B-746

Intermediate

10. The cash offer is better for the target firm shareholders sincethey receive $20.50 per share. In the share offer, the targetfirm’s shareholders will receive:

Equity offer value = (1/2)($18.00) = $9.00 per share

From Problem 9, we know the value of the merged firm’s assets willbe $179,200. The number of shares in the new firm will be:

Shares in new firm = 3,400 + 1,500x

that is, the number of shares outstanding in the bidding firm, plusthe number of shares outstanding in the target firm, times theexchange ratio. This means the post merger share price will be:

P = $179,200/(3,400 + 1,500x)

To make the target firm’s shareholders indifferent, they mustreceive the same wealth, so:

1,500(x)P = 1,500($20.50)

This equation shows that the new offer is the shares outstanding inthe target company times the exchange ratio times the new stockprice. The value under the cash offer is the shares outstandingtimes the cash offer price. Solving this equation for P, we find:

P = $20.50 / x

Combining the two equations, we find:

$179,200/(3,400 + 1,500x) = $20.50 / xx = 0.4695

There is a simpler solution that requires an economic understandingof the merger terms. If the target firm’s shareholders areindifferent, the bidding firm’s shareholders are indifferent aswell. That is, the offer is a zero sum game. Using the new stockprice produced by the cash deal, we find:

Exchange ratio = $20.50/$43.66 = 0.4695

B-747 SOLUTIONS

11. The cost of the acquisition is:

Cost = 200($11) = $2,200

Since the stock price of the acquiring firm is $34, the firm willhave to give up:

Shares offered = $2,200/$34 = 64.71 shares

a. The EPS of the merged firm will be the combined EPS of theexisting firms divided by the new shares outstanding, so:

EPS = ($1,400 + 500)/(500 + 64.71) = $3.36

CHAPTER 26 B-748

b. The PE of the acquiring firm is:

Original P/E = $34/($1,400/500) = 12.14 times

Assuming the PE ratio does not change, the new stock price willbe:

New P = $3.36(12.14) = $40.86

c. If the market correctly analyzes the earnings, the stock pricewill remain unchanged since this is a zero NPV acquisition, so:

New P/E = $34/$3.36 = 10.11 times

d. The new share price will be the combined market value of the twoexisting companies divided by the number of shares outstandingin the merged company. So:

P = [(500)($34) + 200($8)]/(500 + 64.71) = $32.94

And the PE ratio of the merged company will be:

P/E = $32.94/$3.36 = 9.79 times

At the proposed bid price, this is a negative NPV acquisitionfor A since the share price declines. They should revise theirbid downward until the NPV is zero.

12. Beginning with the fact that the NPV of a merger is the value ofthe target minus the cost, we get:

NPV = VB* – Cost

NPV = V + VB – CostNPV = V – (Cost – VB)NPV = V – Merger premium

13. a. The synergy will be the present value of the incremental cashflows of the proposed purchase. Since the cash flows areperpetual, the synergy value is:

Synergy value = $450,000 / .08Synergy value = $5,625,000

B-749 SOLUTIONS

b. The value of Flash-in-the-Pan to Fly-by-Night is the synergyplus the current market value of Flash-in-the-Pan, which is:

Value = $5,625,000 + 14,000,000Value = $19,625,000

c. The value of the cash option is the amount of cash paid, or$18.5 million. The value of the stock acquisition is thepercentage of ownership in the merged company, times the valueof the merged company, so:

Stock acquisition value = .35($19,625,000 + 31,000,000)Stock acquisition value = $17,718,750

CHAPTER 26 B-750

d. The NPV is the value of the acquisition minus the cost, so theNPV of each alternative is:

NPV of cash offer = $19,625,000 – 18,500,000NPV of cash offer = $1,125,000

NPV of stock offer = $19,625,000 – 17,718,780NPV of stock offer = $1,906,250

e. The acquirer should make the stock offer since its NPV isgreater.

14. a. The number of shares after the acquisition will be the currentnumber of shares outstanding for the acquiring firm, plus thenumber of new shares created for the acquisition, which is:

Number of shares after acquisition = 9,000,000 + 2,500,000Number of shares after acquisition = 11,500,000

And the share price will be the value of the combined companydivided by the shares outstanding, which will be:

New stock price = £300,000,000 / 11,500,000New stock price = £26.09

b. Let equal the fraction of ownership for the targetshareholders in the new firm. We can set the percentage ofownership in the new firm equal to the value of the cash offer,so:

(£300,000,000) = £90,000,000 = .30 or 30%

So, the shareholders of the target firm would be equally as welloff if they received 30 percent of the stock in the new companyas if they received the cash offer. The ownership percentage ofthe target firm shareholders in the new firm can be expressedas:

Ownership = New shares issued / (New shares issued + Currentshares of acquiring firm).30 = New shares issued / (New shares issued + 9,000,000)New shares issued = 3,857,143

B-751 SOLUTIONS

To find the exchange ratio, we divide the new shares issued tothe shareholders of the target firm by the existing number ofshares in the target firm, so:

Exchange ratio = New shares / Existing shares in target firmExchange ratio = 3,857,143 / 90,000,000Exchange ratio = 0.4821

An exchange ratio of 0.4821 shares of the merged company foreach share of the target company owned would make the value ofthe stock offer equivalent to the value of the cash offer.

CHAPTER 26 B-752

Challenge

15. a. To find the value of the target to the acquirer, we need to findthe share price with the new growth rate. We begin by findingthe required return for shareholders of the target firm. Theearnings per share of the target are:

EPSP = $680,000/500,000 = $1.36 per share

The price per share is:

PP = 11.5($1.36) = $15.64

And the dividends per share are:

DPSP = $310,000/500,000 = $0.62

The current required return for Pulitzer shareholders, whichincorporates the risk of the company is:

RE = [$0.62(1.05)/$15.64] + .05 = .0916

The price per share of Pulitzer with the new growth rate is:

PP = $0.62(1.07)/(.0916 – .07) = $30.68

The value of the target firm to the acquiring firm is the numberof shares outstanding times the price per share under the newgrowth rate assumptions, so:

VT* = 500,000($30.68) = $15,339,408.63

b. The gain to the acquiring firm will be the value of the targetfirm to the acquiring firm minus the market value of the target,so:

Gain = $15,339,408.63 – 500,000($15.64) = $7,519,408.63

c. The NPV of the acquisition is the value of the target firm tothe acquiring firm minus the cost of the acquisition, so:

NPV = $15,339,408.63 – 500,000($18) = $6,339,408.63

B-753 SOLUTIONS

d. The most the acquiring firm should be willing to pay per shareis the offer price per share plus the NPV per share, so:

Maximum bid price = $18 + ($6,339,408.63/500,000) = $30.68

Notice, this is the same value we calculated earlier in part aas the value of the target to the acquirer.

CHAPTER 26 B-754

e. The price of the stock in the merged firm would be the marketvalue of the acquiring firm plus the value of the target to theacquirer, divided by the number of shares in the merged firm,so:

PFP = ($55,800,000 + 15,339,408.63)/(1,200,000 + 200,000) =$50.81

The NPV of the stock offer is the value of the target to theacquirer minus the value offered to the target shareholders. Thevalue offered to the target shareholders is the stock price ofthe merged firm times the number of shares offered, so:

NPV = $15,339,408.63 – 200,000($50.81) = $5,176,635.97

f. Yes, the acquisition should go forward, and Foxy should offercash since the NPV is higher.

g. Using the new growth rate in the dividend growth model, alongwith the dividend and required return we calculated earlier, theprice of the target under these assumptions is:

PP = $0.62(1.06)/(.0916 – .06) = $20.78

And the value of the target firm to the acquiring firm is:

VP* = 500,000($20.78) = $10,390,828.95

The gain to the acquiring firm will be:

Gain = $10,390,828.95 – 500,000($15.64) = $2,570,828.95

The NPV of the cash offer is now:

NPV cash = $10,390,828.95 – 500,000($18) = $1,390,828.95

And the new price per share of the merged firm will be:

PFP = [$55,800,000 + 10,390,828.95]/(1,200,000 + 200,000) = $47.28

And the NPV of the stock offer under the new assumption will be:

B-755 SOLUTIONS

NPV stock = $10,390,828.95 – 200,000($47.28) = $934,996.25

Even with the lower projected growth rate, the both offers stillhave a positive NPV, although both are significantly lower. Foxyshould purchase Pulitzer with the cash offer.

CHAPTER 27LEASINGAnswers to Concepts Review and Critical Thinking Questions

1. Some key differences are: (1) Lease payments are fully tax-deductible, but only the interest portion of the loan is; (2) Thelessee does not own the asset and cannot depreciate it for taxpurposes; (3) In the event of a default, the lessor cannot forcebankruptcy; and (4) The lessee does not obtain title to the asset atthe end of the lease (absent some additional arrangement).

2. The less profitable one because leasing provides, among other things,a mechanism for transferring tax benefits from entities that valuethem less to entities that value them more.

3. Potential problems include: (1) Care must be taken in interpretingthe IRR (a high or low IRR is preferred depending on the setup of theanalysis); and (2) Care must be taken to ensure the IRR underexamination is not the implicit interest rate just based on the leasepayments.

4. a. Leasing is a form of secured borrowing. It reduces a firm’s costof capital only if it is cheaper than other forms of securedborrowing. The reduction of uncertainty is not particularlyrelevant; what matters is the NAL.

b. The statement is not always true. For example, a lease oftenrequires an advance lease payment or security deposit and may beimplicitly secured by other assets of the firm.

c. Leasing would probably not disappear, since it does reduce theuncertainty about salvage value and the transactions costs oftransferring ownership. However, the use of leasing would begreatly reduced.

5. A lease must be disclosed on the balance sheet if one of thefollowing criteria is met:1. The lease transfers ownership of the asset by the end of the

lease. In this case, the firm essentially owns the asset and willhave access to its residual value.

B-757 SOLUTIONS

2. The lessee can purchase the asset at a price below its fair marketvalue (bargain purchase option) when the lease ends. The firmessentially owns the asset and will have access to most of itsresidual value.

3. The lease term is for 75% or more of the estimated economic lifeof the asset. The firm basically has access to the majority of thebenefits of the asset, without any responsibility for theconsequences of its disposal.

4. The present value of the lease payments is 90% or more of the fairmarket value of the asset at the start of the lease. The firm isessentially purchasing the asset on an installment basis.

6. The lease must meet the following IRS standards for the leasepayments to be tax deductible:1. The lease term must be less than 80% of the economic life of the

asset. If the term is longer, the lease is considered to be aconditional sale.

2. The lease should not contain a bargain purchase option, which theIRS interprets as an equity interest in the asset.

3. The lease payment schedule should not provide for very highpayments early and very low payments late in the life of thelease. This would indicate that the lease is being used simply toavoid taxes.

4. Renewal options should be reasonable and based on the fair marketvalue of the asset at renewal time. This indicates that the leaseis for legitimate business purposes, not tax avoidance.

7. As the term implies, off-balance sheet financing involves financingarrangements that are not required to be reported on the firm’sbalance sheet. Such activities, if reported at all, appear only inthe footnotes to the statements. Operating leases (those that do notmeet the criteria in problem 5) provide off-balance sheet financing.For accounting purposes, total assets will be lower and somefinancial ratios may be artificially high. Financial analysts aregenerally not fooled by such practices. There are no economicconsequences, since the cash flows of the firm are not affected byhow the lease is treated for accounting purposes.

8. The lessee may not be able to take advantage of the depreciation taxshield and may not be able to obtain favorable lease arrangements for“passing on” the tax shield benefits. The lessee might also need thecash flow from the sale to meet immediate needs, but will be able tomeet the lease obligation cash flows in the future.

CHAPTER 27 B-758

9. Since the relevant cash flows are all aftertax, the aftertax discountrate is appropriate.

10. Japan International’s financial position was such that the package ofleasing and buying probably resulted in the overall best aftertaxcost. In particular, Japan International may not have been in aposition to use all of the tax credits and also may not have had thecredit strength to borrow and buy the plane without facing a creditdowngrade and/or substantially higher rates.

11. There is the tax motive, but, beyond this, Genesis Lease Limitedknows that, in the event of a default, Japan International wouldrelinquish the plane, which would then be re-leased. Fungible assets,such as planes, which can be readily reclaimed and redeployed aregood candidates for leasing.

12. The plane will be re-leased to Japan International or another airtransportation firm, used by Genesis Lease Limited, or it will simplybe sold. There is an active market for used aircraft.



Basic

1. We will calculate cash flows from the depreciation tax shield first. The depreciation tax shield is:

Depreciation tax shield = ($5,800,000/4)(.35) = $507,500

The aftertax cost of the lease payments will be:

Aftertax lease payment = ($1,450,000)(1 – .35) = $1,131,000

B-759 SOLUTIONS

So, the total cash flows from leasing are:

OCF = $507,500 + 1,131,000 = $1,638,500

The aftertax cost of debt is:

Aftertax debt cost = .08(1 – .35) = .052

Using all of this information, we can calculate the NAL as:

NAL = $5,800,000 – $1,638,500(PVIFA5.20%,4) = $16,851.25

The NAL is positive so you should lease.

2. If we assume the lessor has the same cost of debt and the same taxrate, the NAL to the lessor is the negative of our company’s NAL, so:

NAL = – $16,851.25

3. To find the maximum lease payment that would satisfy both the lessorand the lessee, we need to find the payment that makes the NAL equalto zero. Using the NAL equation and solving for the OCF, we find:

NAL = 0 = $5,800,000 – OCF(PVIFA5.20%,4) OCF = $1,643,274.35

The OCF for this lease is composed of the depreciation tax shieldcash flow, as well as the aftertax lease payment. Subtracting out thedepreciation tax shield cash flow we calculated earlier, we find:

Aftertax lease payment = $1,643,274.35 – 507,500 = $1,135,774.35

Since this is the aftertax lease payment, we can now calculate thebreakeven pretax lease payment as:

Breakeven lease payment = $1,135,774.35/(1 – .35) = $1,747,345.15

4. If the tax rate is zero, there is no depreciation tax shieldforegone. Also, the aftertax lease payment is the same as the pretaxpayment, and the aftertax cost of debt is the same as the pretaxcost. So:

Cost of debt = .08

CHAPTER 27 B-760

Annual cost of leasing = leasing payment = $1,450,000

The NAL to leasing with these assumptions is:

NAL = $5,800,000 – $1,450,000(PVIFA8%,4) = $36,899.30

B-761 SOLUTIONS

5. We already calculated the breakeven lease payment for the lessor in Problem 3. Since the assumption about the lessor concerning the tax rate have not changed. So, the lessor breaks even with a payment of $1,747,345.15.

For the lessee, we need to calculate the breakeven lease paymentwhich results in a zero NAL. Using the assumptions in Problem 4, wefind:

NAL = 0 = $5,800,000 – PMT(PVIFA8%,4) PMT = $1,751,140.67

So, the range of lease payments that would satisfy both the lesseeand the lessor are:

Total payment range = $1,747,345.15 to $1,751,140.67

6. The appropriate depreciation percentages for a 3-year MACRS classasset can be found in Chapter 10. The depreciation percentagesare .3333, .4445, .1481, and .0741. The cash flows from leasing are:

Year 1: ($5,800,000)(.3333)(.35) + $1,131,000 = $1,807,599Year 2: ($5,800,000)(.4445)(.35) + $1,131,000 = $2,033,335Year 3: ($5,800,000)(.1481)(.35) + $1,131,000 = $1,431,643Year 4: ($5,800,000)(.0741)(.35) + $1,131,000 = $1,281,423

NAL = $5,800,000 – $1,807,599/1.052 – $2,033,335/1.0522 – $1,431,643/1.0523 –

$1,281,423/1.0524

NAL = $31,441.66

The machine should not be leased. This is because of the acceleratedtax benefits due to depreciation, which represents a cost in thedecision to lease compared to an advantage of the decision topurchase.

Intermediate

7. The pretax cost savings are not relevant to the lease versus buydecision, since the firm will definitely use the equipment andrealize the savings regardless of the financing choice made. Thedepreciation tax shield is:

CHAPTER 27 B-762

Depreciation tax shield lost = ($8,000,000/5)(.34) = $544,000

And the aftertax lease payment is:

Aftertax lease payment = $1,900,000(1 – .34) = $1,254,000

The aftertax cost of debt is:

Aftertax debt cost = .09(1 – .34) = .0594 or 5.94%

With these cash flows, the NAL is:

NAL = $8,000,000 – 1,254,000 – $1,254,000(PVIFA5.94%,4) – $544,000(PVIFA5.94%,5) = $99,509.86

The equipment should be leased.

B-763 SOLUTIONS

To find the maximum payment, we find where the NAL is equal to zero,and solve for the payment. Using X to represent the maximum payment:

NAL = 0 = $8,000,000 – X(1.0594)(PVIFA5.94%,5) – $544,000(PVIFA5.94%,5)X = $1,276,262.35

So the maximum pretax lease payment is:

Pretax lease payment = $1,276,262.35/(1 – .34) = $1,933,730.84

8. The aftertax residual value of the asset is an opportunity cost tothe leasing decision, occurring at the end of the project life (year5). Also, the residual value is not really a debt-like cash flow,since there is uncertainty associated with it at year 0.Nevertheless, although a higher discount rate may be appropriate,we’ll use the aftertax cost of debt to discount the residual value asis common in practice. Setting the NAL equal to zero:

NAL = 0 = $8,000,000 – X(1.0594)(PVIFA5.94%,5) – 544,000(PVIFA5.94%,5) –500,000/1.05945

X = $1,192,437.06

So, the maximum pretax lease payment is:

Pretax lease payment = $1,192,437.06/(1 – .34) = $1,806,722.81

9. The security deposit is a cash outflow at the beginning of the leaseand a cash inflow at the end of the lease when it is returned. TheNAL with these assumptions is:

NAL = $8,000,000 – 200,000 – 1,254,000 – $1,254,000(PVIFA5.94%,4) – $544,000(PVIFA5.94%,5)

+ $200,000/1.05945

NAL = $49,385.19

With the security deposit, the firm should still lease the equipmentrather than buy it, because the NAL is greater than zero. We couldalso solve this problem another way. From Problem 7, we know that theNAL without the security deposit is $99,509.86, so, if we find thepresent value of the security deposit, we can simply add this to$99,509.86. The present value of the security deposit is:

PV of security deposit = –$200,000 + $200,000/1.05945 = –$50,124.67

CHAPTER 27 B-764

So, the NAL with the security deposit is:

NAL = $99,509.86 – 50,124.67 = $49,385.19

10. a. The different borrowing rates are irrelevant. A basic tenant ofcapital budgeting is that the return of a project depends on therisk of the project. Since the lease payments are affected by theriskiness of the lessee, the lessee’s cost of debt is theappropriate interest rate for the analysis by both companies.

B-765 SOLUTIONS

b. Since the both companies have the same tax rate, there is only onelease payment that will result in a zero NAL for each company. Wewill calculate cash flows from the depreciation tax shield first.The depreciation tax shield is:

Depreciation tax shield = ($330,000/3)(.34) = $37,400

The aftertax cost of debt is the lessee’s cost of debt, which is:

Aftertax debt cost = .09(1 – .34) = .0594

Using all of this information, we can calculate the lease payment as:

NAL = 0 = $330,000 – PMT(1 – .34)(PVIFA5.94%,3) + $37,400(PVIFA5.94%,3)PMT = $130,180.63

c. Since the lessor’s tax bracket is unchanged, the zero NAL leasepayment is the same as we found in part b. The lessee will notrealize the depreciation tax shield, and the aftertax cost of debtwill be the same as the pretax cost of debt. So, the lessee’smaximum lease payment will be:

NAL = 0 = –$330,000 + PMT(PVIFA9%,3) PMT = $130,368.07

Both parties have positive NAL for lease payments between$130,180.63 and $130,368.07.

11. The APR of the loan is the lease factor times 2,400, so:

APR = 0.00295(2,400) = 7.08%

To calculate the lease payment we first need the net capitalizationcost, which is the base capitalized cost plus any other costs, minusand down payment or rebates. So, the net capitalized cost is:

Net capitalized cost = $35,000 + 450 – 2,000Net capitalized cost = $33,450

The depreciation charge is the net capitalized cost minus theresidual value, divided by the term of the lease, which is:

CHAPTER 27 B-766

Depreciation charge = ($33,450 – 21,500) / 36Depreciation charge = $331.94

Next, we can calculate the finance charge, which is the netcapitalized cost plus the residual value, times the lease factor, or:

Finance charge = ($33,450 + 21,500)(0.00295)Finance charge = $162.10

And the taxes on each monthly payment will be:

Taxes = ($331.94 + 162.10)(0.07)Taxes = $34.58

B-767 SOLUTIONS

The monthly lease payment is the sum of the depreciation charge, thefinance charge, and taxes, which will be:

Lease payment = $331.94 + 162.10 + 34.58Lease payment = $528.63

Challenge

12. With a four-year loan, the annual loan payment will be

$5,800,000 = PMT(PVIFA8%,4) PMT = $1,751,140.67

The aftertax loan payment is found by:

Aftertax payment = Pretax payment – Interest tax shield

So, we need to find the interest tax shield. To find this, we need aloan amortization table since the interest payment each year is thebeginning balance times the loan interest rate of 8 percent. Theinterest tax shield is the interest payment times the tax rate. Theamortization table for this loan is:

YearBeginningbalance

Totalpayment

Interestpayment

Principalpayment

Endingbalance

1$5,800,000.

00$1,751,140

.67$464,000.

00$1,287,140

.67$4,512,859

.33

24,512,859.3

31,751,140.

67361,028.7

51,390,111.

923,122,747.

42

33,122,747.4

21,751,140.

67249,819.7

91,501,320.

871,621,426.

54

41,621,426.5

41,751,140.

67129,714.1

21,621,426.

54 0.00

So, the total cash flows each year are:

Year Beginning balance

Aftertaxloan

payment OCFTotal

cash flow

1$1,751,140.67 – $5,800,000(.08)(.35)

$1,588,740.67

–$1,638,50

0= –$49,759.33

CHAPTER 27 B-768

2

$1,751,140.67 – $4,512,859.33 (.08)(.35) $1,624,780

.60

–$1,638,50

0= –$13,719.40

3

$1,751,140.67 – $3,122,747.42 (.08)(.35) $1,663,703

.74

–$1,638,50

0= $25,203.74

4

$1,751,140.67 – $1,621,426.54(.08)(.35) $1,705,740

.72

–$1,638,50

0= $67,240.72

So, the NAL with the loan payments is:

NAL = 0 – $49,759.33/1.052 – $13,719.40/1.0522 + $25,203.74/1.0523 +$67,240.72/1.0524

NAL = $16,851.25

The NAL is the same because the present value of the aftertax loanpayments, discounted at the aftertax cost of capital (which is theaftertax cost of debt) equals $5,800,000.

Revised SM 9th Ed (2)

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