As the COVID-19 pandemic continues to impact the U.S. business environment, businesses will need to increase focus on managing cash and liquidity in the near term and revise financial plans to strengthen their resiliency through recovery.

Below are some corporate finance tips and solutions for businesses to consider through this uncertain time:

##### Managing cash and liquidity:

- Implement a rapid cash management/13-week cash flow model diagnostic to understand your cash position and short-term working capital needs. (Learn more about our financial resilience solutions)
- Assess the potential for changes to your cash conversion cycle based on the current environment’s impact on your suppliers and customers.
- Consider changes to working capital practices (including inventory management) with a focus on business continuity and resilience.
- Reduce all non-essential expenses.

##### Revise financial plans for resilience

- Once your short-term financial needs are mapped out, consider assessing your longer-term financing alternatives.
- Begin financial modeling of multiple scenarios to assess the potential impact on 2020 budgets and projections for 2021 and beyond.
- Perform financial and liquidity stress tests and ensure appropriate contingencies are in place for downside scenarios.

##### Restructure existing debt and/or seek alternative financing options

- Consider renegotiating and restructuring loans and loan covenants if potential issues are uncovered in the financial scenario modeling.
- Prepare to raise new capital, including bank debt (credit lines/revolvers, commercial loans or asset-based loans), subordinated debt, minority equity and alternative financing (i.e., factoring, term-B loans, sale/lease backs) in the event of potential liquidity issues.
- Seek state tax credits and incentives available to businesses impacted by COVID-19.
- Assess divesting non-core corporate businesses or assets to generate cash.

As your business continues to navigate its financial needs during this time, Baker Tilly and Baker Tilly Capital, LLC can provide assistance in implementing these corporate finance tips and solutions for your business.

**For more information on this topic, or to learn how Baker Tilly Capital specialists can help, contact our team.**

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## About us

Mergence Corporate Solutions (Pty) Ltd, previously Cadiz Corporate Solutions, had its origins in Cadiz Holdings Limited, a financial services group which was listed on the JSE from 1999 to 2015.

We focus on providing best-in-class advisory and capital solutions, that are innovative and outcomes orientated, to our clients for their specific needs and requirements. Our integrated, solutions-driven approach is a strong differentiator. We partner with clients to unlock long-term value and provide across-the-board execution capabilities.

Over the years we have developed a track record of raising capital and unlocking opportunities for clients. Our relationship driven business has enabled us over the years to be particularly successful on the sell-side.

We have formed partnerships with leading M&A practices in China and India, and we have advised on M&A deals worth over R110 billion in value in the past ten years.

**COVID-19:**Click here to visit South Africa’s official Coronavirus (Covid-19) online news and information portal.

## Franklin

### Login, libraries-wide information

- Author/Creator:
- Crabb, Peter.
- Publication:
- New York : McGraw-Hill-Irwin, 2014.
- Format/Description:
- Book

1 v. (various pagings) : ill. ; 28 cm. - Status/Location:
Loading...

### Details

- Other Title:
- Principles of corporate finance
- Subjects:
- Corporations -- Finance -- Problems, exercises, etc.
- Contents:
- Ch. 1 Introduction to Corporate Finance

ch. 2 How to Calculate Present Values

ch. 3 Valuing Bonds

ch. 4 The Value of Common Stocks

ch. 5 Net Present Value and Other Investment Criteria

ch. 6 Making Investment Decisions with the Net Present Value Rule

ch. 7 Introduction to Risk and Return

ch. 8 Portfolio Theory and the Capital Asset Pricing Model?

ch. 9 Risk and the Cost of Capital

ch. 10 Project Analysis

ch. 11 Investment, Strategy, and Economic Rents

ch. 12 Agency Problems, Compensation, and Performance Measurement

ch. 13 Efficient Markets and Behavioral Finance

ch. 14 An Overview of Corporate Financing

ch. 15 How Corporations Issue Securities

ch. 16 Payout Policy

ch. 17 Does Debt Policy Matter?

ch. 18 How Much Should a Corporation Borrow?

ch. 19 Financing and Valuation

ch. 20 Understanding Options

ch. 21 Valuing Options

ch. 22 Real Options

ch. 23 Credit Risk and the Value of Corporate Debt

ch. 24 The Many Different Kinds of Debt

ch. 25 Leasing

ch. 26 Managing Risk

ch. 27 Managing International Risks

ch. 28 Financial Analysis

ch. 29 Financial Planning

ch. 30 Working Capital Management

ch. 31 Mergers

ch. 32 Corporate, Restructuring. - Notes:
- Textbook published: Principles of corporate finance / Richard A. Brealey, Stewart C. Myers, Franklin Allen. New York : McGraw-Hill Irwin,c2014.
- Local notes:
- Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.
- Contributor:
- Brealey, Richard A.

Myers, Stewart C.

Allen, Franklin, 1956-

Rosengarten Family Fund. - ISBN:
- 9780077502478

0077502477 - OCLC:
- 854579249
- Web link:
- The Rosengarten Family Fund Home Page

### M&A strategy and advisory

The decision to embark upon an M&A transaction is not one to be taken lightly. Whether pursuing a merger or acquisition, or considering a disposal or divestiture, the complexities of an M&A transaction require careful planning and execution.

Deloitte's global network of member firms works with clients ranging from corporates and privately-owned companies to private equity and management teams. Our member firms advise on all aspects of M&A strategy, planning, and deal execution -- and provide access to an international network of contacts, in-depth industry knowledge and local market insight.

Clients can depend on specialist teams to lead or support transactions requiring specific expertise on issues including business modelling, fast-track M&A, economic consulting, and government/public sector advice. With deep experience, knowledge, and tools, Deloitte's network of member firms delivers insights and advice that can help clients avoid the many pitfalls of complex deals and benefit from an M&A transaction that can meet stakeholder targets and expectations.

## Finance solutions corporate

## End of Chapter Solutions Corporate Finance 8th edition Ross ...

**End****of****Chapter****Solutions****Corporate****Finance** 8 th **edition****Ross**, Westerfield, and JaffeUpdated 11-21-2006

CHAPTER 1INTRODUCTION TO CORPORATEFINANCEAnswers to Concept Questions1. In the corporate form **of** ownership, the shareholders are the owners **of** the firm. The shareholderselect the directors **of** the corporation, who in turn appoint the firm’s management. This separation **of**ownership from control in the corporate form **of** organization is what causes agency problems toexist. Management may act in its own or someone else’s best interests, rather than those **of** theshareholders. If such events occur, they may contradict the goal **of** maximizing the share price **of** theequity **of** the firm.2. Such organizations frequently pursue social or political missions, so many different goals areconceivable. One goal that is **of**ten cited is revenue minimization; i.e., provide whatever goods andservices are **of**fered at the lowest possible cost to society. A better approach might be to observe thateven a not-for-pr**of**it business has equity. Thus, one answer is that the appropriate goal is tomaximize the value **of** the equity.3. Presumably, the current stock value reflects the risk, timing, and magnitude **of** all future cash flows,both short-term and long-term. If this is correct, then the statement is false.4. An argument can be made either way. At the one extreme, we could argue that in a market economy,all **of** these things are priced. There is thus an optimal level **of**, for example, ethical and/or illegalbehavior, and the framework **of** stock valuation explicitly includes these. At the other extreme, wecould argue that these are non-economic phenomena and are best handled through the politicalprocess. A classic (and highly relevant) thought question that illustrates this debate goes somethinglike this: “A firm has estimated that the cost **of** improving the safety **of** one **of** its products is $30million. However, the firm believes that improving the safety **of** the product will only save $20million in product liability claims. What should the firm do?”5. The goal will be the same, but the best course **of** action toward that goal may be different because **of**differing social, political, and economic institutions.6. The goal **of** management should be to maximize the share price for the current shareholders. Ifmanagement believes that it can improve the pr**of**itability **of** the firm so that the share price willexceed $35, then they should fight the **of**fer from the outside company. If management believes thatthis bidder or other unidentified bidders will actually pay more than $35 per share to acquire thecompany, then they should still fight the **of**fer. However, if the current management cannot increasethe value **of** the firm beyond the bid price, and no other higher bids come in, then management is notacting in the interests **of** the shareholders by fighting the **of**fer. Since current managers **of**ten losetheir jobs when the corporation is acquired, poorly monitored managers have an incentive to fightcorporate takeovers in situations such as this.

B-2 SOLUTIONS7. We would expect agency problems to be less severe in other countries, primarily due to the relativelysmall percentage **of** individual ownership. Fewer individual owners should reduce the number **of**diverse opinions concerning corporate goals. The high percentage **of** institutional ownership mightlead to a higher degree **of** agreement between owners and managers on decisions concerning riskyprojects. In addition, institutions may be better able to implement effective monitoring mechanismson managers than can individual owners, based on the institutions’ deeper resources and experienceswith their own management.8. The increase in institutional ownership **of** stock in the United States and the growing activism **of**these large shareholder groups may lead to a reduction in agency problems for U.S. corporations anda more efficient market for corporate control. However, this may not always be the case. If themanagers **of** the mutual fund or pension plan are not concerned with the interests **of** the investors, theagency problem could potentially remain the same, or even increase since there is the possibility **of**agency problems between the fund and its investors.9. How much is too much? Who is worth more, Jack Welch or Tiger Woods? The simplest answer isthat there is a market for executives just as there is for all types **of** labor. Executive compensation isthe price that clears the market. The same is true for athletes and performers. Having said that, oneaspect **of** executive compensation deserves comment. A primary reason executive compensation hasgrown so dramatically is that companies have increasingly moved to stock-based compensation.Such movement is obviously consistent with the attempt to better align stockholder and managementinterests. In recent years, stock prices have soared, so management has cleaned up. It is sometimesargued that much **of** this reward is simply due to rising stock prices in general, not managerialperformance. Perhaps in the future, executive compensation will be designed to reward onlydifferential performance, i.e., stock price increases in excess **of** general market increases.10. Maximizing the current share price is the same as maximizing the future share price at any futureperiod. The value **of** a share **of** stock depends on all **of** the future cash flows **of** company. Anotherway to look at this is that, barring large cash payments to shareholders, the expected price **of** thestock must be higher in the future than it is today. Who would buy a stock for $100 today when theshare price in one year is expected to be $80?

CHAPTER 2ACCOUNTING STATEMENTS, TAXES,AND CASH FLOWAnswers to Concepts Review and Critical Thinking Questions1. True. Every asset can be converted to cash at some price. However, when we are referring to a liquidasset, the added assumption that the asset can be converted cash at or near market value is important.2. The recognition and matching principles in financial accounting call for revenues, and the costsassociated with producing those revenues, to be “booked” when the revenue process is essentiallycomplete, not necessarily when the cash is collected or bills are paid. Note that this way is notnecessarily correct; it’s the way accountants have chosen to do it.3. The bottom line number shows the change in the cash balance on the balance sheet. As such, it is nota useful number for analyzing a company.4. The major difference is the treatment **of** interest expense. The accounting statement **of** cash flowstreats interest as an operating cash flow, while the financial cash flows treat interest as a financingcash flow. The logic **of** the accounting statement **of** cash flows is that since interest appears on theincome statement, which shows the operations for the period, it is an operating cash flow. In reality,interest is a financing expense, which results from the company’s choice **of** debt and equity. We willhave more to say about this in a later chapter. When comparing the two cash flow statements, thefinancial statement **of** cash flows is a more appropriate measure **of** the company’s performancebecause **of** its treatment **of** interest.5. Market values can never be negative. Imagine a share **of** stock selling for –$20. This would meanthat if you placed an order for 100 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** corporate and individualbankruptcy laws, net worth for a person or a corporation cannot be negative, implying that liabilitiescannot exceed assets in market value.6. For a successful company that is rapidly expanding, for example, capital outlays will be large,possibly leading to negative cash flow from assets. In general, what matters is whether the money isspent wisely, not whether cash flow from assets is positive or negative.7. It’s probably not a good sign for an established company to have negative cash flow from assets, butit would be fairly ordinary for a start-up, so it depends.

B-4 SOLUTIONS8. For example, if a company were to become more efficient in inventory management, the amount **of**inventory needed would decline. The same might be true if the company becomes better at collectingits receivables. In general, anything that leads to a decline in ending NWC relative to beginningwould have this effect. Negative net capital spending would mean more long-lived assets wereliquidated than purchased.9. If a company raises more money from selling stock than it pays in dividends in a particular period,its cash flow to stockholders will be negative. If a company borrows more than it pays in interest andprincipal, its cash flow to creditors will be negative.10. The adjustments discussed were purely accounting changes; they had no cash flow or market valueconsequences unless the new accounting information caused stockholders to revalue the derivatives.**Solutions** to Questions and ProblemsNOTE: 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 each problem isfound without rounding during any step in the problem.Basic1. To find owner’s equity, we must construct a balance sheet as follows:Balance SheetCA $5,000 CL $4,300NFA 23,000 LTD 13,000OE ??TA $28,000 TL & OE $28,000We know that total liabilities and owner’s equity (TL & OE) must equal total assets **of** $28,000. Wealso know that TL & OE is equal to current liabilities plus long-term debt plus owner’s equity, soowner’s equity is:OE = $28,000 –13,000 – 4,300 = $10,700NWC = CA – CL = $5,000 – 4,300 = $7002. The income statement for the company is:Income StatementSales $527,000Costs 280,000Depreciation 38,000EBIT $209,000Interest 15,000EBT $194,000Taxes (35%) 67,900Net income $126,100

CHAPTER 2 B- 5One equation for net income is:Net income = Dividends + Addition to retained earningsRearranging, we get:Addition to retained earnings = Net income – DividendsAddition to retained earnings = $126,100 – 48,000Addition to retained earnings = $78,1003. To find the book value **of** current assets, we use: NWC = CA – CL. Rearranging to solve for currentassets, we get:CA = NWC + CL = $900K + 2.2M = $3.1MThe market value **of** current assets and fixed assets is given, so:Book value CA = $3.1M Market value CA = $2.8MBook value NFA = $4.0M Market value NFA = $3.2MBook value assets = $3.1M + 4.0M = $7.1M Market value assets = $2.8M + 3.2M = $6.0M4. Taxes = 0.15($50K) + 0.25($25K) + 0.34($25K) + 0.39($273K – 100K)Taxes = $89,720The average tax rate is the total tax paid divided by net income, so:Average tax rate = $89,720 / $273,000Average tax rate = 32.86%The marginal tax rate is the tax rate on the next $1 **of** earnings, so the marginal tax rate = 39%.5. To calculate OCF, we first need the income statement:Income StatementSales $13,500Costs 5,400Depreciation 1,200EBIT $6,900Interest 680Taxable income $6,220Taxes (35%) 2,177Net income $4,043OCF = EBIT + Depreciation – TaxesOCF = $6,900 + 1,200 – 2,177OCF = $5,9236. Net capital spending = NFA end – NFA beg + DepreciationNet capital spending = $4,700,000 – 4,200,000 + 925,000Net capital spending = $1,425,000

B-8 SOLUTIONSCapital spending**End**ing fixed assets $250Beginning fixed assets (150)Depreciation 75Capital spending $175Now we can calculate the cash flow generated by the firm’s assets, which is:Cash flow from assetsOperating cash flow $200Capital spending (175)Change in NWC (50)Cash flow from assets $(25)Notice that the accounting statement **of** cash flows shows a positive cash flow, but the financialcash flows show a negative cash flow. The cash flow generated by the firm’s assets is a betternumber for analyzing the firm’s performance.12. With the information provided, the cash flows from the firm are the capital spending and the changein net working capital, so:Cash flows from the firmCapital spending $(3,000)Additions to NWC (1,000)Cash flows from the firm $(4,000)And the cash flows to the investors **of** the firm are:Cash flows to investors **of** the firmSale **of** short-term debt $(7,000)Sale **of** long-term debt (18,000)Sale **of** common stock (2,000)Dividends paid 23,000Cash flows to investors **of** the firm $(4,000)

CHAPTER 2 B- 913. a. The interest expense for the company is the amount **of** debt times the interest rate on the debt.So, the income statement for the company is:Income StatementSales $1,000,000Cost **of** goods sold 300,000Selling costs 200,000Depreciation 100,000EBIT $400,000Interest 100,000Taxable income $300,000Taxes (35%) 105,000Net income $195,000b. And the operating cash flow is:OCF = EBIT + Depreciation – TaxesOCF = $400,000 + 100,000 – 105,000OCF = $395,00014. To find the OCF, we first calculate net income.Income StatementSales $145,000Costs 86,000Depreciation 7,000Other expenses 4,900EBIT $47,100Interest 15,000Taxable income $32,100Taxes (40%) 12,840Net income $19,260Dividends $8,700Additions to RE $10,560a. OCF = EBIT + Depreciation – TaxesOCF = $47,100 + 7,000 – 12,840OCF = $41,260b. CFC = Interest – Net new LTDCFC = $15,000 – (–$6,500)CFC = $21,500Note that the net new long-term debt is negative because the company repaid part **of** its longtermdebt.c. CFS = Dividends – Net new equityCFS = $8,700 – 6,450CFS = $2,250

B-10SOLUTIONSd. We know that CFA = CFC + CFS, so:CFA = $21,500 + 2,250 = $23,750CFA is also equal to OCF – Net capital spending – Change in NWC. We already know OCF.Net capital spending is equal to:Net capital spending = Increase in NFA + DepreciationNet capital spending = $5,000 + 7,000Net capital spending = $12,000Now we can use:CFA = OCF – Net capital spending – Change in NWC$23,750 = $41,260 – 12,000 – Change in NWC.Solving for the change in NWC gives $5,510, meaning the company increased its NWC by$5,510.15. The solution to this question works the income statement backwards. Starting at the bottom:Net income = Dividends + Addition to ret. earningsNet income = $900 + 4,500Net income = $5,400Now, looking at the income statement:EBT – (EBT × Tax rate) = Net incomeRecognize that EBT × tax rate is simply the calculation for taxes. Solving this for EBT yields:EBT = NI / (1– Tax rate)EBT = $5,400 / 0.65EBT = $8,308Now we can calculate:EBIT = EBT + InterestEBIT = $8,308 + 1,600EBIT = $9,908The last step is to use:EBIT = Sales – Costs – Depreciation$9,908 = $29,000 – 13,000 – DepreciationDepreciation = $6,092Solving for depreciation, we find that depreciation = $6,092

CHAPTER 2 B- 1116. The balance sheet for the company looks like this:Balance SheetCash $175,000 Accounts payable $430,000Accounts receivable 140,000 Notes payable 180,000Inventory 265,000 Current liabilities $610,000Current assets $580,000 Long-term debt 1,430,000Total liabilities $2,040,000Tangible net fixed assets 2,900,000Intangible net fixed assets 720,000 Common stock ??Accumulated ret. earnings 1,240,000Total assets $4,200,000 Total liab. & owners’ equity $4,200,000Total liabilities and owners’ equity is:TL & OE = CL + LTD + Common stockSolving for this equation for equity gives us:Common stock = $4,200,000 – 1,240,000 – 2,040,000Common stock = $920,00017. The market value **of** shareholders’ equity cannot be zero. A negative market value in this case wouldimply that the company would pay you to own the stock. The market value **of** shareholders’ equitycan be stated as: Shareholders’ equity = Max [(TA – TL), 0]. So, if TA is $4,300, equity is equal to$800, and if TA is $3,200, equity is equal to $0. We should note here that while the market value **of**equity cannot be negative, the book value **of** shareholders’ equity can be negative.18. a. Taxes Growth = 0.15($50K) + 0.25($25K) + 0.34($10K) = $17,150Taxes Income = 0.15($50K) + 0.25($25K) + 0.34($25K) + 0.39($235K) + 0.34($8.165M)= $2,890,000b. Each firm has a marginal tax rate **of** 34% on the next $10,000 **of** taxable income, despite theirdifferent average tax rates, so both firms will pay an additional $3,400 in taxes.19. Income StatementSales $850,000COGS 630,000A&S expenses 120,000Depreciation 130,000EBIT ($30,000)Interest 85,000Taxable income ($115,000)Taxes (35%) 0a. Net income ($115,000)

B-12SOLUTIONSb. OCF = EBIT + Depreciation – TaxesOCF = ($30,000) + 130,000 – 0OCF = $100,000c. Net income was negative because **of** the tax deductibility **of** depreciation and interest expense.However, the actual cash flow from operations was positive because depreciation is a non-cashexpense and interest is a financing expense, not an operating expense.20. A firm can still pay out dividends if net income is negative; it just has to be sure there is sufficientcash flow to make the dividend payments.Change in NWC = Net capital spending = Net new equity = 0. (Given)Cash flow from assets = OCF – Change in NWC – Net capital spendingCash flow from assets = $100,000 – 0 – 0 = $100,000Cash flow to stockholders = Dividends – Net new equityCash flow to stockholders = $30,000 – 0 = $30,000Cash flow to creditors = Cash flow from assets – Cash flow to stockholdersCash flow to creditors = $100,000 – 30,000Cash flow to creditors = $70,000Cash flow to creditors is also:Cash flow to creditors = Interest – Net new LTDSo:Net new LTD = Interest – Cash flow to creditorsNet new LTD = $85,000 – 70,000Net new LTD = $15,00021. a. The income statement is:Income StatementSales $12,800Cost **of** good sold 10,400Depreciation 1,900EBIT $ 500Interest 450Taxable income $ 50Taxes (34%) 17Net income $33b. OCF = EBIT + Depreciation – TaxesOCF = $500 + 1,900 – 17OCF = $2,383

CHAPTER 2 B- 13c. Change in NWC = NWC end – NWC beg= (CA end – CL end ) – (CA beg – CL beg )= ($3,850 – 2,100) – ($3,200 – 1,800)= $1,750 – 1,400 = $350Net capital spending = NFA end – NFA beg + Depreciation= $9,700 – 9,100 + 1,900= $2,500CFA = OCF – Change in NWC – Net capital spending= $2,383 – 350 – 2,500= –$467The cash flow from assets can be positive or negative, since it represents whether the firm raisedfunds or distributed funds on a net basis. In this problem, even though net income and OCF arepositive, the firm invested heavily in both fixed assets and net working capital; it had to raise a net$467 in funds from its stockholders and creditors to make these investments.d. Cash flow to creditors = Interest – Net new LTD= $450 – 0= $450Cash flow to stockholders = Cash flow from assets – Cash flow to creditors= –$467 – 450= –$917We can also calculate the cash flow to stockholders as:Cash flow to stockholders = Dividends – Net new equitySolving for net new equity, we get:Net new equity = $500 – (–917)= $1,417The firm had positive earnings in an accounting sense (NI > 0) and had positive cash flow fromoperations. The firm invested $350 in new net working capital and $2,500 in new fixed assets. Thefirm had to raise $467 from its stakeholders to support this new investment. It accomplished this byraising $1,417 in the form **of** new equity. After paying out $500 **of** this in the form **of** dividends toshareholders and $450 in the form **of** interest to creditors, $467 was left to meet the firm’s cashflow needs for investment.22. a. Total assets 2006 = $650 + 2,900 = $3,550Total liabilities 2006 = $265 + 1,500 = $1,765Owners’ equity 2006 = $3,550 – 1,765 = $1,785Total assets 2007 = $705 + 3,400 = $4,105Total liabilities 2007 = $290 + 1,720 = $2,010Owners’ equity 2007 = $4,105 – 2,010 = $2,095

B-14SOLUTIONSb. NWC 2006 = CA06 – CL06 = $650 – 265 = $385NWC 2007 = CA07 – CL07 = $705 – 290 = $415Change in NWC = NWC07 – NWC065 = $415 – 385 = $30c. We can calculate net capital spending as:Net capital spending = Net fixed assets 2007 – Net fixed assets 2006 + DepreciationNet capital spending = $3,400 – 2,900 + 800Net capital spending = $1,300So, the company had a net capital spending cash flow **of** $1,300. We also know that net capitalspending is:Net capital spending = Fixed assets bought – Fixed assets sold$1,300 = $1,500 – Fixed assets soldFixed assets sold = $1,500 – 1,300 = $200To calculate the cash flow from assets, we must first calculate the operating cash flow. Theoperating cash flow is calculated as follows (you can also prepare a traditional incomestatement):EBIT = Sales – Costs – DepreciationEBIT = $8,600 – 4,150 – 800EBIT = $3,650EBT = EBIT – InterestEBT = $3,650 – 216EBT = $3,434Taxes = EBT .35Taxes = $3,434 .35Taxes = $1,202OCF = EBIT + Depreciation – TaxesOCF = $3,650 + 800 – 1,202OCF = $3,248Cash flow from assets = OCF – Change in NWC – Net capital spending.Cash flow from assets = $3,248 – 30 – 1,300Cash flow from assets = $1,918d. Net new borrowing = LTD07 – LTD06Net new borrowing = $1,720 – 1,500Net new borrowing = $220Cash flow to creditors = Interest – Net new LTDCash flow to creditors = $216 – 220Cash flow to creditors = –$4Net new borrowing = $220 = Debt issued – Debt retiredDebt retired = $300 – 220 = $80

CHAPTER 2 B- 1523.Balance sheet as **of** Dec. 31, 2006Cash $2,107 Accounts payable $2,213Accounts receivable 2,789 Notes payable 407Inventory 4,959 Current liabilities $2,620Current assets $9,855Long-term debt $7,056Net fixed assets $17,669 Owners' equity $17,848Total assets $27,524 Total liab. & equity $27,524Balance sheet as **of** Dec. 31, 2007Cash $2,155 Accounts payable $2,146Accounts receivable 3,142 Notes payable 382Inventory 5,096 Current liabilities $2,528Current assets $10,393Long-term debt $8,232Net fixed assets $18,091 Owners' equity $17,724Total assets $28,484 Total liab. & equity $28,4842006 Income Statement 2007 Income StatementSales $4,018.00 Sales $4,312.00COGS 1,382.00 COGS 1,569.00Other expenses 328.00 Other expenses 274.00Depreciation 577.00 Depreciation 578.00EBIT $1,731.00 EBIT $1,891.00Interest 269.00 Interest 309.00EBT $1,462.00 EBT $1,582.00Taxes (34%) 497.08 Taxes (34%) 537.88Net income $ 964.92 Net income $1,044.12Dividends $490.00 Dividends $539.00Additions to RE $474.92 Additions to RE $505.1224. OCF = EBIT + Depreciation – TaxesOCF = $1,891 + 578 – 537.88OCF = $1,931.12Change in NWC = NWC end – NWC beg = (CA – CL) end – (CA – CL) begChange in NWC = ($10,393 – 2,528) – ($9,855 – 2,620)Change in NWC = $7,865 – 7,235 = $630Net capital spending = NFA end – NFA beg + DepreciationNet capital spending = $18,091 – 17,669 + 578Net capital spending = $1,000

B-16SOLUTIONSCash flow from assets = OCF – Change in NWC – Net capital spendingCash flow from assets = $1,931.12 – 630 – 1,000Cash flow from assets = $301.12Cash flow to creditors = Interest – Net new LTDNet new LTD = LTD end – LTD begCash flow to creditors = $309 – ($8,232 – 7,056)Cash flow to creditors = –$867Net new equity = Common stock end – Common stock begCommon stock + Retained earnings = Total owners’ equityNet new equity = (OE – RE) end – (OE – RE) begNet new equity = OE end – OE beg + RE beg – RE endRE end = RE beg + Additions to RE Net new equity = OE end – OE beg + RE beg – (RE beg + Additions to RE)= OE end – OE beg – Additions to RENet new equity = $17,724 – 17,848 – 505.12 = –$629.12Cash flow to stockholders = Dividends – Net new equityCash flow to stockholders = $539 – (–$629.12)Cash flow to stockholders = $1,168.12As a check, cash flow from assets is $301.12.Cash flow from assets = Cash flow from creditors + Cash flow to stockholdersCash flow from assets = –$867 + 1,168.12Cash flow from assets = $301.12Challenge25. We will begin by calculating the operating cash flow. First, we need the EBIT, which can becalculated as:EBIT = Net income + Current taxes + Deferred taxes + InterestEBIT = $192 + 110 + 21 + 57EBIT = $380Now we can calculate the operating cash flow as:Operating cash flowEarnings before interest and taxes $380Depreciation 105Current taxes (110)Operating cash flow $375

CHAPTER 2 B- 17The cash flow from assets is found in the investing activities portion **of** the accounting statement **of**cash flows, so:Cash flow from assetsAcquisition **of** fixed assets $198Sale **of** fixed assets (25)Capital spending $173The net working capital cash flows are all found in the operations cash flow section **of** theaccounting statement **of** cash flows. However, instead **of** calculating the net working capital cashflows as the change in net working capital, we must calculate each item individually. Doing so, wefind:Net working capital cash flowCash $140Accounts receivable 31Inventories (24)Accounts payable (19)Accrued expenses 10Notes payable (6)Other (2)NWC cash flow $130Except for the interest expense and notes payable, the cash flow to creditors is found in the financingactivities **of** the accounting statement **of** cash flows. The interest expense from the income statementis given, so:Cash flow to creditorsInterest $57Retirement **of** debt 84Debt service $141Proceeds from sale **of** long-term debt (129)Total $12And we can find the cash flow to stockholders in the financing section **of** the accounting statement **of**cash flows. The cash flow to stockholders was:Cash flow to stockholdersDividends $94Repurchase **of** stock 15Cash to stockholders $109Proceeds from new stock issue (49)Total $60

B-18SOLUTIONS26. Net capital spending = NFA end – NFA beg + Depreciation= (NFA end – NFA beg ) + (Depreciation + AD beg ) – AD beg= (NFA end – NFA beg )+ AD end – AD beg= (NFA end + AD end ) – (NFA beg + AD beg ) = FA end – FA beg27. a. The tax bubble causes average tax rates to catch up to marginal tax rates, thus eliminating thetax advantage **of** low marginal rates for high income corporations.b. Assuming a taxable income **of** $100,000, the taxes will be:Taxes = 0.15($50K) + 0.25($25K) + 0.34($25K) + 0.39($235K) = $113.9KAverage tax rate = $113.9K / $335K = 34%The marginal tax rate on the next dollar **of** income is 34 percent.For corporate taxable income levels **of** $335K to $10M, average tax rates are equal to marginaltax rates.Taxes = 0.34($10M) + 0.35($5M) + 0.38($3.333M) = $6,416,667Average tax rate = $6,416,667 / $18,333,334 = 35%The marginal tax rate on the next dollar **of** income is 35 percent. For corporate taxable incomelevels over $18,333,334, average tax rates are again equal to marginal tax rates.c. Taxes = 0.34($200K) = $68K = 0.15($50K) + 0.25($25K) + 0.34($25K) + X($100K);X($100K) = $68K – 22.25K = $45.75KX = $45.75K / $100KX = 45.75%

CHAPTER 3LONG-TERM FINANCIAL PLANNINGAND GROWTHAnswers to Concepts Review and Critical Thinking Questions1. Time trend analysis gives a picture **of** changes in the company’s financial situation over time.Comparing a firm to itself over time allows the financial manager to evaluate whether some aspects**of** the firm’s operations, finances, or investment activities have changed. Peer group analysisinvolves comparing the financial ratios and operating performance **of** a particular firm to a set **of**peer group firms in the same industry or line **of** business. Comparing a firm to its peers allows thefinancial manager to evaluate whether some aspects **of** the firm’s operations, finances, or investmentactivities are out **of** line with the norm, thereby providing some guidance on appropriate actions totake to adjust these ratios if appropriate. Both allow an investigation into what is different about acompany from a financial perspective, but neither method gives an indication **of** whether thedifference is positive or negative. For example, suppose a company’s current ratio is increasing overtime. It could mean that the company had been facing liquidity problems in the past and is rectifyingthose problems, or it could mean the company has become less efficient in managing its currentaccounts. Similar arguments could be made for a peer group comparison. A company with a currentratio lower than its peers could be more efficient at managing its current accounts, or it could befacing liquidity problems. Neither analysis method tells us whether a ratio is good or bad, bothsimply show that something is different, and tells us where to look.2. If a company is growing by opening new stores, then presumably total revenues would be rising.Comparing total sales at two different points in time might be misleading. Same-store sales controlfor this by only looking at revenues **of** stores open within a specific period.3. The reason is that, ultimately, sales are the driving force behind a business. A firm’s assets,employees, and, in fact, just about every aspect **of** its operations and financing exist to directly orindirectly 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.4. Two assumptions **of** the sustainable growth formula are that the company does not want to sell newequity, and that financial policy is fixed. If the company raises outside equity, or increases its debtequityratio, it can grow at a higher rate than the sustainable growth rate. Of course, the companycould also grow faster than its pr**of**it margin increases, if it changes its dividend policy by increasingthe retention ratio, or its total asset turnover increases.

B-20SOLUTIONS5. The sustainable growth rate is greater than 20 percent, because at a 20 percent growth rate thenegative EFN indicates that there is excess financing still available. If the firm is 100 percent equityfinanced, then the sustainable and internal growth rates are equal and the internal growth rate wouldbe greater than 20 percent. However, when the firm has some debt, the internal growth rate is alwaysless than the sustainable growth rate, so it is ambiguous whether the internal growth rate would begreater than or less than 20 percent. If the retention ratio is increased, the firm will have more internalfunding sources available, 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 theretention rate is zero, both the internal and sustainable growth rates are zero, and the EFN will rise tothe change in total assets.6. Common-size financial statements provide the financial manager with a ratio analysis **of** thecompany. The common-size income statement can show, for example, that cost **of** goods sold as apercentage **of** sales is increasing. The common-size balance sheet can show a firm’s increasingreliance on debt as a form **of** financing. Common-size statements **of** cash flows are not calculated fora simple reason: There is no possible denominator.7. It would reduce the external funds needed. If the company is not operating at full capacity, it wouldbe able to increase sales without a commensurate increase in fixed assets.8. ROE is a better measure **of** the company’s performance. ROE shows the percentage return for theyear earned on shareholder investment. Since the goal **of** a company is to maximize shareholderwealth, this ratio shows the company’s performance in achieving this goal over the period.9. The EBITD/Assets ratio shows the company’s operating performance before interest, taxes, anddepreciation. This ratio would show how a company has controlled costs. While taxes are a cost, anddepreciation and amortization can be considered costs, they are not as easily controlled by companymanagement. Conversely, depreciation and amortization can be altered by accounting choices. Thisratio only uses costs directly related to operations in the numerator. As such, it gives a better metricto measure management performance over a period than does ROA.10. Long-term liabilities and equity are investments made by investors in the company, either in theform **of** a loan or ownership. Return on investment is intended to measure the return the companyearned from these investments. Return on investment will be higher than the return on assets for acompany with current liabilities. To see this, realize that total assets must equal total debt and equity,and total debt and equity is equal to current liabilities plus long-term liabilities plus equity. So, returnon investment could be calculated as net income divided by total assets minus current liabilities.11. Presumably not, but, **of** course, if the product had been much less popular, then a similar fate wouldhave awaited due to lack **of** sales.12. 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 flowdeclined. The firm’s capital spending was also rising. Thus, all three components **of** cash flow fromassets were negatively impacted.13. Financing possibly could have been arranged if the company had taken quick enough action.Sometimes it becomes apparent that help is needed only when it is too late, again emphasizing theneed for planning.

CHAPTER 3 B- 2114. All three were important, but the lack **of** cash or, more generally, financial resources ultimatelyspelled doom. An inadequate cash resource is usually cited as the most common cause **of** smallbusiness failure.15. Demanding cash upfront, increasing prices, subcontracting production, and improving financialresources via new owners or new sources **of** credit are some **of** the options. When orders exceedcapacity, price increases may be especially beneficial.**Solutions** to Questions and ProblemsNOTE: 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 each problem isfound without rounding during any step in the problem.Basic1. ROE = (PM)(TAT)(EM)ROE = (.085)(1.30)(1.75) = 19.34%2. The equity multiplier is:EM = 1 + D/EEM = 1 + 1.40 = 2.40One formula to calculate return on equity is:ROE = (ROA)(EM)ROE = .087(2.40) = 20.88%ROE can also be calculated as:ROE = NI / TESo, net income is:NI = ROE(TE)NI = (.2088)($520,000) = $108,5763. This is a multi-step problem involving several ratios. The ratios given are all part **of** the Du PontIdentity. The only Du Pont Identity ratio not given is the pr**of**it margin. If we know the pr**of**it margin,we can find the net income since sales are given. So, we begin with the Du Pont Identity:ROE = 0.16 = (PM)(TAT)(EM) = (PM)(S / TA)(1 + D/E)Solving the Du Pont Identity for pr**of**it margin, we get:PM = [(ROE)(TA)] / [(1 + D/E)(S)]PM = [(0.16)($1,185)] / [(1 + 1)( $2,700)] = .0351

B-22SOLUTIONSNow that we have the pr**of**it margin, we can use this number and the given sales figure to solve fornet income:PM = .0351 = NI / SNI = .0351($2,700) = $94.804. An increase **of** sales to $23,040 is an increase **of**:Sales increase = ($23,040 – 19,200) / $19,200Sales increase = .20 or 20%Assuming costs and assets increase proportionally, the pro forma financial statements will look likethis:Pro forma income statementPro forma balance sheetSales $23,040.00 Assets $ 111,600 Debt $ 20,400.00Costs 18,660.00 Equity 74,334.48EBIT 4,380.00 Total $ 111,600 Total $ 94,734.48Taxes (34%) 1,489.20Net income $ 2,890.80The payout ratio is constant, so the dividends paid this year is the payout ratio from last year timesnet income, or:Dividends = ($963.60 / $2,409)($2,890.80)Dividends = $1,156.32The addition to retained earnings is:Addition to retained earnings = $2,890.80 – 1,156.32Addition to retained earnings = $1,734.48And the new equity balance is:Equity = $72,600 + 1,734.48Equity = $74,334.48So the EFN is:EFN = Total assets – Total liabilities and equityEFN = $111,600 – 94,734.48EFN = $16,865.52

CHAPTER 3 B- 235. The maximum percentage sales increase is the sustainable growth rate. To calculate the sustainablegrowth rate, we first need to calculate the ROE, which is:ROE = NI / TEROE = $12,672 / $73,000ROE = .1736The plowback ratio, b, is one minus the payout ratio, so:b = 1 – .30b = .70Now we can use the sustainable growth rate equation to get:Sustainable growth rate = (ROE × b) / [1 – (ROE × b)]Sustainable growth rate = [.1736(.70)] / [1 – .1736(.70)]Sustainable growth rate = .1383 or 13.83%So, the maximum dollar increase in sales is:Maximum increase in sales = $54,000(.1383)Maximum increase in sales = $7,469.276. We need to calculate the retention ratio to calculate the sustainable growth rate. The retention ratiois:b = 1 – .25b = .75Now we can use the sustainable growth rate equation to get:Sustainable growth rate = (ROE × b) / [1 – (ROE × b)]Sustainable growth rate = [.19(.75)] / [1 – .19(.75)]Sustainable growth rate = .1662 or 16.62%7. We must first calculate the ROE using the Du Pont ratio to calculate the sustainable growth rate. TheROE is:ROE = (PM)(TAT)(EM)ROE = (.076)(1.40)(1.50)ROE = 15.96%The plowback ratio is one minus the dividend payout ratio, so:b = 1 – .40b = .60

B-24SOLUTIONSNow, we can use the sustainable growth rate equation to get:Sustainable growth rate = (ROE × b) / [1 – (ROE × b)]Sustainable growth rate = [.1596(.60)] / [1 – .1596(.60)]Sustainable growth rate = 10.59%8. An increase **of** sales to $5,192 is an increase **of**:Sales increase = ($5,192 – 4,400) / $4,400Sales increase = .18 or 18%Assuming costs and assets increase proportionally, the pro forma financial statements will look likethis:Pro forma income statementPro forma balance sheetSales $ 5,192 Assets $ 15,812 Debt $ 9,100Costs 3,168 Equity 6,324Net income $ 2,024 Total $ 15,812 Total $ 15,424If no dividends are paid, the equity account will increase by the net income, so:Equity = $4,300 + 2,024Equity = $6,324So the EFN is:EFN = Total assets – Total liabilities and equityEFN = $15,812 – 15,424 = $3889. a. First, we need to calculate the current sales and change in sales. The current sales are nextyear’s sales divided by one plus the growth rate, so:Current sales = Next year’s sales / (1 + g)Current sales = $440,000,000 / (1 + .10)Current sales = $400,000,000And the change in sales is:Change in sales = $440,000,000 – 400,000,000Change in sales = $40,000,000

CHAPTER 3 B- 25We can now complete the current balance sheet. The current assets, fixed assets, and short-termdebt are calculated as a percentage **of** current sales. The long-term debt and par value **of** stockare given. The plug variable is the additions to retained earnings. So:AssetsLiabilities and equityCurrent assets $80,000,000 Short-term debt $60,000,000Long-term debt $145,000,000Fixed assets 560,000,000 Common stock $60,000,000Accumulated retained earnings 375,000,000Total equity $435,000,000Total assets $640,000,000 Total liabilities and equity $640,000,000b. We can use the equation from the text to answer this question. The assets/sales and debt/salesare the percentages given in the problem, so: Assets Debt EFN = × Sales – × Sales – (p × Projected sales) × (1 – d) Sales Sales EFN = (.20 + 1.40) × $40,000,000 – (.15 × $40,000,000) – [(.12 × $440,000,000) × (1 – .40)]EFN = $26,320,000c. The current assets, fixed assets, and short-term debt will all increase at the same percentage assales. The long-term debt and common stock will remain constant. The accumulated retainedearnings will increase by the addition to retained earnings for the year. We can calculate theaddition to retained earnings for the year as:Net income = Pr**of**it margin × SalesNet income = .12($440,000,000)Net income = $52,800,000The addition to retained earnings for the year will be the net income times one minus thedividend payout ratio, which is:Addition to retained earnings = Net income(1 – d)Addition to retained earnings = $52,800,000(1 – .40)Addition to retained earnings = $31,680,000So, the new accumulated retained earnings will be:Accumulated retained earnings = $375,000,000 + 31,680,000Accumulated retained earnings = $406,680,000

B-26SOLUTIONSThe pro forma balance sheet will be:AssetsLiabilities and equityCurrent assets $88,000,000 Short-term debt $66,000,000Long-term debt $145,000,000Fixed assets 616,000,000 Common stock $60,000,000Accumulated retained earnings 406,680,000Total equity $466,680,000Total assets $704,000,000 Total liabilities and equity $677,680,000The EFN is:EFN = Total assets – Total liabilities and equityEFN = $704,000,000 – 677,680,000EFN = $26,320,00010. a. The sustainable growth is:ROE bSustainable growth rate =1- ROE bwhere:b = Retention ratio = 1 – Payout ratio = .65So:Sustainable growth rate =.0850 .651-.0850 .65Sustainable growth rate = .0585 or 5.85%b. It is possible for the sustainable growth rate and the actual growth rate to differ. If any **of** theactual parameters in the sustainable growth rate equation differs from those used to computethe sustainable growth rate, the actual growth rate will differ from the sustainable growth rate.Since the sustainable growth rate includes ROE in the calculation, this also implies that changesin the pr**of**it margin, total asset turnover, or equity multiplier will affect the sustainable growthrate.c. The company can increase its growth rate by doing any **of** the following:- Increase the debt-to-equity ratio by selling more debt or repurchasing stock- Increase the pr**of**it margin, most likely by better controlling costs.- Decrease its total assets/sales ratio; in other words, utilize its assets more efficiently.- Reduce the dividend payout ratio.

CHAPTER 3 B- 27Intermediate11. The solution requires substituting two ratios into a third ratio. Rearranging D/TA:Firm AFirm BD / TA = .60 D / TA = .40(TA – E) / TA = .60 (TA – E) / TA = .40(TA / TA) – (E / TA) = .60 (TA / TA) – (E / TA) = .401 – (E / TA) = .60 1 – (E / TA) = .40E / TA = .40 E / TA = .60E = .40(TA)E = .60(TA)Rearranging ROA, we find:NI / TA = .20 NI / TA = .30NI = .20(TA)NI = .30(TA)Since ROE = NI / E, we can substitute the above equations into the ROE formula, which yields:ROE = .20(TA) / .40(TA) = .20 / .40 = 50% ROE = .30(TA) / .60 (TA) = .30 / .60 = 50%12. PM = NI / S = –£13,156 / £147,318 = –8.93%As long as both net income and sales are measured in the same currency, there is no problem; in fact,except for some market value ratios like EPS and BVPS, none **of** the financial ratios discussed in thetext are measured in terms **of** currency. This is one reason why financial ratio analysis is widely usedin international finance to compare the business operations **of** firms and/or divisions across nationaleconomic borders. The net income in dollars is:NI = PM × SalesNI = –0.0893($267,661) = –$23,90313. a. The equation for external funds needed is: Assets EFN = Sales where: Debt × Sales – Sales × Sales – (PM × Projected sales) × (1 – d)Assets/Sales = $31,000,000/$38,000,000 = 0.82Sales = Current sales × Sales growth rate = $38,000,000(.20) = $7,600,000Debt/Sales = $8,000,000/$38,000,000 = .2105p = Net income/Sales = $2,990,000/$38,000,000 = .0787Projected sales = Current sales × (1 + Sales growth rate) = $38,000,000(1 + .20) = $45,600,000d = Dividends/Net income = $1,196,000/$2,990,000 = .40so:EFN = (.82 × $7,600,000) – (.2105 × $7,600,000) – (.0787 × $45,600,000) × (1 – .40)EFN = $2,447,200

B-28SOLUTIONSb. The current assets, fixed assets, and short-term debt will all increase at the same percentage assales. The long-term debt and common stock will remain constant. The accumulated retainedearnings will increase by the addition to retained earnings for the year. We can calculate theaddition to retained earnings for the year as:Net income = Pr**of**it margin × SalesNet income = .0787($45,600,000)Net income = $3,588,000The addition to retained earnings for the year will be the net income times one minus thedividend payout ratio, which is:Addition to retained earnings = Net income(1 – d)Addition to retained earnings = $3,588,000(1 – .40)Addition to retained earnings = $2,152,800So, the new accumulated retained earnings will be:Accumulated retained earnings = $13,000,000 + 2,152,800Accumulated retained earnings = $15,152,800The pro forma balance sheet will be:AssetsLiabilities and equityCurrent assets $10,800,000 Short-term debt $9,600,000Long-term debt $6,000,000Fixed assets 26,400,000 Common stock $4,000,000Accumulated retained earnings 15,152,800Total equity $19,152,800Total assets $37,200,000 Total liabilities and equity $34,752,800The EFN is:EFN = Total assets – Total liabilities and equityEFN = $37,200,000 – 34,752,800EFN = $2,447,200

CHAPTER 3 B- 29c. The sustainable growth is:ROE bSustainable growth rate =1- ROE bwhere:ROE = Net income/Total equity = $2,990,000/$17,000,000 = .1759b = Retention ratio = Retained earnings/Net income = $1,794,000/$2,990,000 = .60So:.1759.60Sustainable growth rate =1-.1759 .60Sustainable growth rate = .1180 or 11.80%d. The company cannot just cut its dividends to achieve the forecast growth rate. As shown below,even with a zero dividend policy, the EFN will still be $1,012,000.AssetsLiabilities and equityCurrent assets $10,800,000 Short-term debt $9,600,000Long-term debt $6,000,000Fixed assets 26,400,000 Common stock $4,000,000Accumulated retained earnings 16,588,000Total equity $20,588,000Total assets $37,200,000 Total liabilities and equity $36,188,000The EFN is:EFN = Total assets – Total liabilities and equityEFN = $37,200,000 – 36,188,000EFN = $1,012,000The company does have several alternatives. It can increase its asset utilization and/or its pr**of**itmargin. The company could also increase the debt in its capital structure. This will decrease theequity account, thereby increasing ROE.14. This is a multi-step problem involving several ratios. It is **of**ten easier to look backward to determinewhere to start. We need receivables turnover to find days’ sales in receivables. To calculatereceivables turnover, we need credit sales, and to find credit sales, we need total sales. Since we aregiven the pr**of**it margin and net income, we can use these to calculate total sales as:PM = 0.086 = NI / Sales = $173,000 / Sales; Sales = $2,011,628Credit sales are 75 percent **of** total sales, so:Credit sales = $2,011,628(0.75) = $1,508,721

B-30SOLUTIONSNow we can find receivables turnover by:Receivables turnover = Sales / Accounts receivable = $1,508,721 / $143,200 = 10.54 timesDays’ sales in receivables = 365 days / Receivables turnover = 365 / 10.54 = 34.64 days15. The solution to this problem requires a number **of** steps. First, remember that CA + NFA = TA. So, ifwe find the CA and the TA, we can solve for NFA. Using the numbers given for the current ratio andthe current liabilities, we solve for CA:CR = CA / CLCA = CR(CL) = 1.20($850) = $1,020To find the total assets, we must first find the total debt and equity from the information given. So,we find the net income using the pr**of**it margin:PM = NI / SalesNI = Pr**of**it margin × Sales = .095($4,310) = $409.45We now use the net income figure as an input into ROE to find the total equity:ROE = NI / TETE = NI / ROE = $409.45 / .215 = $1,904.42Next, we need to find the long-term debt. The long-term debt ratio is:Long-term debt ratio = 0.70 = LTD / (LTD + TE)Inverting both sides gives:1 / 0.70 = (LTD + TE) / LTD = 1 + (TE / LTD)Substituting the total equity into the equation and solving for long-term debt gives the following:1 + $1,904.42 / LTD = 1.429LTD = $1,904.42 / .429 = $4,443.64Now, we can find the total debt **of** the company:TD = CL + LTD = $850 + 4,443.64 = $5,293.64And, with the total debt, we can find the TD&E, which is equal to TA:TA = TD + TE = $5,293.64 + 1,904.42 = $7,198.06And finally, we are ready to solve the balance sheet identity as:NFA = TA – CA = $7,198.06 – 1,020 = $6,178.06

CHAPTER 3 B- 3116. This problem requires you to work backward through the income statement. First, recognize thatNet income = (1 – t C )EBT. Plugging in the numbers given and solving for EBT, we get:EBT = $7,850 / 0.66 = $11,893.94Now, we can add interest to EBIT to get EBIT as follows:EBIT = EBT + Interest paid = $11,893.94 + 2,108 = $14,001.94To get EBITD (earnings before interest, taxes, and depreciation), the numerator in the cash coverageratio, add depreciation to EBIT:EBITD = EBIT + Depreciation = $14,001.94 + 1,687 = $15,688.94Now, simply plug the numbers into the cash coverage ratio and calculate:Cash coverage ratio = EBITD / Interest = $15,688.94 / $2,108 = 7.44 times17. The only ratio given which includes cost **of** goods sold is the inventory turnover ratio, so it is the lastratio used. Since current liabilities are given, we start with the current ratio:Current ratio = 3.3 = CA / CL = CA / $340,000CA = $1,122,000Using the quick ratio, we solve for inventory:Quick ratio = 1.8 = (CA – Inventory) / CL = ($1,122,000 – Inventory) / $340,000Inventory = CA – (Quick ratio × CL)Inventory = $1,122,000 – (1.8 × $340,000)Inventory = $510,000Inventory turnover = 4.2 = COGS / Inventory = COGS / $510,000COGS = $2,142,000

B-32SOLUTIONS18. Common Common Common-2005 size 2006 size base yearAssetsCurrent assetsCash $ 10,168 2.54% $ 10,683 2.37% 1.0506Accounts receivable 27,145 6.77% 28,613 6.34% 1.0541Inventory 59,324 14.80% 64,853 14.37% 1.0932Total $ 96,637 24.11% $104,419 23.08% 1.0777Fixed assetsNet plant and equipment 304,165 75.89% 347,168 76.92% 1.1414Total assets $400,802 100% $451,317 100% 1.1260Liabilities and Owners’ EquityCurrent liabilitiesAccounts payable $ 73,185 18.26% $ 59,309 13.14% 0.8104Notes payable 39,125 9.76% 48,168 10.67% 1.2311Total $112,310 28.02% $107,477 23.81% 0.9570Long-term debt $ 50,000 12.47% $ 62,000 13.74% 1.2400Owners’ equityCommon stock & paid-in surplus $ 80,000 19.96% $ 80,000 17.73% 1.0000Accumulated retained earnings 158,492 39.54% 201,840 44.72% 1.2735Total $238,492 59.50% $281,840 62.45% 1.1818Total liabilities and owners’ equity $400,802 100% $451,317 100% 1.1260The common-size balance sheet answers are found by dividing each category by total assets. Forexample, the cash percentage for 2005 is:$10,168 / $400,802 = .0254 or 2.54%This means that cash is 2.54% **of** total assets.The common-base year answers are found by dividing each category value for 2006 by the samecategory value for 2005. For example, the cash common-base year number is found by:$10,683 / $10,168 = 1.050619. To determine full capacity sales, we divide the current sales by the capacity the company is currentlyusing, so:Full capacity sales = $510,000 / .85Full capacity sales = $600,000So, the dollar growth rate in sales is:Sales growth = $600,000 – 510,000Sales growth = $90,000

CHAPTER 3 B- 3320. To find the new level **of** fixed assets, we need to find the current percentage **of** fixed assets to fullcapacity sales. Doing so, we find:Fixed assets / Full capacity sales = $415,000 / $600,000Fixed assets / Full capacity sales = .6917Next, we calculate the total dollar amount **of** fixed assets needed at the new sales figure.Total fixed assets = .6917($680,000)Total fixed assets = $470,333.33The new fixed assets necessary is the total fixed assets at the new sales figure minus the current level**of** fixed assets.New fixed assets = $470,333.33 – 415,000New fixed assets = $55,333.3321. Assuming costs vary with sales and a 20 percent increase in sales, the pro forma income statementwill look like this:MOOSE TOURS INC.Pro Forma Income StatementSales $ 1,086,000Costs 852,000Other expenses 14,400EBIT $ 219,600Interest 19,700Taxable income $ 199,900Taxes(35%) 69,965Net income $ 129,935The payout ratio is constant, so the dividends paid this year is the payout ratio from last year timesnet income, or:Dividends = ($42,458/$106,145)($129,935)Dividends = $51,974And the addition to retained earnings will be:Addition to retained earnings = $129,935 – 51,974Addition to retained earnings = $77,961The new accumulated retained earnings on the pro forma balance sheet will be:New accumulated retained earnings = $257,000 + 77,961New accumulated retained earnings = $334,961

B-34SOLUTIONSThe pro forma balance sheet will look like this:MOOSE TOURS INC.Pro Forma Balance SheetAssetsLiabilities and Owners’ EquityCurrent assetsCurrent liabilitiesCash $ 30,000 Accounts payable $ 78,000Accounts receivable 51,600 Notes payable 9,000Inventory 91,200 Total $ 87,000Total $ 172,800 Long-term debt 156,000Fixed assetsNet plant andOwners’ equityequipment 436,800 Common stock andpaid-in surplus $ 21,000Retained earnings 334,961Total $ 355,961Total liabilities and owners’Total assets $ 609,600 equity $ 598,961So, the EFN is:EFN = Total assets – Total liabilities and equityEFN = $609,600 – 598,961EFN = $10,63922. First, we need to calculate full capacity sales, which is:Full capacity sales = $905,000 / .80Full capacity sales = $1,131,250The capital intensity ratio at full capacity sales is:Capital intensity ratio = Fixed assets / Full capacity salesCapital intensity ratio = $364,000 / $1,131,250Capital intensity ratio = .32177The fixed assets required at full capacity sales is the capital intensity ratio times the projected saleslevel:Total fixed assets = .32177($1,086,000) = $349,440So, EFN is:EFN = ($172,800 + 349,440) – $598,961 = –$76,721Note that this solution assumes that fixed assets are decreased (sold) so the company has a 100percent fixed asset utilization. If we assume fixed assets are not sold, the answer becomes:EFN = ($172,800 + 364,000) – $598,961 = –$62,161

CHAPTER 3 B- 3523. The D/E ratio **of** the company is:D/E = ($156,000 + 74,000) / $278,000D/E = .82734So the new total debt amount will be:New total debt = .82734($355,961)New total debt = $294,500.11So, the EFN is:EFN = $609,600 – ($294,500.11 + 355,961) = –$40,861.11An interpretation **of** the answer is not that the company has a negative EFN. Looking back atProblem 21, we see that for the same sales growth, the EFN is $10,639. The negative number in thiscase means the company has too much capital. There are two possible solutions. First, the companycan put the excess funds in cash, which has the effect **of** changing the current asset growth rate.Second, the company can use the excess funds to repurchase debt and equity. To maintain the currentcapital structure, the repurchase must be in the same proportion as the current capital structure.Challenge24. The pro forma income statements for all three growth rates will be:MOOSE TOURS INC.Pro Forma Income Statement15 % SalesGrowth20% SalesGrowth25% SalesGrowthSales $1,040,750 $1,086,000 $1,131,250Costs 816,500 852,000 887,500Other expenses 13,800 14,400 15,000EBIT $ 210,450 $ 219,600 $ 228,750Interest 19,700 19,700 19,700Taxable income $ 190,750 $ 199,900 $ 209,050Taxes (35%) 66,763 69,965 73,168Net income $ 123,988 $ 129,935 $ 135,883Dividends $ 49,595 $ 51,974 $ 54,353Add to RE 74,393 77,961 81,530We will calculate the EFN for the 15 percent growth rate first. Assuming the payout ratio is constant,the dividends paid will be:Dividends = ($42,458/$106,145)($123,988)Dividends = $49,595

B-36SOLUTIONSAnd the addition to retained earnings will be:Addition to retained earnings = $123,988 – 49,595Addition to retained earnings = $74,393The new accumulated retained earnings on the pro forma balance sheet will be:New accumulated retained earnings = $257,000 + 74,393New accumulated retained earnings = $331,393The pro forma balance sheet will look like this:15% Sales Growth:MOOSE TOURS INC.Pro Forma Balance SheetAssetsLiabilities and Owners’ EquityCurrent assetsCurrent liabilitiesCash $ 28,750 Accounts payable $ 74,750Accounts receivable 49,450 Notes payable 9,000Inventory 87,400 Total $ 83,750Total $ 165,600 Long-term debt 156,000Fixed assetsNet plant andOwners’ equityequipment 418,600 Common stock andpaid-in surplus $ 21,000Retained earnings 331,393Total $ 352,393Total liabilities and owners’Total assets $ 584,200 equity $ 592,143So, the EFN is:EFN = Total assets – Total liabilities and equityEFN = $584,200 – 592,143EFN = –$7,943At a 20 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be:Dividends = ($42,458/$106,145)($129,935)Dividends = $51,974And the addition to retained earnings will be:Addition to retained earnings = $129,935 – 51,974Addition to retained earnings = $77,961

CHAPTER 3 B- 37The new accumulated retained earnings on the pro forma balance sheet will be:New accumulated retained earnings = $257,000 + 77,961New accumulated retained earnings = $334,961The pro forma balance sheet will look like this:20% Sales Growth:MOOSE TOURS INC.

B-38SOLUTIONSThe pro forma balance sheet will look like this:25% Sales Growth:MOOSE TOURS INC.Pro Forma Balance SheetAssetsLiabilities and Owners’ EquityCurrent assetsCurrent liabilitiesCash $ 31,250 Accounts payable $ 81,250Accounts receivable 53,750 Notes payable 9,000Inventory 95,000 Total $ 90,250Total $ 180,000 Long-term debt 156,000Fixed assetsNet plant andOwners’ equityequipment 455,000 Common stock andpaid-in surplus $ 21,000Retained earnings 338,530Total $ 359,530Total liabilities and owners’Total assets $ 635,000 equity $ 605,780So, the EFN is:EFN = Total assets – Total liabilities and equityEFN = $635,000 – 605,780EFN = $29,22125. The pro forma income statements for all three growth rates will be:MOOSE TOURS INC.Pro Forma Income Statement20% SalesGrowth30% SalesGrowth35% SalesGrowthSales $1,086,000 $1,176,500 $1,221,750Costs 852,000 923,000 958,500Other expenses 14,400 15,600 16,200EBIT $ 219,600 $ 237,900 $ 247,050Interest 19,700 19,700 19,700Taxable income $ 199,900 $ 218,200 $ 227,350Taxes (35%) 69,965 76,370 79,573Net income $ 129,935 $ 141,830 $ 147,778Dividends $ 51,974 $ 56,732 $ 59,111Add to RE 77,961 85,098 88,667

CHAPTER 3 B- 39Under the sustainable growth rate assumption, the company maintains a constant debt-equity ratio.The D/E ratio **of** the company is:D/E = ($156,000 + 74,000) / $278,000D/E = .82734At a 20 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be:Dividends = ($42,458/$106,145)($129,935)Dividends = $51,974And the addition to retained earnings will be:Addition to retained earnings = $129,935 – 51,974Addition to retained earnings = $77,961The total equity on the pro forma balance sheet will be:New total equity = $21,000 + 257,000 + 77,961New total equity = $355,961The new total debt will be:New total debt = .82734($355,961)New total debt = $294,500So, the new long-term debt will be the new total debt minus the new short-term debt, or:New long-term debt = $294,500 – 87,000New long-term debt = $207,500

B-40SOLUTIONSThe pro forma balance sheet will look like this:Sales growth rate = 20% and Debt/Equity ratio = .82734:MOOSE TOURS INC.82734($363,098)New total debt = $300,405

CHAPTER 3 B- 41So, the new long-term debt will be the new total debt minus the new short-term debt, or:New long-term debt = $300,405 – 93,500New long-term debt = $206,905Sales growth rate = 30% and debt/equity ratio = .82734:MOOSE TOURS INC.Pro Forma Balance SheetAssetsLiabilities and Owners’ EquityCurrent assetsCurrent liabilitiesCash $ 32,500 Accounts payable $ 84,500Accounts receivable 55,900 Notes payable 9,000Inventory 98,800 Total $ 93,500Total $ 187,200 Long-term debt 206,905Fixed assetsNet plant andOwners’ equityequipment 473,200 Common stock andpaid-in surplus $ 21,000Retained earnings 342,098Total $ 363,098Total liabilities and owners’Total assets $ 660,400 equity $ 663,503So, the EFN is:EFN = Total assets – Total liabilities and equityEFN = $660,400 – 663,503EFN = –$3,103At a 35 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be:Dividends = ($42,458/$106,145)($147,778)Dividends = $59,111And the addition to retained earnings will be:Addition to retained earnings = $147,778 – 59,111Addition to retained earnings = $88,667The new total equity on the pro forma balance sheet will be:New total equity = $21,000 + 257,000 + 88,667New total equity = $366,667

B-42SOLUTIONSThe new total debt will be:New total debt = .82734($366,667)New total debt = $303,357So, the new long-term debt will be the new total debt minus the new short-term debt, or:New long-term debt = $303,357 – 96,750New long-term debt = $206,607Sales growth rate = 35% and debt/equity ratio = .82734:MOOSE TOURS INC.Pro Forma Balance SheetAssetsLiabilities and Owners’ EquityCurrent assetsCurrent liabilitiesCash $ 33,750 Accounts payable $ 87,750Accounts receivable 58,050 Notes payable 9,000Inventory 102,600 Total $ 96,750Total $ 194,400 Long-term debt 206,607Fixed assetsNet plant andOwners’ equityequipment 491,400 Common stock andpaid-in surplus $ 21,000Retained earnings 345,667Total $ 366,667Total liabilities and owners’Total assets $ 685,800 equity $ 670,024So the EFN is:EFN = Total assets – Total liabilities and equityEFN = $685,800 – 670,024EFN = $15,77626. We must need the ROE to calculate the sustainable growth rate. The ROE is:ROE = (PM)(TAT)(EM)ROE = (.062)(1 / 1.55)(1 + 0.3)ROE = .0520 or 5.20%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 = .14 = [.0520(b)] / [1 – .0520(b)]b = 2.36

CHAPTER 3 B- 43This implies the payout ratio is:Payout ratio = 1 – bPayout ratio = 1 – 2.36Payout ratio = –1.36This is a negative dividend payout ratio **of** 136 percent, which is impossible. The growth rate is notconsistent with the other constraints. The lowest possible payout rate is 0, which corresponds toretention 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 = [.0520(1)] / [1 – .0520(1)]Maximum sustainable growth rate = .0549 or 5.49%27. We know that EFN is:EFN = Increase in assets – Addition to retained earningsThe increase in assets is the beginning assets times the growth rate, so:Increase in assets = A gThe addition to retained earnings next year is the current net income times the retention ratio, timesone plus the growth rate, so:Addition to retained earnings = (NI b)(1 + g)And rearranging the pr**of**it margin to solve for net income, we get:NI = PM(S)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]g28. We start with the EFN equation we derived in Problem 27 and set it equal to zero:EFN = 0 = – PM(S)b + [A – PM(S)b]gSubstituting the rearranged pr**of**it margin equation into the internal growth rate equation, we have:Internal growth rate = [PM(S)b ] / [A – PM(S)b]

B-44SOLUTIONSSince:ROA = NI / AROA = PM(S) / AWe can substitute this into the internal growth rate equation and divide both the numerator anddenominator by A. This gives:Internal growth rate = {[PM(S)b] / A} / {[A – PM(S)b] / A}Internal growth rate = b(ROA) / [1 – b(ROA)]To derive the sustainable growth rate, we must realize that to maintain a constant D/E ratio with noexternal 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)]29. In the following derivations, the subscript “E” refers to end **of** period numbers, and the subscript “B”refers to beginning **of** period numbers. TE is total equity and TA is total assets.For the sustainable growth rate:Sustainable growth rate = (ROE E × b) / (1 – ROE E × b)Sustainable growth rate = (NI/TE E × b) / (1 – NI/TE E × b)We multiply this equation by:(TE E / TE E )Sustainable growth rate = (NI / TE E × b) / (1 – NI / TE E × b) × (TE E / TE E )Sustainable growth rate = (NI × b) / (TE E – NI × b)Recognize that the denominator is equal to beginning **of** period equity, that is:(TE E – NI × b) = TE BSubstituting this into the previous equation, we get:Sustainable rate = (NI × b) / TE B

CHAPTER 3 B- 45Which is equivalent to:Sustainable rate = (NI / TE B ) × bSince ROE B = NI / TE BThe sustainable growth rate equation is:Sustainable growth rate = ROE B × bFor the internal growth rate:Internal growth rate = (ROA E × b) / (1 – ROA E × b)Internal growth rate = (NI / TA E × b) / (1 – NI / TA E × b)We multiply this equation by:(TA E / TA E )Internal growth rate = (NI / TA E × b) / [(1 – NI / TA E × b) × (TA E / TA E )]Internal growth rate = (NI × b) / (TA E – NI × b)Recognize that the denominator is equal to beginning **of** period assets, that is:(TA E – NI × b) = TA BSubstituting this into the previous equation, we get:Internal growth rate = (NI × b) / TA BWhich is equivalent to:Internal growth rate = (NI / TA B ) × bSince ROA B = NI / TA BThe internal growth rate equation is:Internal growth rate = ROA B × b30. Since the company issued no new equity, shareholders’ equity increased by retained earnings.Retained earnings for the year were:Retained earnings = NI – DividendsRetained earnings = $80,000 – 49,000Retained earnings = $31,000

B-46SOLUTIONSSo, the equity at the end **of** the year was:**End**ing equity = $165,000 + 31,000**End**ing equity = $196,000The ROE based on the end **of** period equity is:ROE = $80,000 / $196,000ROE = 40.82%The plowback ratio is:Plowback ratio = Addition to retained earnings/NIPlowback ratio = $31,000 / $80,000Plowback ratio = .3875 or = 38.75%Using the equation presented in the text for the sustainable growth rate, we get:Sustainable growth rate = (ROE × b) / [1 – (ROE × b)]Sustainable growth rate = [.4082(.3875)] / [1 – .4082(.3875)]Sustainable growth rate = .1879 or 18.79%The ROE based on the beginning **of** period equity isROE = $80,000 / $165,000ROE = .4848 or 48.48%Using the shortened equation for the sustainable growth rate and the beginning **of** period ROE, weget:Sustainable growth rate = ROE × bSustainable growth rate = .4848 × .3875Sustainable growth rate = .1879 or 18.79%Using the shortened equation for the sustainable growth rate and the end **of** period ROE, we get:Sustainable growth rate = ROE × bSustainable growth rate = .4082 × .3875Sustainable growth rate = .1582 or 15.82%Using the end **of** period ROE in the shortened sustainable growth rate results in a growth rate that istoo low. This will always occur whenever the equity increases. If equity increases, the ROE based onend **of** period equity is lower than the ROE based on the beginning **of** period equity. The ROE (andsustainable growth rate) in the abbreviated equation is based on equity that did not exist when the netincome was earned.

CHAPTER 4DISCOUNTED CASH FLOW VALUATIONAnswers to Concepts Review and Critical Thinking Questions1. Assuming positive cash flows and interest rates, the future value increases and the present valuedecreases.2. Assuming positive cash flows and interest rates, the present value will fall and the future value willrise.3. The better deal is the one with equal installments.4. Yes, they should. APRs generally don’t provide the relevant rate. The only advantage is that they areeasier to compute, but, with modern computing equipment, that advantage is not very important.5. A freshman does. The reason is that the freshman gets to use the money for much longer beforeinterest starts to accrue.6. It’s a reflection **of** the time value **of** money. GMAC gets to use the $500 immediately. If GMAC usesit wisely, it will be worth more than $10,000 in thirty years.7. Oddly enough, it actually makes it more desirable since GMAC only has the right to pay the full$10,000 before it is due. This is an example **of** a “call” feature. Such features are discussed at lengthin a later chapter.8. The key considerations would be: (1) Is the rate **of** return implicit in the **of**fer attractive relative toother, similar risk investments? and (2) How risky is the investment; i.e., how certain are we that wewill actually get the $10,000? Thus, our answer does depend on who is making the promise to repay.9. The Treasury security would have a somewhat higher price because the Treasury is the strongest **of**all borrowers.10. The price would be higher because, as time passes, the price **of** the security will tend to rise toward$10,000. This rise is just a reflection **of** the time value **of** money. As time passes, the time untilreceipt **of** the $10,000 grows shorter, and the present value rises. In 2010, the price will probably behigher for the same reason. We cannot be sure, however, because interest rates could be muchhigher, or GMAC’s financial position could deteriorate. Either event would tend to depress thesecurity’s price.

B-48SOLUTIONS**Solutions** to Questions and ProblemsNOTE: 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 each problem isfound without rounding during any step in the problem.Basic1. The simple interest per year is:$5,000 × .07 = $350So, after 10 years, you will have:$350 × 10 = $3,500 in interest.The total balance will be $5,000 + 3,500 = $8,500With compound interest, we use the future value formula:FV = PV(1 +r) tFV = $5,000(1.07) 10 = $9,835.76The difference is:$9,835.76 – 8,500 = $1,335.762. To find the FV **of** a lump sum, we use:FV = PV(1 + r) ta. FV = $1,000(1.05) 10 = $1,628.89b. FV = $1,000(1.07) 10 = $1,967.15c. FV = $1,000(1.05) 20 = $2,653.30d. Because interest compounds on the interest already earned, the future value in part c is morethan twice the future value in part a. With compound interest, future values grow exponentially.3. To find the PV **of** a lump sum, we use:PV = FV / (1 + r) tPV = $15,451 / (1.05) 6 = $11,529.77PV = $51,557 / (1.11) 9 = $20,154.91PV = $886,073 / (1.16) 18 = $61,266.87PV = $550,164 / (1.19) 23 = $10,067.28

CHAPTER 4 B- 494. To answer this question, we can use either the FV or the PV formula. Both will give the same answersince they are the inverse **of** each other. We will use the FV formula, that is:FV = PV(1 + r) tSolving for r, we get:r = (FV / PV) 1 / t – 1FV = $307 = $265(1 + r) 2 ; r = ($307 / $265) 1/2 – 1 = 7.63%FV = $896 = $360(1 + r) 9 ; r = ($896 / $360) 1/9 – 1 = 10.66%FV = $162,181 = $39,000(1 + r) 15 ; r = ($162,181 / $39,000) 1/15 – 1 = 9.97%FV = $483,500 = $46,523(1 + r) 30 ; r = ($483,500 / $46,523) 1/30 – 1 = 8.12%5. To answer this question, we can use either the FV or the PV formula. Both will give the same answersince they are the inverse **of** each other. We will use the FV formula, that is:FV = PV(1 + r) tSolving for t, we get:t = ln(FV / PV) / ln(1 + r)FV = $1,284 = $625(1.08) t ;FV = $4,341 = $810(1.07) t ;FV = $402,662 = $18,400(1.21) t ;FV = $173,439 = $21,500(1.29) t ;t = ln($1,284/ $625) / ln 1.08 = 9.36 yrst = ln($4,341/ $810) / ln 1.07 = 24.81 yrst = ln($402,662 / $18,400) / ln 1.21 = 16.19 yrst = ln($173,439 / $21,500) / ln 1.29 = 8.20 yrs6. To find the length **of** time for money to double, triple, etc., the present value and future value areirrelevant as long as the future value is twice the present value for doubling, three times as large fortripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give thesame answer since they are the inverse **of** each other. We will use the FV formula, that is:FV = PV(1 + r) tSolving for t, we get:t = ln(FV / PV) / ln(1 + r)The length **of** time to double your money is:FV = $2 = $1(1.07) tt = ln 2 / ln 1.07 = 10.24 yearsThe length **of** time to quadruple your money is:FV = $4 = $1(1.07) tt = ln 4 / ln 1.07 = 20.49 years

B-50SOLUTIONSNotice that the length **of** time to quadruple your money is twice as long as the time needed to doubleyour money (the difference in these answers is due to rounding). This is an important concept **of** timevalue **of** money.7. To find the PV **of** a lump sum, we use:PV = FV / (1 + r) tPV = $800,000,000 / (1.095) 20 = $130,258,959.128. To answer this question, we can use either the FV or the PV formula. Both will give the same answersince they are the inverse **of** each other. We will use the FV formula, that is:FV = PV(1 + r) tSolving for r, we get: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 FV is less than the PV.9. A consol is a perpetuity. To find the PV **of** a perpetuity, we use the equation:PV = C / rPV = $120 / .15PV = $800.0010. To find the future value with continuous compounding, we use the equation:FV = PVe Rta. FV = $1,000e .12(5) = $1,822.12b. FV = $1,000e .10(3) = $1,349.86c. FV = $1,000e .05(10) = $1,648.72d. FV = $1,000e .07(8) = $1,750.6711. To solve this problem, we must find the PV **of** each cash flow and add them. To find the PV **of** alump sum, we use:PV = FV / (1 + r) [email protected]% = $1,200 / 1.10 + $600 / 1.10 2 + $855 / 1.10 3 + $1,480 / 1.10 4 = $3,[email protected]% = $1,200 / 1.18 + $600 / 1.18 2 + $855 / 1.18 3 + $1,480 / 1.18 4 = $2,[email protected]% = $1,200 / 1.24 + $600 / 1.24 2 + $855 / 1.24 3 + $1,480 / 1.24 4 = $2,432.40

CHAPTER 4 B- 5112. To find the PVA, we use the equation:PVA = C({1 – [1/(1 + r)] t } / r )At a 5 percent interest rate:[email protected]%: PVA = $4,000{[1 – (1/1.05) 9 ] / .05 } = $28,[email protected]%: PVA = $6,000{[1 – (1/1.05) 5 ] / .05 } = $25,976.86And at a 22 percent interest rate:[email protected]%: PVA = $4,000{[1 – (1/1.22) 9 ] / .22 } = $15,[email protected]%: PVA = $6,000{[1 – (1/1.22) 5 ] / .22 } = $17,181.84Notice that the PV **of** Cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a22 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, thetotal cash flow is more important since the cost **of** waiting (the interest rate) is not as great. At ahigher interest rate, Y is more valuable since it has larger cash flows. At a higher interest rate, thesebigger cash flows early are more important since the cost **of** waiting (the interest rate) is so muchgreater.13. To find the PVA, we use the equation:PVA = C({1 – [1/(1 + r)] t } / r )[email protected] yrs: PVA = $3,600{[1 – (1/1.10) 15 ] / .10} = $27,[email protected] yrs: PVA = $3,600{[1 – (1/1.10) 40 ] / .10} = $35,[email protected] yrs: PVA = $3,600{[1 – (1/1.10) 75 ] / .10} = $35,971.70To find the PV **of** a perpetuity, we use the equation:PV = C / rPV = $3,600 / .10PV = $36,000.00Notice that as the length **of** the annuity payments increases, the present value **of** the annuityapproaches the present value **of** the perpetuity. The present value **of** the 75-year annuity and thepresent value **of** the perpetuity imply that the value today **of** all perpetuity payments beyond 75 yearsis only $28.30.14. This cash flow is a perpetuity. To find the PV **of** a perpetuity, we use the equation:PV = C / rPV = $15,000 / .08 = $187,500.00

B-52SOLUTIONSTo find the interest rate that equates the perpetuity cash flows with the PV **of** the cash flows. Usingthe PV **of** a perpetuity equation:PV = C / r$195,000 = $15,000 / rWe can now solve for the interest rate as follows:r = $15,000 / $195,000 = 7.69%15. For discrete compounding, to find the EAR, we use the equation:EAR = [1 + (APR / m)] m – 1EAR = [1 + (.11 / 4)] 4 – 1 = 11.46%EAR = [1 + (.07 / 12)] 12 – 1 = 7.23%EAR = [1 + (.09 / 365)] 365 – 1 = 9.42%To find the EAR with continuous compounding, we use the equation:EAR = e q – 1EAR = e .17 – 1 = 18.53%16. Here, we are given the EAR and need to find the APR. Using the equation for discretecompounding:EAR = [1 + (APR / m)] m – 1We can now solve for the APR. Doing so, we get:APR = m[(1 + EAR) 1/m – 1]EAR = .081 = [1 + (APR / 2)] 2 – 1 APR = 2[(1.081) 1/2 – 1] = 7.94%EAR = .076 = [1 + (APR / 12)] 12 – 1 APR = 12[(1.076) 1/12 – 1] = 7.35%EAR = .168 = [1 + (APR / 52)] 52 – 1 APR = 52[(1.168) 1/52 – 1] = 15.55%Solving the continuous compounding EAR equation:EAR = e q – 1We get:APR = ln(1 + EAR)APR = ln(1 + .262)APR = 23.27%

CHAPTER 4 B- 5317. For discrete compounding, to find the EAR, we use the equation:EAR = [1 + (APR / m)] m – 1So, for each bank, the EAR is:First National: EAR = [1 + (.122 / 12)] 12 – 1 = 12.91%First United: EAR = [1 + (.124 / 2)] 2 – 1 = 12.78%Notice that the higher APR does not necessarily mean the higher EAR. The number **of** compoundingperiods within a year will also affect the EAR.18. The cost **of** a case **of** wine is 10 percent less than the cost **of** 12 individual bottles, so the cost **of** acase will be:Cost **of** case = (12)($10)(1 – .10)Cost **of** case = $108Now, we need to find the interest rate. The cash flows are an annuity due, so:PVA = (1 + r) C({1 – [1/(1 + r)] t } / r)$108 = (1 + r) $10({1 – [1 / (1 + r) 12 ] / r )Solving for the interest rate, we get:r = .0198 or 1.98% per weekSo, the APR **of** this investment is:APR = .0198(52)APR = 1.0277 or 102.77%And the EAR is:EAR = (1 + .0198) 52 – 1EAR = 1.7668 or 176.68%The analysis appears to be correct. He really can earn about 177 percent buying wine by the case.The only question left is this: Can you really find a fine bottle **of** Bordeaux for $10?19. Here, we need to find the length **of** an annuity. We know the interest rate, the PV, and the payments.Using the PVA equation:PVA = C({1 – [1/(1 + r)] t } / r)$16,500 = $500{ [1 – (1/1.009) t ] / .009}

B-54SOLUTIONSNow, we solve for t:1/1.009 t = 1 – [($16,500)(.009) / ($500)]1.009 t = 1/(0.703) = 1.422t = ln 1.422 / ln 1.009 = 39.33 months20. Here, we are trying to find the interest rate when we know the PV and FV. Using the FV equation:FV = PV(1 + r)$4 = $3(1 + r)r = 4/3 – 1 = 33.33% per weekThe interest rate is 33.33% per week. To find the APR, we multiply this rate by the number **of** weeksin 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%Intermediate21. To find the FV **of** a lump sum with discrete compounding, we use:FV = PV(1 + r) ta. FV = $1,000(1.08) 3 = $1,259.71b. FV = $1,000(1 + .08/2) 6 = $1,265.32c. FV = $1,000(1 + .08/12) 36 = $1,270.24To find the future value with continuous compounding, we use the equation:FV = PVe Rtd. FV = $1,000e .08(3) = $1,271.25e. The future value increases when the compounding period is shorter because interest is earnedon previously accrued interest. The shorter the compounding period, the more frequentlyinterest is earned, and the greater the future value, assuming the same stated interest rate.22. The total interest paid by First Simple Bank is the interest rate per period times the number **of**periods. In other words, the interest by First Simple Bank paid over 10 years will be:.08(10) = .8

CHAPTER 4 B- 55First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor**of** $1, or:(1 + r) 10Setting the two equal, we get:(.08)(10) = (1 + r) 10 – 1r = 1.8 1/10 – 1 = 6.05%23. We need to find the annuity payment in retirement. Our retirement savings ends at the same time theretirement withdrawals begin, so the PV **of** the retirement withdrawals will be the FV **of** theretirement savings. So, we 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)] = $1,963,163.82Bond account: FVA = $300[{[1 + (.07/12) ] 360 – 1} / (.07/12)] = $365,991.30So, the total amount saved at retirement is:$1,963,163.82 + 365,991.30 = $2,329,155.11Solving for the withdrawal amount in retirement using the PVA equation gives us:PVA = $2,329,155.11 = C[1 – {1 / [1 + (.09/12)] 300 } / (.09/12)]C = $2,329,155.11 / 119.1616 = $19,546.19 withdrawal per month24. Since we are looking to triple our money, the PV and FV are irrelevant as long as the FV is threetimes as large as the PV. The number **of** periods is four, the number **of** quarters per year. So:FV = $3 = $1(1 + r) (12/3)r = 31.61%25. Here, we need to find the interest rate for two possible investments. Each investment is a lump sum,so:G: PV = $50,000 = $85,000 / (1 + r) 5(1 + r) 5 = $85,000 / $50,000r = (1.70) 1/5 – 1 = 11.20%H: PV = $50,000 = $175,000 / (1 + r) 11(1 + r) 11 = $175,000 / $50,000r = (3.50) 1/11 – 1 = 12.06%

B-56SOLUTIONS26. This is a growing perpetuity. The present value **of** a growing perpetuity is:PV = C / (r – g)PV = $200,000 / (.10 – .05)PV = $4,000,000It is important to recognize that when dealing with annuities or perpetuities, the present valueequation calculates the present value one period before the first payment. In this case, since the firstpayment is in two years, we have calculated the present value one year from now. To find the valuetoday, we simply discount this value as a lump sum. Doing so, we find the value **of** the cash flowstream today is:PV = FV / (1 + r) tPV = $4,000,000 / (1 + .10) 1PV = $3,636,363.6427. The dividend payments are made quarterly, so we must use the quarterly interest rate. The quarterlyinterest rate is:Quarterly rate = Stated rate / 4Quarterly rate = .12 / 4Quarterly rate = .03Using the present value equation for a perpetuity, we find the value today **of** the dividends paid mustbe:PV = C / rPV = $10 / .03PV = $333.3328. We can use the PVA annuity equation to answer this question. The annuity has 20 payments, not 19payments. Since there is a payment made in Year 3, the annuity actually begins in Year 2. So, thevalue **of** the annuity in Year 2 is:PVA = C({1 – [1/(1 + r)] t } / r )PVA = $2,000({1 – [1/(1 + .08)] 20 } / .08)PVA = $19,636.29This is the value **of** the annuity one period before the first payment, or Year 2. So, the value **of** thecash flows today is:PV = FV/(1 + r) tPV = $19,636.29/(1 + .08) 2PV = $16,834.9629. We need to find the present value **of** an annuity. Using the PVA equation, and the 15 percent interestrate, we get:PVA = C({1 – [1/(1 + r)] t } / r )PVA = $500({1 – [1/(1 + .15)] 15 } / .15)PVA = $2,923.69

CHAPTER 4 B- 57This is the value **of** the annuity in Year 5, one period before the first payment. Finding the value **of**this amount today, we find:PV = FV/(1 + r) tPV = $2,923.69/(1 + .12) 5PV = $1,658.9830. The amount borrowed is the value **of** the home times one minus the down payment, or:Amount borrowed = $400,000(1 – .20)Amount borrowed = $320,000The monthly payments with a balloon payment loan are calculated assuming a longer amortizationschedule, in this case, 30 years. The payments based on a 30-year repayment schedule would be:PVA = $320,000 = C({1 – [1 / (1 + .08/12)] 360 } / (.08/12))C = $2,348.05Now, at time = 8, we need to find the PV **of** the payments which have not been made. The balloonpayment will be:PVA = $2,348.05({1 – [1 / (1 + .08/12)] 22(12) } / (.08/12))PVA = $291,256.6331. Here, we need to find the FV **of** a lump sum, with a changing interest rate. We must do this problemin two parts. After the first six months, the balance will be:FV = $4,000 [1 + (.019/12)] 6 = $4,038.15This is the balance in six months. The FV in another six months will be:FV = $4,038.15 [1 + (.16/12)] 6 = $4,372.16The problem asks for the interest accrued, so, to find the interest, we subtract the beginning balancefrom the FV. The interest accrued is:Interest = $4,372.16 – 4,000.00 = $372.1632. The company would be indifferent at the interest rate that makes the present value **of** the cash flowsequal to the cost today. Since the cash flows are a perpetuity, we can use the PV **of** a perpetuityequation. Doing so, we find:PV = C / r$240,000 = $21,000 / rr = $21,000 / $240,000r = .0875 or 8.75%

B-58SOLUTIONS33. The company will accept the project if the present value **of** the increased cash flows is greater thanthe cost. The cash flows are a growing perpetuity, so the present value is:PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)] t }PV = $12,000{[1/(.11 – .06)] – [1/(.11 – .06)] × [(1 + .06)/(1 + .11)] 5 }PV = $49,398.78The company should not accept the project since the cost is greater than the increased cash flows.34. Since your salary grows at 4 percent per year, your salary next year will be:Next year’s salary = $50,000 (1 + .04)Next year’s salary = $52,000This means your deposit next year will be:Next year’s deposit = $52,000(.02)Next year’s deposit = $1,040Since your salary grows at 4 percent, you deposit will also grow at 4 percent. We can use the presentvalue **of** a growing perpetuity equation to 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 = $1,040{[1/(.08 – .04)] – [1/(.08 – .04)] × [(1 + .04)/(1 + .08)] 40 }PV = $20,254.12Now, we can find the future value **of** this lump sum in 40 years. We find:FV = PV(1 + r) tFV = $20,254.12(1 + .08) 40FV = $440,011.02This is the value **of** your savings in 40 years.35. 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 decreasesThe present values **of** $5,000 per year for 10 years at the various interest rates given are:[email protected]% = $5,000{[1 – (1/1.10) 10 ] / .10} = $30,[email protected]% = $5,000{[1 – (1/1.05) 10 ] / .05} = $38,[email protected]% = $5,000{[1 – (1/1.15) 10 ] / .15} = $25,093.84

CHAPTER 4 B- 5936. Here, we are given the FVA, the interest rate, and the amount **of** the annuity. We need to solve forthe number **of** payments. Using the FVA equation:FVA = $20,000 = $125[{[1 + (.10/12)] t – 1 } / (.10/12)]Solving for t, we get:1.00833 t = 1 + [($20,000)(.10/12) / 125]t = ln 2.33333 / ln 1.00833 = 102.10 payments37. Here, we are given the PVA, number **of** periods, and the amount **of** the annuity. We need to solve forthe interest rate. Using the PVA equation:PVA = $45,000 = $950[{1 – [1 / (1 + r)] 60 }/ r]To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,or by trial and error. If you use trial and error, remember that increasing the interest rate lowers thePVA, and increasing the interest rate decreases the PVA. Using a spreadsheet, we find:r = 0.810%The APR is the periodic interest rate times the number **of** periods in the year, so:APR = 12(0.810) = 9.72%38. The amount **of** principal paid on the loan is the PV **of** the monthly payments you make. So, thepresent value **of** the $1,000 monthly payments is:PVA = $1,000[(1 – {1 / [1 + (.068/12)]} 360 ) / (.068/12)] = $153,391.83The monthly payments **of** $1,000 will amount to a principal payment **of** $153,391.83. The amount **of**principal you will still owe is:$200,000 – 153,391.83 = $46,608.17This remaining principal amount will increase at the interest rate on the loan until the end **of** the loanperiod. So the balloon payment in 30 years, which is the FV **of** the remaining principal will be:Balloon payment = $46,608.17 [1 + (.068/12)] 360 = $356,387.1039. We are given the total PV **of** all four cash flows. If we find the PV **of** the three cash flows we know, andsubtract them from the total PV, the amount left over must be the PV **of** the missing cash flow. So, thePV **of** the cash flows we know are:PV **of** Year 1 CF: $1,000 / 1.10 = $909.09PV **of** Year 3 CF: $2,000 / 1.10 3 = $1,502.63PV **of** Year 4 CF: $2,000 / 1.10 4 = $1,366.03

B-60SOLUTIONSSo, the PV **of** the missing CF is:$5,979 – 909.09 – 1,502.63 – 1,366.03 = $2,201.25The question asks for the value **of** the cash flow in Year 2, so we must find the future value **of** thisamount. The value **of** the missing CF is:$2,201.25(1.10) 2 = $2,663.5240. To solve this problem, we simply need to find the PV **of** each lump sum and add them together. It isimportant to note that the first cash flow **of** $1 million occurs today, so we do not need to discountthat cash flow. The PV **of** the lottery winnings is:$1,000,000 + $1,400,000/1.10 + $1,800,000/1.10 2 + $2,200,000/1.10 3 + $2,600,000/1.10 4 +$3,000,000/1.10 5 + $3,400,000/1.10 6 + $3,800,000/1.10 7 + $4,200,000/1.10 8 +$4,600,000/1.10 9 + $5,000,000/1.10 10 = $18,758,930.7941. Here, we are finding interest rate for an annuity cash flow. We are given the PVA, number **of**periods, and the amount **of** the annuity. We need to solve for the number **of** payments. We shouldalso note that the PV **of** the annuity is not the amount borrowed since we are making a downpayment on the warehouse. The amount borrowed is:Amount borrowed = 0.80($1,600,000) = $1,280,000Using the PVA equation:PVA = $1,280,000 = $10,000[{1 – [1 / (1 + r)] 360 }/ r]Unfortunately, this equation cannot be solved to find the interest rate using algebra. To find theinterest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trialand error. If you use trial and error, remember that increasing the interest rate decreases the PVA,and decreasing the interest rate increases the PVA. Using a spreadsheet, we find:r = 0.7228%The APR is the monthly interest rate times the number **of** months in the year, so:APR = 12(0.7228) = 8.67%And the EAR is:EAR = (1 + .007228) 12 – 1 = 9.03%42. The pr**of**it the firm earns is just the PV **of** the sales price minus the cost to produce the asset. We findthe PV **of** the sales price as the PV **of** a lump sum:PV = $115,000 / 1.13 3 = $79,700.77

CHAPTER 4 B- 61And the firm’s pr**of**it is:Pr**of**it = $79,700.77 – 72,000.00 = $7,700.77To find the interest rate at which the firm will break even, we need to find the interest rate using thePV (or FV) **of** a lump sum. Using the PV equation for a lump sum, we get:$72,000 = $115,000 / ( 1 + r) 3r = ($115,000 / $72,000) 1/3 – 1 = 16.89%43. We want to find the value **of** the cash flows today, so we will find the PV **of** the annuity, and thenbring the lump sum PV back to today. The annuity has 17 payments, so the PV **of** the annuity is:PVA = $2,000{[1 – (1/1.12) 17 ] / .12} = $14,239.26Since this is an ordinary annuity equation, this is the PV one period before the first payment, so it isthe PV at t = 8. To find the value today, we find the PV **of** this lump sum. The value today is:PV = $14,239.26 / 1.12 8 = $5,751.0044. This question is asking for the present value **of** an annuity, but the interest rate changes during thelife **of** the annuity. We need to find the present value **of** the cash flows for the last eight years first.The PV **of** these cash flows is:PVA 2 = $1,500 [{1 – 1 / [1 + (.12/12)] 96 } / (.12/12)] = $92,291.55Note that this is the PV **of** this annuity exactly seven years from today. Now, we can discount thislump sum to today. The value **of** this cash flow today is:PV = $92,291.55 / [1 + (.15/12)] 84 = $32,507.18Now, we need to find the PV **of** the annuity for the first seven years. The value **of** these cash flowstoday is:PVA 1 = $1,500 [{1 – 1 / [1 + (.15/12)] 84 } / (.15/12)] = $77,733.28The value **of** the cash flows today is the sum **of** these two cash flows, so:PV = $77,733.28 + 32,507.18 = $110,240.4645. Here, we are trying to find the dollar amount invested today that will equal the FVA with a knowninterest rate, and payments. First, we need to determine how much we would have in the annuityaccount. Finding the FV **of** the annuity, we get:FVA = $1,000 [{[ 1 + (.105/12)] 180 – 1} / (.105/12)] = $434,029.81Now, we need to find the PV **of** a lump sum that will give us the same FV. So, using the FV **of** alump sum with continuous compounding, we get:FV = $434,029.81 = PVe .09(15)PV = $434,029.81 e –1.35 = $112,518.00

B-62SOLUTIONS46. To find the value **of** the perpetuity at t = 7, we first need to use the PV **of** a perpetuity equation.Using this equation we find:PV = $3,000 / .065 = $46,153.85Remember that the PV **of** a perpetuity (and annuity) equations give the PV one period before the firstpayment, so, this is the value **of** the perpetuity at t = 14. To find the value at t = 7, we find the PV **of**this lump sum as:PV = $46,153.85 / 1.065 7 = $29,700.2947. To find the APR and EAR, we need to use the actual cash flows **of** the loan. In other words, theinterest rate quoted in the problem is only relevant to determine the total interest under the termsgiven. The interest rate for the cash flows **of** the loan is:PVA = $20,000 = $1,900{(1 – [1 / (1 + r)] 12 ) / r }Again, we cannot solve this equation for r, so we need to solve this equation on a financialcalculator, using a spreadsheet, or by trial and error. Using a spreadsheet, we find:r = 2.076% per monthSo the APR is:APR = 12(2.076%) = 24.91%And the EAR is:EAR = (1.0276) 12 – 1 = 27.96%48. The cash flows in this problem are semiannual, so we need the effective semiannual rate. Theinterest rate given is the APR, so the monthly interest rate is:Monthly rate = .12 / 12 = .01To get the semiannual interest rate, we can use the EAR equation, but instead **of** using 12 months asthe exponent, we will use 6 months. The effective semiannual rate is:Semiannual rate = (1.01) 6 – 1 = 6.15%We can now use this rate to find the PV **of** the annuity. The PV **of** the annuity is:PVA @ t = 9: $6,000{[1 – (1 / 1.0615) 10 ] / .0615} = $43,844.21Note, that this is the value one period (six months) before the first payment, so it is the value at t = 9.So, the value at the various times the questions asked for uses this value 9 years from now.PV @ t = 5: $43,844.21 / 1.0615 8 = $27,194.83

CHAPTER 4 B- 63Note, that you can also calculate this present value (as well as the remaining present values) usingthe number **of** years. To do this, you need the EAR. The EAR is:EAR = (1 + .01) 12 – 1 = 12.68%So, we can find the PV at t = 5 using the following method as well:PV @ t = 5: $43,844.21 / 1.1268 4 = $27,194.83The value **of** the annuity at the other times in the problem is:PV @ t = 3: $43,844.21 / 1.0615 12 = $21,417.72PV @ t = 3: $43,844.21 / 1.1268 6 = $21,417.72PV @ t = 0: $43,844.21 / 1.0615 18 = $14,969.38PV @ t = 0: $43,844.21 / 1.1268 9 = $14,969.3849. a. Calculating the PV **of** an ordinary annuity, we get:PVA = $525{[1 – (1/1.095) 6 ] / .095} = $2,320.41b. To calculate the PVA due, we calculate the PV **of** an ordinary annuity for t – 1 payments, andadd the payment that occurs today. So, the PV **of** the annuity due is:PVA = $525 + $525{[1 – (1/1.095) 5 ] / .095} = $2,540.8550. We need to use the PVA due equation, that is:PVA due = (1 + r) PVAUsing this equation:PVA due = $56,000 = [1 + (.0815/12)] × C[{1 – 1 / [1 + (.0815/12)] 48 } / (.0815/12)$55,622.23 = C{1 – [1 / (1 + .0815/12) 48 ]} / (.0815/12)C = $1,361.82Notice, that when we find the payment for the PVA due, we simply discount the PV **of** the annuitydue back one period. We then use this value as the PV **of** an ordinary annuity.Challenge51. The monthly interest rate is the annual interest rate divided by 12, or:Monthly interest rate = .12 / 12Monthly interest rate = .01

B-64SOLUTIONSNow we can set the present value **of** the lease payments equal to the cost **of** the equipment, or$4,000. The lease payments are in the form **of** an annuity due, so:PVA due = (1 + r) C({1 – [1/(1 + r)] t } / r )$4,000 = (1 + .01) C({1 – [1/(1 + .01)] 24 } / .01 )C = $186.4352. First, we will calculate the present value if the college expenses for each child. The expenses are anannuity, so the present value **of** the college expenses is:PVA = C({1 – [1/(1 + r)] t } / r )PVA = $23,000({1 – [1/(1 + .065)] 4 } / .065)PVA = $78,793.37This is the cost **of** each child’s college expenses one year before they enter college. So, the cost **of**the oldest child’s college expenses today will be:PV = FV/(1 + r) tPV = $78,793.37/(1 + .065) 14PV = $32,628.35And the cost **of** the youngest child’s college expenses today will be:PV = FV/(1 + r) tPV = $78,793.37/(1 + .065) 16PV = $28,767.09Therefore, the total cost today **of** your children’s college expenses is:Cost today = $32,628.35 + 28,767.09Cost today = $61,395.44This is the present value **of** your annual savings, which are an annuity. So, the amount you must saveeach year will be:PVA = C({1 – [1/(1 + r)] t } / r )$61,395.44 = C({1 – [1/(1 + .065)] 15 } / .065)C = $6,529.5853. The salary is a growing annuity, so using the equation for the present value **of** a growing annuity.The salary growth rate is 4 percent and the discount rate is 12 percent, so the value **of** the salary **of**fertoday is:PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)] t }PV = $35,000{[1/(.12 – .04)] – [1/(.12 – .04)] × [(1 + .04)/(1 + .12)] 25 }PV = $368,894.18The yearly bonuses are 10 percent **of** the annual salary. This means that next year’s bonus will be:Next year’s bonus = .10($35,000)Next year’s bonus = $3,500

CHAPTER 4 B- 65Since the salary grows at 4 percent, the bonus will grow at 4 percent as well. Using the growingannuity equation, with a 4 percent growth rate and a 12 percent discount rate, the present value **of** theannual bonuses is:PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)] t }PV = $3,500{[1/(.12 – .04)] – [1/(.12 – .04)] × [(1 + .04)/(1 + .12)] 25 }PV = $36,889.42Notice the present value **of** the bonus is 10 percent **of** the present value **of** the salary. The presentvalue **of** the bonus will always be the same percentage **of** the present value **of** the salary as the bonuspercentage. So, the total value **of** the **of**fer is:PV = PV(Salary) + PV(Bonus) + Bonus paid todayPV = $368,894.18 + 36,889.42 + 10,000PV = $415,783.6054. Here, we need to compare to options. In order to do so, we must get the value **of** the two cash flowstreams to the same time, so we will find the value **of** each today. We must also make sure to use theaftertax cash flows, since it is more relevant. For Option A, the aftertax cash flows are:Aftertax cash flows = Pretax cash flows(1 – tax rate)Aftertax cash flows = $160,000(1 – .28)Aftertax cash flows = $115,200The aftertax cash flows from Option A are in the form **of** an annuity due, so the present value **of** thecash flow today is:PVA due = (1 + r) C({1 – [1/(1 + r)] t } / r )PVA due = (1 + .10) $115,200({1 – [1/(1 + .10)] 31 } / .10 )PVA due = $1,201,180.55For Option B, the aftertax cash flows are:Aftertax cash flows = Pretax cash flows(1 – tax rate)Aftertax cash flows = $101,055(1 – .28)Aftertax cash flows = $72,759.60The aftertax cash flows from Option B are an ordinary annuity, plus the cash flow today, so thepresent value:PV = C({1 – [1/(1 + r)] t } / r ) + CF 0PV = $72,759.60({1 – [1/(1 + .10)] 30 } / .10 ) + $446,000PV = $1,131,898.53You should choose Option A because it has a higher present value on an aftertax basis.

B-66SOLUTIONS55. We need to find the first payment into the retirement account. The present value **of** the desiredamount at retirement is:PV = FV/(1 + r) tPV = $1,000,000/(1 + .10) 30PV = $57,308.55This is the value today. Since the savings are in the form **of** a growing annuity, we can use thegrowing annuity equation and solve for the payment. Doing so, we get:PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)] t }$57,308.55 = C{[1/(.10 – .03)] – [1/(.10 – .03)] × [(1 + .03)/(1 + .10)] 30 }C = $4,659.79This is the amount you need to save next year. So, the percentage **of** your salary is:Percentage **of** salary = $4,659.79/$55,000Percentage **of** salary = .0847 or 8.47%Note that this is the percentage **of** your salary you must save each year. Since your salary isincreasing at 3 percent, and the savings are increasing at 3 percent, the percentage **of** salary willremain constant.56. Since she put $1,000 down, the amount borrowed will be:Amount borrowed = $15,000 – 1,000Amount borrowed = $14,000So, the monthly payments will be:PVA = C({1 – [1/(1 + r)] t } / r )$14,000 = C[{1 – [1/(1 + .096/12)] 60 } / (.096/12)]C = $294.71The amount remaining on the loan is the present value **of** the remaining payments. Since the firstpayment was made on October 1, 2004, and she made a payment on October 1, 2006, there are 35payments remaining, with the first payment due immediately. So, we can find the present value **of**the remaining 34 payments after November 1, 2006, and add the payment made on this date. So theremaining principal owed on the loan is:PV = C({1 – [1/(1 + r)] t } / r ) + C 0PV = $294.71[{1 – [1/(1 + .096/12)] 34 } / (.096/12)] + $294.71C = $9,037.33She must also pay a one percent prepayment penalty, so the total amount **of** the payment is:Total payment = Amount due(1 + Prepayment penalty)Total payment = $9,037.33(1 + .01)Total payment = $9,127.71

CHAPTER 4 B- 6757. The cash flows for this problem occur monthly, and the interest rate given is the EAR. Since the cashflows occur monthly, we must get the effective monthly rate. One way to do this is to find the APRbased on monthly compounding, and then divide by 12. So, the pre-retirement APR is:EAR = .1011 = [1 + (APR / 12)] 12 – 1; APR = 12[(1.11) 1/12 – 1] = 10.48%And the post-retirement APR is:EAR = .08 = [1 + (APR / 12)] 12 – 1; APR = 12[(1.08) 1/12 – 1] = 7.72%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 = $25,000{1 – [1 / (1 + .0772/12) 12(20) ]} / (.0772/12) = $3,051,943.26PV = $750,000 / [1 + (.0772/12)] 240 = $160,911.16So, at retirement, he needs:$3,051,943.26 + 160,911.16 = $3,212,854.42He will be saving $2,100 per month for the next 10 years until he purchases the cabin. The value **of**his savings after 10 years will be:FVA = $2,100[{[ 1 + (.1048/12)] 12(10) – 1} / (.1048/12)] = $442,239.69After he purchases the cabin, the amount he will have left is:$442,239.69 – 350,000 = $92,239.69He still has 20 years until retirement. When he is ready to retire, this amount will have grown to:FV = $92,239.69[1 + (.1048/12)] 12(20) = $743,665.12So, when he is ready to retire, based on his current savings, he will be short:$3,212,854.41 – 743,665.12 = $2,469,189.29This amount is the FV **of** the monthly savings he must make between years 10 and 30. So, findingthe annuity payment using the FVA equation, we find his monthly savings will need to be:FVA = $2,469,189.29 = C[{[ 1 + (.1048/12)] 12(20) – 1} / (.1048/12)]C = $3,053.8758. To answer this question, we should find the PV **of** both options, and compare them. Since we arepurchasing the car, the lowest PV is the best option. The PV **of** the leasing is simply the PV **of** thelease payments, plus the $1. The interest rate we would use for the leasing option is the same as theinterest rate **of** the loan. The PV **of** leasing is:PV = $1 + $450{1 – [1 / (1 + .08/12) 12(3) ]} / (.08/12) = $14,361.31

B-68SOLUTIONSThe 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 + (.08/12)] 12(3) = $18,106.86The PV **of** the decision to purchase is:$35,000 – $18,106.86 = $16,893.14In this case, it is cheaper to lease the car than buy it since the PV **of** the leasing cash flows is lower.To find the breakeven resale price, we need to find the resale price that makes the PV **of** the twooptions the same. In other words, the PV **of** the decision to buy should be:$35,000 – PV **of** resale price = $14,361.31PV **of** resale price = $20,638.69The resale price that would make the PV **of** the lease versus buy decision is the FV **of** this value, so:Breakeven resale price = $20,638.69[1 + (.08/12)] 12(3) = $26,216.0359. To find the quarterly salary for the player, we first need to find the PV **of** the current contract. Thecash flows for the contract are annual, and we are given a daily interest rate. We need to find theEAR so the interest compounding is the same as the timing **of** the cash flows. The EAR is:EAR = [1 + (.045/365)] 365 – 1 = 4.60%The PV **of** the current contract **of**fer is the sum **of** the PV **of** the cash flows. So, the PV is:PV = $8,000,000 + $4,000,000/1.046 + $4,800,000/1.046 2 + $5,700,000/1.046 3 + $6,400,000/1.046 4+ $7,000,000/1.046 5 + $7,500,000/1.046 6PV = $37,852,037.91The player wants the contract increased in value by $750,000, so the PV **of** the new contract will be:PV = $37,852,037.91 + 750,000 = $38,602,037.91The player has also requested a signing bonus payable today in the amount **of** $9 million. We cansimply subtract this amount from the PV **of** the new contract. The remaining amount will be the PV**of** the future quarterly paychecks.$38,602,037.91 – 9,000,000 = $29,602,037.91To find the quarterly payments, first realize that the interest rate we need is the effective quarterlyrate. Using the daily interest rate, we can find the quarterly interest rate using the EAR equation,with the number **of** days being 91.25, the number **of** days in a quarter (365 / 4). The effectivequarterly rate is:Effective quarterly rate = [1 + (.045/365)] 91.25 – 1 = 1.131%

CHAPTER 4 B- 69Now, we have the interest rate, the length **of** the annuity, and the PV. Using the PVA equation andsolving for the payment, we get:PVA = $29,602,037.91 = C{[1 – (1/1.01131) 24 ] / .01131}C = $1,415,348.3760. To find the APR and EAR, we need to use the actual cash flows **of** the loan. In other words, theinterest rate quoted in the problem is only relevant to determine the total interest under the termsgiven. The cash flows **of** the loan are the $20,000 you must repay in one year, and the $17,600 youborrow today. The interest rate **of** the loan is:$20,000 = $17,600(1 + r)r = ($20,000 / 17,600) – 1 = 13.64%Because **of** the discount, you only get the use **of** $17,600, and the interest you pay on that amount is13.64%, not 12%.61. Here, we have cash flows that would have occurred in the past and cash flows that would occur inthe future. We need to bring both cash flows to today. Before we calculate the value **of** the cashflows today, we must adjust the interest rate, so we have the effective monthly interest rate. Findingthe APR with monthly compounding and dividing by 12 will give us the effective monthly rate. TheAPR with monthly compounding is:APR = 12[(1.09) 1/12 – 1] = 8.65%To find the value today **of** the back pay from two years ago, we will find the FV **of** the annuity, andthen find the FV **of** the lump sum. Doing so gives us:FVA = ($40,000/12) [{[ 1 + (.0865/12)] 12 – 1} / (.0865/12)] = $41,624.33FV = $41,624.33(1.09) = $45,370.52Notice we found the FV **of** the annuity with the effective monthly rate, and then found the FV **of** thelump sum with the EAR. Alternatively, we could have found the FV **of** the lump sum with theeffective monthly rate as 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 = ($43,000/12) [{[ 1 + (.0865/12)] 12 – 1} / (.0865/12)] = $44,746.15Next, we find the value today **of** the five year’s future salary:PVA = ($45,000/12){[{1 – {1 / [1 + (.0865/12)] 12(5) }] / (.0865/12)}= $182,142.14The value today **of** the jury award is the sum **of** salaries, plus the compensation for pain andsuffering, and court costs. The award should be for the amount **of**:Award = $45,370.52 + 44,746.15 + 182,142.14 + 100,000 + 20,000Award = $392,258.81

B-70SOLUTIONSAs the plaintiff, you would prefer a lower interest rate. In this problem, we are calculating both thePV and FV **of** annuities. A lower interest rate will decrease the FVA, but increase the PVA. So, by alower interest rate, we are lowering the value **of** the back pay. But, we are also increasing the PV **of**the future salary. Since the future salary is larger and has a longer time, this is the more importantcash flow to the plaintiff.62. Again, to find the interest rate **of** a loan, we need to look at the cash flows **of** the loan. Since this loanis in the form **of** a lump sum, the amount you will repay is the FV **of** the principal amount, whichwill be:Loan repayment amount = $10,000(1.10) = $11,000The amount you will receive today is the principal amount **of** the loan times one minus the points.Amount received = $10,000(1 – .03) = $9,700Now, we simply find the interest rate for this PV and FV.$11,000 = $9,700(1 + r)r = ($11,000 / $9,700) – 1 = 13.40%With a 13 percent quoted interest rate loan and two points, the EAR is:Loan repayment amount = $10,000(1.13) = $11,300Amount received = $10,000(1 – .02) = $9,800$11,300 = $9,800(1 + r)r = ($11,300 / $9,800) – 1 = 15.31%The effective rate is not affected by the loan amount, since it drops out when solving for r.63. First, we will find the APR and EAR for the loan with the refundable fee. Remember, we need to usethe actual cash flows **of** the loan to find the interest rate. With the $1,500 application fee, you willneed to borrow $201,500 to have $200,000 after deducting the fee. Solving for the payment underthese circumstances, we get:PVA = $201,500 = C {[1 – 1/(1.00625) 360 ]/.00625} where .00625 = .075/12C = $1,408.92We can now use this amount in the PVA equation with the original amount we wished to borrow,$200,000. Solving for r, we find:PVA = $200,000 = $1,408.92[{1 – [1 / (1 + r)] 360 }/ r]

CHAPTER 4 B- 71Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:r = 0.6314% per monthAPR = 12(0.6314%) = 7.58%EAR = (1 + .006314) 12 – 1 = 7.85%With the nonrefundable fee, the APR **of** the loan is simply the quoted APR since the fee is notconsidered part **of** the loan. So:APR = 7.50%EAR = [1 + (.075/12)] 12 – 1 = 7.76%64. Be careful **of** interest rate quotations. The actual interest rate **of** a loan is determined by the cashflows. Here, we are told that the PV **of** the loan is $1,000, and the payments are $42.25 per month forthree years, so the interest rate on the loan is:PVA = $1,000 = $42.25[ {1 – [1 / (1 + r)] 36 } / r ]Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:r = 2.47% per monthAPR = 12(2.47%) = 29.63%EAR = (1 + .0247) 12 – 1 = 34.00%It’s called add-on interest because the interest amount **of** the loan is added to the principal amount **of**the loan before the loan payments are calculated.65. Here, we are solving a two-step time value **of** money problem. Each question asks for a differentpossible cash flow to fund the same retirement plan. Each savings possibility has the same FV, thatis, the PV **of** the retirement spending when your friend is ready to retire. The amount needed whenyour friend is ready to retire is:PVA = $90,000{[1 – (1/1.08) 15 ] / .08} = $770,353.08This 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 equalto the amount needed in retirement. The required savings each year will be:FVA = $770,353.08 = C[(1.08 30 – 1) / .08]C = $6,800.24b. Here we need to find a lump sum savings amount. Using the FV for a lump sum equation, we get:FV = $770,353.08 = PV(1.08) 30PV = $76,555.63

B-72SOLUTIONSc. In this problem, we have a lump sum savings in addition to an annual deposit. Since we alreadyknow the value needed at retirement, we can subtract the value **of** the lump sum savings atretirement to find out how much your friend is short. Doing so gives us:FV **of** trust fund deposit = $25,000(1.08) 10 = $53,973.12So, the amount your friend still needs at retirement is:FV = $770,353.08 – 53,973.12 = $716,379.96Using the FVA equation, and solving for the payment, we get:$716,379.96 = C[(1.08 30 – 1) / .08]C = $6,323.80This is the total annual contribution, but your friend’s employer will contribute $1,500 per year,so your friend must contribute:Friend's contribution = $6,323.80 – 1,500 = $4,823.8066. We will calculate the number **of** periods necessary to repay the balance with no fee first. We simplyneed to use the PVA equation and solve for the number **of** payments.Without fee and annual rate = 19.20%:PVA = $10,000 = $200{[1 – (1/1.016) t ] / .016 } where .016 = .192/12Solving for t, we get:t = ln{1 / [1 – ($10,000/$200)(.016)]} / ln(1.016)t = ln 5 / ln 1.016t = 101.39 monthsWithout fee and annual rate = 9.20%:PVA = $10,000 = $200{[1 – (1/1.0076667) t ] / .0076667 } where .0076667 = .092/12Solving for t, we get:t = ln{1 / [1 – ($10,000/$200)(.0076667)]} / ln(1.0076667)t = ln 1.6216 / ln 1.0076667t = 63.30 monthsNote that we do not need to calculate the time necessary to repay your current credit card with a feesince no fee will be incurred. The time to repay the new card with a transfer fee is:

CHAPTER 4 B- 73With fee and annual rate = 9.20%:PVA = $10,200 = $200{ [1 – (1/1.0076667) t ] / .0076667 } where .0076667 = .092/12Solving for t, we get:t = ln{1 / [1 – ($10,200/$200)(.0076667)]} / ln(1.0076667)t = ln 1.6420 / ln 1.0076667t = 64.94 months67. We need to find the FV **of** the premiums to compare with the cash payment promised at age 65. Wehave to find the value **of** the premiums at year 6 first since the interest rate changes at that time. So:FV 1 = $750(1.11) 5 = $1,263.79FV 2 = $750(1.11) 4 = $1,138.55FV 3 = $850(1.11) 3 = $1,162.49FV 4 = $850(1.11) 2 = $1,047.29FV 5 = $950(1.11) 1 = $1,054.50Value at year six = $1,263.79 + 1,138.55 + 1,162.49 + 1,047.29 + 1,054.50 + 950.00 = $6,616.62Finding the FV **of** this lump sum at the child’s 65 th birthday:FV = $6,616.62(1.07) 59 = $358,326.50The policy is not worth buying; the future value **of** the policy is $358,326.50, but the policy contractwill pay **of**f $250,000. The premiums are worth $108,326.50 more than the policy pay**of**f.Note, we could also compare the PV **of** the two cash flows. The PV **of** the premiums is:PV = $750/1.11 + $750/1.11 2 + $850/1.11 3 + $850/1.11 4 + $950/1.11 5 + $950/1.11 6 = $3,537.51And the value today **of** the $250,000 at age 65 is:PV = $250,000/1.07 59 = $4,616.33PV = $4,616.33/1.11 6 = $2,468.08The premiums still have the higher cash flow. At time zero, the difference is $2,148.25. Wheneveryou are comparing two or more cash flow streams, the cash flow with the highest value at one timewill have the highest value at any other time.Here is a question for you: Suppose you invest $2,148.25, the difference in the cash flows at timezero, for six years at an 11 percent interest rate, and then for 59 years at a seven percent interest rate.How much will it be worth? Without doing calculations, you know it will be worth $108,326.50, thedifference in the cash flows at time 65!

B-74SOLUTIONS68. Since the payments occur at six month intervals, we need to get the effective six-month interest rate.We can calculate the daily interest rate since we have an APR compounded daily, so the effectivesix-month interest rate is:Effective six-month rate = (1 + Daily rate) 180 – 1Effective six-month rate = (1 + .09/365) 180 – 1Effective six-month rate = .0454 or 4.54%Now, we can use the PVA equation to find the present value **of** the semi-annual payments. Doing so,we find:PVA = C({1 – [1/(1 + r)] t } / r )PVA = $500,000({1 – [1/(1 + .0454] 40 } / .0454)PVA = $9,151,418.61This is the value six months from today, which is one period (six months) prior to the first payment.So, the value today is:PV = $9,151,418.61 / (1 + .0454)PV = $8,754,175.76This means the total value **of** the lottery winnings today is:Value **of** winnings today = $8,754,175.76 + 1,000,000Value **of** winnings today = $9,754,175.76You should take the **of**fer since the value **of** the **of**fer is greater than the present value **of** thepayments.69. Here, we need to find the interest rate that makes the PVA, the college costs, equal to the FVA, thesavings. The PV **of** the college costs are:PVA = $20,000[{1 – [1 / (1 + r)] 4 } / r ]And the FV **of** the savings is:FVA = $8,000{[(1 + r) 6 – 1 ] / r }Setting these two equations equal to each other, we get:$20,000[{1 – [1 / (1 + r)] 4 } / r ] = $8,000{[ (1 + r) 6 – 1 ] / r }Reducing the equation gives us:(1 + r) 10 – 4.00(1 + r) 4 + 40.00 = 0Now, we need to find the roots **of** this equation. We can solve using trial and error, a root-solvingcalculator routine, or a spreadsheet. Using a spreadsheet, we find:r = 10.57%

CHAPTER 4 B- 7570. Here, we need to find the interest rate that makes us indifferent between an annuity and a perpetuity.To solve this problem, we need to find the PV **of** the two options and set them equal to each other.The PV **of** the perpetuity is:PV = $10,000 / rAnd the PV **of** the annuity is:PVA = $22,000[{1 – [1 / (1 + r)] 10 } / r ]Setting them equal and solving for r, we get:$10,000 / r = $22,000[{1 – [1 / (1 + r)] 10 } / r ]$10,000 / $22,000 = 1 – [1 / (1 + r)] 10.5455 1/10 = 1 / (1 + r)r = 1 / .5455 1/10 – 1r = .0625 or 6.25%71. The cash flows in this problem occur every two years, so we need to find the effective two year rate.One way to find the effective two year 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** days in two years since it isdaily compounding; if monthly compounding was assumed, we would use the number **of** months intwo years.) So, the effective two-year interest rate is:Effective 2-year rate = [1 + (.13/365)] 365(2) – 1 = 29.69%We can use this interest rate to find the PV **of** the perpetuity. Doing so, we find:PV = $6,700 /.2969 = $22,568.80This is an important point: Remember that the PV equation for a perpetuity (and an ordinaryannuity) tells you the PV one period before the first cash flow. In this problem, since the cash flowsare two years apart, we have found the value **of** the perpetuity one period (two years) before the firstpayment, which is one year ago. We need to compound this value for one year to find the valuetoday. The value **of** the cash flows today is:PV = $22,568.80(1 + .13/365) 365 = $25,701.39The second part **of** the question assumes the perpetuity cash flows begin in four years. In this case,when we use the PV **of** a perpetuity equation, we find the value **of** the perpetuity two years fromtoday. So, the value **of** these cash flows today is:PV = $22,568.80 / (1 + .13/365) 2(365) = $17,402.51

B-76SOLUTIONS72. To solve for the PVA due:C CCPVA = .... 2 t(1r)(1r)(1r)CCPVA due = C .... t -1(1r)(1r) C CCPVA due = (1 )....2(1 ) (1 ) (1 ) r t r r r PVA due = (1 + r) PVAAnd the FVA due is:FVA = C + C(1 + r) + C(1 + r) 2 + …. + C(1 + r) t – 1FVA due = C(1 + r) + C(1 + r) 2 + …. + C(1 + r) tFVA due = (1 + r)[C + C(1 + r) + …. + C(1 + r) t – 1 ]FVA due = (1 + r)FVA73. a. The APR is the interest rate per week times 52 weeks in a year, so:APR = 52(10%) = 520%EAR = (1 + .10) 52 – 1 = 14,104.29%b. In a discount loan, the amount you receive is lowered by the discount, and you repay the fullprincipal. With a 10 percent discount, you would receive $9 for every $10 in principal, so theweekly interest rate would be:$10 = $9(1 + r)r = ($10 / $9) – 1 = 11.11%Note the dollar amount we use is irrelevant. In other words, we could use $0.90 and $1, $90 and$100, or any other combination and we would get the same interest rate. Now we can find theAPR and the EAR:APR = 52(11.11%) = 577.78%EAR = (1 + .1111) 52 – 1 = 23,854.63%

CHAPTER 4 B- 77c. Using the cash flows from the loan, we have the PVA and the annuity payments and need to findthe interest rate, so:PVA = $58.84 = $25[{1 – [1 / (1 + r)] 4 }/ r ]Using a spreadsheet, trial and error, or a financial calculator, we find:r = 25.19% per weekAPR = 52(25.19%) = 1,309.92%EAR = 1.2518 52 – 1 = 11,851,501.94%74. To answer this, we can diagram the perpetuity cash flows, which are: (Note, the subscripts are onlyto differentiate when the cash flows begin. The cash flows are all the same amount.)…..C 3C 2 C 2C 1 C 1 C 1Thus, each **of** the increased cash flows is a perpetuity in itself. So, we can write the cash flowsstream as:C 1 /R C 2 /R C 3 /R C 4 /R ….So, we can write the cash flows as the present value **of** a perpetuity with a perpetuity payment **of**:C 2 /R C 3 /R C 4 /R ….The present value **of** this perpetuity is:PV = (C/R) / R = C/R 2So, the present value equation **of** a perpetuity that increases by C each period is:PV = C/R + C/R 2

B-78SOLUTIONS75. Since it is only an approximation, we know the Rule **of** 72 is exact for only one interest rate. Usingthe basic future value equation for an amount that doubles in value and solving for t, we find:FV = PV(1 + R) t$2 = $1(1 + R) tln(2) = t ln(1 + R)t = ln(2) / ln(1 + R)We also know the Rule **of** 72 approximation is:t = 72 / RWe can set these two equations equal to each other and solve for R. We also need to remember thatthe exact future value equation uses decimals, so the equation becomes:.72 / R = ln(2) / ln(1 + R)0 = (.72 / R) / [ ln(2) / ln(1 + R)]It is not possible to solve this equation directly for R, but using Solver, we find the interest rate forwhich the Rule **of** 72 is exact is 7.846894 percent.76. We are only concerned with the time it takes money to double, so the dollar amounts are irrelevant.So, we can write the future value **of** a lump sum with continuously compounded interest as:$2 = $1e Rt2 = e RtRt = ln(2)Rt = .693147t = .691347 / RSince we are using percentage interest rates while the equation uses decimal form, to make theequation correct with percentages, we can multiply by 100:t = 69.1347 / R

CHAPTER 4 B- 79Calculator **Solutions**1.Enter 10 7% $5,000N I/Y PV PMT FVSolve for $9,835.76$9,835.76 – 8,500 = $1,335.762.Enter 10 5% $1,000N I/Y PV PMT FVSolve for $1,628.89Enter 10 7% $1,000N I/Y PV PMT FVSolve for $1,967.15Enter 20 5% $1,000N I/Y PV PMT FVSolve for $2,653.303.Enter 6 5% $15,451N I/Y PV PMT FVSolve for $11,529.77Enter 9 11% $51,557N I/Y PV PMT FVSolve for $20,154.91Enter 23 16% $886,073N I/Y PV PMT FVSolve for $29,169.95Enter 18 19% $550,164N I/Y PV PMT FVSolve for $24,024.094.Enter 2 $265 $307N I/Y PV PMT FVSolve for 7.63%

B-80SOLUTIONSEnter 9 $360 $896N I/Y PV PMT FVSolve for 10.66%Enter 15 $39,000 $162,181N I/Y PV PMT FVSolve for 9.97%Enter 30 $46,523 $483,500N I/Y PV PMT FVSolve for 8.12%5.Enter 8% $625 $1,284N I/Y PV PMT FVSolve for 9.36Enter 7% $810 $4,341N I/Y PV PMT FVSolve for 24.81Enter 21% $18,400 $402,662N I/Y PV PMT FVSolve for 16.19Enter 29% $21,500 $173,439N I/Y PV PMT FVSolve for 8.206.Enter 7% $1 $2N I/Y PV PMT FVSolve for 10.24Enter 7% $1 $4N I/Y PV PMT FVSolve for 20.497.Enter 20 9.5% $800,000,000N I/Y PV PMT FVSolve for $130,258,959.12

CHAPTER 4 B- 818.Enter 4 $12,377,500 $10,311,500N I/Y PV PMT FVSolve for –4.46%11.CFo $0 CFo $0 CFo $0C01 $1,200 C01 $1,200 C01 $1,200F01 1 F01 1 F01 1C02 $600 C02 $600 C02 $600F02 1 F02 1 F02 1C03 $855 C03 $855 C03 $855F03 1 F03 1 F03 1C04 $1,480 C04 $1,480 C04 $1,480F04 1 F04 1 F04 1I = 10 I = 18 I = 24NPV CPT NPV CPT NPV CPT$3,240.01 $2,731.61 $2,432.4012.Enter 9 5% $4,000N I/Y PV PMT FVSolve for $28,431.29Enter 5 5% $6,000N I/Y PV PMT FVSolve for $25,976.86Enter 9 22% $4,000N I/Y PV PMT FVSolve for $15,145.14Enter 5 22% $6,000N I/Y PV PMT FVSolve for $17,181.8413.Enter 15 10% $3,600N I/Y PV PMT FVSolve for $27,381.89Enter 40 10% $3,600N I/Y PV PMT FVSolve for $35,204.58

B-82SOLUTIONSEnter 75 10% $3,600N I/Y PV PMT FVSolve for $35,971.7015.Enter 11% 4NOM EFF C/YSolve for 11.46%Enter 7% 12NOM EFF C/YSolve for 7.23%Enter 9% 365NOM EFF C/YSolve for 9.42%16.Enter 8.1% 2NOM EFF C/YSolve for 7.94%Enter 7.6% 12NOM EFF C/YSolve for 7.35%Enter 16.8% 52NOM EFF C/YSolve for 15.55%17.Enter 12.2% 12NOM EFF C/YSolve for 12.91%Enter 12.4% 2NOM EFF C/YSolve for 12.78%18. 2 nd BGN 2 nd SETEnter 12 $108 $10N I/Y PV PMT FVSolve for 1.98%APR = 1.98% × 52 = 102.77%

CHAPTER 4 B- 83Enter 102.77% 52NOM EFF C/YSolve for 176.68%19.Enter 0.9% $16,500 $500N I/Y PV PMT FVSolve for 39.3320.Enter 1,733.33% 52NOM EFF C/YSolve for 313,916,515.69%21.Enter 3 8% $1,000N I/Y PV PMT FVSolve for $1,259.71Enter 3 × 2 8%/2 $1,000N I/Y PV PMT FVSolve for $1,265.32Enter 3 × 12 8%/12 $1,000N I/Y PV PMT FVSolve for $1,270.2423. Stock account:Enter 360 11% / 12 $700N I/Y PV PMT FVSolve for $1,963,163.82Bond account:Enter 360 7% / 12 $300N I/Y PV PMT FVSolve for $365,991.30Savings at retirement = $1,963,163.82 + 365,991.30 = $2,329,155.11Enter 300 9% / 12 $2,329,155.11N I/Y PV PMT FVSolve for $19,546.19

B-84SOLUTIONS24.Enter 12 / 3 $1 $3N I/Y PV PMT FVSolve for 31.61%25.Enter 5 $50,000 $85,000N I/Y PV PMT FVSolve for 11.20%Enter 11 50,000 $175,000N I/Y PV PMT FVSolve for 12.06%28.Enter 20 8% $2,000N I/Y PV PMT FVSolve for $19,636.29Enter 2 8% $19,636.29N I/Y PV PMT FVSolve for $16,834.9629.Enter 15 15% $500N I/Y PV PMT FVSolve for $2,923.66Enter 5 12% $2,923.66N I/Y PV PMT FVSolve for $1,658.9830.Enter 360 8%/12 .80($400,000)N I/Y PV PMT FVSolve for $2,348.05Enter 22 × 12 8%/12 $2,348.05N I/Y PV PMT FVSolve for $291,256.6331.Enter 6 1.90% / 12 $4,000N I/Y PV PMT FVSolve for $4,038.15

CHAPTER 4 B- 85Enter 6 16% / 12 $4,038.15N I/Y PV PMT FVSolve for $4,372.16$4,372.16 – 4,000 = $372.1635.Enter 10 10% $5,000N I/Y PV PMT FVSolve for $30,722.84Enter 10 5% $5,000N I/Y PV PMT FVSolve for $38,608.67Enter 10 15% $5,000N I/Y PV PMT FVSolve for $25,093.8436.Enter 10% / 12 $125 $20,000N I/Y PV PMT FVSolve for 102.1037.Enter 60 $45,000 $950N I/Y PV PMT FVSolve for 0.810%0.810% 12 = 9.72%38.Enter 360 6.8% / 12 $1,000N I/Y PV PMT FVSolve for $153,391.83$200,000 – 153,391.83 = $46,608.17Enter 360 6.8% / 12 $46,608.17N I/Y PV PMT FVSolve for $356,387.10

B-86SOLUTIONS39.CFo $0C01 $1,000F01 1C02 $0F02 1C03 $2,000F03 1C04 $2,000F04 1I = 10%NPV CPT$3,777.75PV **of** missing CF = $5,979 – 3,777.75 = $2,201.25Value **of** missing CF:Enter 2 10% $2,201.25N I/Y PV PMT FVSolve for $2,663.5240.CFo $1,000,000C01 $1,400,000F01 1C02 $1,800,000F02 1C03 $2,200,000F03 1C04 $2,600,000F04 1C05 $3,000,000F05 1C06 $3,400,000F06 1C07 $3,800,000F07 1C08 $4,200,000F08 1C09 $4,600,000F09 1C010 $5,000,000I = 10%NPV CPT$18,758,930.79

CHAPTER 4 B- 8741.Enter 360 .80($1,600,000) $10,000N I/Y PV PMT FVSolve for 0.7228%APR = 0.7228% 12 = 8.67%Enter 8.67% 12NOM EFF C/YSolve for 9.03%42.Enter 3 13% $115,000N I/Y PV PMT FVSolve for $79,700.77Pr**of**it = $79,700.77 – 72,000 = $7,700.77Enter 3 $72,000 $115,000N I/Y PV PMT FVSolve for 16.89%43.Enter 17 12% $2,000N I/Y PV PMT FVSolve for $14,239.26Enter 8 12% $14,239.26N I/Y PV PMT FVSolve for $5,751.0044.Enter 84 15% / 12 $1,500N I/Y PV PMT FVSolve for $77,733.28Enter 96 12% / 12 $1,500N I/Y PV PMT FVSolve for $92,291.55Enter 84 15% / 12 $92,291.55N I/Y PV PMT FVSolve for $32,507.18$77,733.28 + 32,507.18 = $110,240.46

B-88SOLUTIONS45.Enter 15 × 12 10.5%/12 $1,000N I/Y PV PMT FVSolve for $434,029.81FV = $434,029.81 = PV e .09(15) ; PV = $434,029.81 e –1.35 = $112,518.0046. [email protected] t = 14: $3,000 / 0.065 = $46,153.85Enter 7 6.5% $46,153.85N I/Y PV PMT FVSolve for $29,700.2947.Enter 12 $20,000 $1,900N I/Y PV PMT FVSolve for 2.076%APR = 2.076% 12 = 24.91%Enter 24.91% 12NOM EFF C/YSolve for 27.96%48. Monthly rate = .12 / 12 = .01; semiannual rate = (1.01) 6 – 1 = 6.15%Enter 10 6.15% $6,000N I/Y PV PMT FVSolve for $43,844.21Enter 8 6.15% $43,844.21N I/Y PV PMT FVSolve for $27,194.83Enter 12 6.15% $43,844.21N I/Y PV PMT FVSolve for $21,417.72Enter 18 6.15% $43,844.21N I/Y PV PMT FVSolve for $14,969.38

CHAPTER 4 B- 8949.a.Enter 6 9.5% $525N I/Y PV PMT FVSolve for $2,320.41b. 2 nd BGN 2 nd SETEnter 6 9.5% $525N I/Y PV PMT FVSolve for $2,540.8550. 2 nd BGN 2 nd SETEnter 48 8.15% / 12 $56,000N I/Y PV PMT FVSolve for $1,361.8251. 2 nd BGN 2 nd SETEnter 2 × 12 12% / 12 $4,000N I/Y PV PMT FVSolve for $186.4352. PV **of** college expenses:Enter 4 6.5% $23,000N I/Y PV PMT FVSolve for $78,793.37Cost today **of** oldest child’s expenses:Enter 14 6.5% $78,793.37N I/Y PV PMT FVSolve for $32,628.35Cost today **of** youngest child’s expenses:Enter 16 6.5% $78,793.37N I/Y PV PMT FVSolve for $25,767.09Total cost today = $32,628.35 + 25,767.09 = $61,395.44Enter 15 6.5% $61,395.44N I/Y PV PMT FVSolve for $6,529.58

B-90SOLUTIONS54. Option A:Aftertax cash flows = Pretax cash flows(1 – tax rate)Aftertax cash flows = $160,000(1 – .28)Aftertax cash flows = $115,2002 ND BGN 2 nd SETEnter 31 10% $115,200N I/Y PV PMT FVSolve for $1,201,180.55Option B:Aftertax cash flows = Pretax cash flows(1 – tax rate)Aftertax cash flows = $101,055(1 – .28)Aftertax cash flows = $72,759.602 ND BGN 2 nd SETEnter 30 10% $446,000 $72,759.60N I/Y PV PMT FVSolve for $1,131,898.5356.Enter 5 × 12 9.6% / 12 $14,000N I/Y PV PMT FVSolve for $294.712 nd BGN 2 nd SETEnter 35 9.6% / 12 $294.71N I/Y PV PMT FVSolve for $9,073.33Total payment = Amount due(1 + Prepayment penalty)Total payment = $9,073.33(1 + .01)Total payment = $9,127.7157. Pre-retirement APR:Enter 11% 12NOM EFF C/YSolve for 10.48%Post-retirement APR:Enter 8% 12NOM EFF C/YSolve for 7.72%

CHAPTER 4 B- 91At retirement, he needs:Enter 240 7.72% / 12 $25,000 $750,000N I/Y PV PMT FVSolve for $3,3212,854.41In 10 years, his savings will be worth:Enter 120 10.48% / 12 $2,100N I/Y PV PMT FVSolve for $442,239.69After purchasing the cabin, he will have: $442,239.69 – 350,000 = $92,239.69Each month between years 10 and 30, he needs to save:Enter 240 10.48% / 12 $92,239.69 $3,212,854.42N I/Y PV PMT FVSolve for $3,053.8758. PV **of** purchase:Enter 36 8% / 12 $23,000N I/Y PV PMT FVSolve for $18,106.86$35,000 – 18,106.86 = $16,893.14PV **of** lease:Enter 36 8% / 12 $450N I/Y PV PMT FVSolve for $14,360.31$14,360.31 + 1 = $14,361.31Lease the car.You would be indifferent when the PV **of** the two cash flows are equal. The present value **of** thepurchase decision must be $14,361.31. Since the difference in the two cash flows is $35,000 –14,361.31 = $20,638.69, this must be the present value **of** the future resale price **of** the car. Thebreak-even resale price **of** the car is:Enter 36 8% / 12 $20,638.69N I/Y PV PMT FVSolve for $26,216.0359.Enter 4.50% 365NOM EFF C/YSolve for 4.60%

B-92SOLUTIONSCFo $8,000,000C01 $4,000,000F01 1C02 $4,800,000F02 1C03 $5,700,000F03 1C04 $6,400,000F04 1C05 $7,000,000F05 1C06 $7,500,000F06 1I = 4.60%NPV CPT$37,852,037.91New contract value = $37,852,037.91 + 750,000 = $38,602,037.91PV **of** payments = $38,602,037.91 – 9,000,000 = $29,602,037.91Effective quarterly rate = [1 + (.045/365)] 91.25 – 1 = 1.131%Enter 24 1.131% $29,602,037.91N I/Y PV PMT FVSolve for $1,415,348.3760.Enter 1 $17,600 $20,000N I/Y PV PMT FVSolve for 13.64%61.Enter 9% 12NOM EFF C/YSolve for 8.65%Enter 12 8.65% / 12 $40,000 / 12N I/Y PV PMT FVSolve for $41,624.33Enter 1 9% $41,624.33N I/Y PV PMT FVSolve for $45,370.52

CHAPTER 4 B- 93Enter 12 8.65% / 12 $43,000 / 12N I/Y PV PMT FVSolve for $44,746.15Enter 60 8.65% / 12 $45,000 / 12N I/Y PV PMT FVSolve for $182,142.14Award = $45,370.52 + 44,746.15 + 182,142.14 + 100,000 + 20,000 = $392,258.8162.Enter 1 $9,700 $11,000N I/Y PV PMT FVSolve for 13.40%Enter 1 $9,800 $11,300N I/Y PV PMT FVSolve for 15.31%63. Refundable fee: With the $1,500 application fee, you will need to borrow $201,500 to have$200,000 after deducting the fee. Solve for the payment under these circumstances.Enter 30 12 7.50% / 12 $201,500N I/Y PV PMT FVSolve for $1,408.92Enter 30 12 $200,000 $1,408.92N I/Y PV PMT FVSolve for 0.6314%APR = 0.6314% 12 = 7.58%Enter 7.58% 12NOM EFF C/YSolve for 7.85%Without refundable fee: APR = 7.50%Enter 7.50% 12NOM EFF C/YSolve for 7.76%

B-94SOLUTIONS64.Enter 36 $1,000 $42.25N I/Y PV PMT FVSolve for 2.47%APR = 2.47% 12 = 29.63%Enter 29.63% 12NOM EFF C/YSolve for 34.00%65. What she needs at age 65:Enter 15 8% $90,000N I/Y PV PMT FVSolve for $770,353.08a.Enter 30 8% $770,353.08N I/Y PV PMT FVSolve for $6,800.24b.Enter 30 8% $770,353.08N I/Y PV PMT FVSolve for $76,555.63c.Enter 10 8% $25,000N I/Y PV PMT FVSolve for $53,973.12At 65, she is short: $770,353.08 – 53,973.12 = $716,379.96Enter 30 8% ±$716,379.96N I/Y PV PMT FVSolve for $6,323.80Her employer will contribute $1,500 per year, so she must contribute:$6,323.80 – 1,500 = $4,823.80 per year66. Without fee:Enter 19.2% / 12 $10,000 $200N I/Y PV PMT FVSolve for 101.39

CHAPTER 4 B- 95Enter 9.2% / 12 $10,000 $200N I/Y PV PMT FVSolve for 63.30With fee:Enter 9.2% / 12 $10,200 $200N I/Y PV PMT FVSolve for 64.9467. Value at Year 6:Enter 5 11% $750N I/Y PV PMT FVSolve for $1,263.79Enter 4 11% $750N I/Y PV PMT FVSolve for $1,138.55Enter 3 11% $850N I/Y PV PMT FVSolve for $1,162.49Enter 2 11% $850N I/Y PV PMT FVSolve for $1,047.29Enter 1 11% $950N I/Y PV PMT FVSolve for $1,054.50So, at Year 5, the value is: $1,263.79 + 1,138.55 + 1,162.49 + 1,047.29 + 1,054.50+ 950 = $6,612.62At Year 65, the value is:Enter 59 7% $6,612.62N I/Y PV PMT FVSolve for $358,326.50The policy is not worth buying; the future value **of** the policy is $358K, but the policy contractwill pay **of**f $250K.

B-96SOLUTIONS68. Effective six-month rate = (1 + Daily rate) 180 – 1Effective six-month rate = (1 + .09/365) 180 – 1Effective six-month rate = .0454 or 4.54%Enter 40 4.54% $500,000N I/Y PV PMT FVSolve for $9,151,418.61Enter 1 4.54% $9,089,929.35N I/Y PV PMT FVSolve for $8,754,175.76Value **of** winnings today = $8,754,175.76 + 1,000,000Value **of** winnings today = $9,754,175.7669.CFo $8,000C01 $8,000F01 5C02 $20,000F02 4IRR CPT10.57%73.a. APR = 10% 52 = 520%Enter 520% 52NOM EFF C/YSolve for 14,104.29%b.Enter 1 $9.00 $10.00N I/Y PV PMT FVSolve for 11.11%APR = 11.11% 52 = 577.78%Enter 577.78% 52NOM EFF C/YSolve for 23,854.63%c.Enter 4 $58.84 $25N I/Y PV PMT FVSolve for 25.19%

CHAPTER 4 B- 97APR = 25.19% 52 = 1,309.92%Enter 1,309.92 % 52NOM EFF C/YSolve for 11,851,501.94%

CHAPTER 4, APPENDIXNET PRESENT VALUE: FIRSTPRINCIPLES OF FINANCE**Solutions** to Questions and ProblemsNOTE: 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 each problem isfound without rounding during any step in the problem.1. The potential consumption for a borrower next year is the salary during the year, minus therepayment **of** the loan and interest to fund the current consumption. The amount that must beborrowed to fund this year’s consumption is:Amount to borrow = $100,000 – 80,000 = $20,000Interest will be charged the amount borrowed, so the repayment **of** this loan next year will be:Loan repayment = $20,000(1.10) = $22,000So, the consumption potential next year is the salary minus the loan repayment, or:Consumption potential = $90,000 – 22,000 = $68,0002. The potential consumption for a saver next year is the salary during the year, plus the savings fromthe current year and the interest earned. The amount saved this year is:Amount saved = $50,000 – 35,000 = $15,000The saver will earn interest over the year, so the value **of** the savings next year will be:Savings value in one year = $15,000(1.12) = $16,800So, the consumption potential next year is the salary plus the value **of** the savings, or:Consumption potential = $60,000 – 16,800 = $76,8003. Financial markets arise to facilitate borrowing and lending between individuals. By borrowing andlending, people can adjust their pattern **of** consumption over time to fit their particular preferences.This allows corporations to accept all positive NPV projects, regardless **of** the inter-temporalconsumption preferences **of** the shareholders.

CHAPTER 4 APPENDIX B- 994. a. The present value **of** labor income is the total **of** the maximum current consumption. So,solving for the interest rate, we find:$86 = $40 + $50/(1 + R)R = .0870 or 8.70%b. The NPV **of** the investment is the difference between the new maximum current consumptionminus the old maximum current consumption, or:NPV = $98 – 86 = $12c. The total maximum current consumption amount must be the present value **of** the equal annualconsumption amount. If C is the equal annual consumption amount, we find:$98 = C + C/(1 + R)$98 = C + C/(1.0870)C = $51.045. a. The market interest rate must be the increase in the maximum current consumption to themaximum consumption next year, which is:Market interest rate = $90,000/$80,000 – 1 = 0.1250 or 12.50%b. Harry will invest $10,000 in financial assets and $30,000 in productive assets today.c. NPV = –$30,000 + $56,250/1.125NPV = $20,000

CHAPTER 5HOW TO VALUE STOCKS AND BONDSAnswers to Concepts Review and Critical Thinking Questions1. Bond issuers look at outstanding bonds **of** similar maturity and risk. The yields on such bonds areused to establish the coupon rate necessary for a particular issue to initially sell for par value. Bondissuers also simply ask potential purchasers what coupon rate would be necessary to attract them.The coupon rate is fixed and simply determines what the bond’s coupon payments will be. Therequired return is what investors actually demand on the issue, and it will fluctuate through time. Thecoupon rate and required return are equal only if the bond sells exactly at par.2. Lack **of** transparency means that a buyer or seller can’t see recent transactions, so it is much harderto determine what the best price is at any point in time.3. The value **of** any investment depends on the present value **of** its cash flows; i.e., what investors willactually receive. The cash flows from a share **of** stock are the dividends.4. Investors believe the company will eventually start paying dividends (or be sold to anothercompany).5. In general, companies that need the cash will **of**ten forgo dividends since dividends are a cashexpense. Young, growing companies with pr**of**itable investment opportunities are one example;another example is a company in financial distress. This question is examined in depth in a laterchapter.6. The general method for valuing a share **of** stock is to find the present value **of** all expected futuredividends. The dividend growth model presented in the text is only valid (i) if dividends are expectedto occur forever; that is, the stock provides dividends in perpetuity, and (ii) if a constant growth rate**of** dividends occurs forever. A violation **of** the first assumption might be a company that is expectedto cease operations and dissolve itself some finite number **of** years from now. The stock **of** such acompany would be valued by applying the general method **of** valuation explained in this chapter. Aviolation **of** the second assumption might be a start-up firm that isn’t currently paying any dividends,but is expected to eventually start making dividend payments some number **of** years from now. Thisstock would also be valued by the general dividend valuation method explained in this chapter.7. The common stock probably has a higher price because the dividend can grow, whereas it is fixed onthe preferred. However, the preferred is less risky because **of** the dividend and liquidationpreference, so it is possible the preferred could be worth more, depending on the circumstances.8. Yes. If the dividend grows at a steady rate, so does the stock price. In other words, the dividendgrowth rate and the capital gains yield are the same.9. The three factors are: 1) The company’s future growth opportunities. 2) The company’s level **of** risk,which determines the interest rate used to discount cash flows. 3) The accounting method used.

CHAPTER 5 B- 10110. Presumably, the current stock value reflects the risk, timing and magnitude **of** all future cash flows,both short-term and long-term. If this is correct, then the statement is false.**Solutions** to Questions and ProblemsNOTE: 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 each problem isfound without rounding during any step in the problem.NOTE: Most problems do not explicitly list a par value for bonds. Even though a bond can have any parvalue, in general, corporate bonds in the United States will have a par value **of** $1,000. We will use thispar value in all problems unless a different par value is explicitly stated.Basic1. The price **of** a pure discount (zero coupon) bond is the present value **of** the par. Even though thebond makes no coupon payments, the present value is found using semiannual compounding periods,consistent with coupon bonds. This is a bond pricing convention. So, the price **of** the bond for eachYTM is:a. P = $1,000/(1 + .025) 20 = $610.27b. P = $1,000/(1 + .05) 20 = $376.89c. P = $1,000/(1 + .075) 20 = $235.412. The price **of** any bond is the PV **of** the interest payment, plus the PV **of** the par value. Notice thisproblem assumes an annual coupon. The price **of** the bond at each YTM will be:a. P = $40({1 – [1/(1 + .04)] 40 } / .04) + $1,000[1 / (1 + .04) 40 ]P = $1,000.00When the YTM and the coupon rate are equal, the bond will sell at par.b. P = $40({1 – [1/(1 + .05)] 40 } / .05) + $1,000[1 / (1 + .05) 40 ]P = $828.41When the YTM is greater than the coupon rate, the bond will sell at a discount.c. P = $40({1 – [1/(1 + .03)] 40 } / .03) + $1,000[1 / (1 + .03) 40 ]P = $1,231.15When the YTM is less than the coupon rate, the bond will sell at a premium.

B-102 SOLUTIONSWe would like to introduce shorthand notation here. Rather than write (or type, as the case may be)the entire equation for the PV **of** a lump sum, or the PVA equation, it is common to abbreviate theequations as:PVIF R,t = 1 / (1 + r) twhich stands for Present Value Interest Factor, and:PVIFA R,t = ({1 – [1/(1 + r)] t } / r )which stands for Present Value Interest Factor **of** an AnnuityThese abbreviations are short hand notation for the equations in which the interest rate and thenumber **of** periods are substituted into the equation and solved. We will use this shorthand notationin the remainder **of** the solutions key.3. Here we are finding the YTM **of** a semiannual coupon bond. The bond price equation is:P = $970 = $43(PVIFA R%,20 ) + $1,000(PVIF R%,20 )Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trialand error, we find:R = 4.531%Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR**of** the bond, so:YTM = 2 4.531% = 9.06%4. The constant dividend growth model is:P t = D t × (1 + g) / (R – g)So, the price **of** the stock today is:P 0 = D 0 (1 + g) / (R – g) = $1.40 (1.06) / (.12 – .06) = $24.73The dividend at year 4 is the dividend today times the FVIF for the growth rate in dividends and fouryears, so:P 3 = D 3 (1 + g) / (R – g) = D 0 (1 + g) 4 / (R – g) = $1.40 (1.06) 4 / (.12 – .06) = $29.46We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so:P 15 = D 15 (1 + g) / (R – g) = D 0 (1 + g) 16 / (R – g) = $1.40 (1.06) 16 / (.12 – .06) = $59.27

CHAPTER 5 B- 103There is another feature **of** the constant dividend growth model: The stock price grows at thedividend growth rate. So, if we know the stock price today, we can find the future value for any timein the future we want to calculate the stock price. In this problem, we want to know the stock price inthree years, and we have already calculated the stock price today. The stock price in three years willbe:P 3 = P 0 (1 + g) 3 = $24.73(1 + .06) 3 = $29.46And the stock price in 15 years will be:P 15 = P 0 (1 + g) 15 = $24.73(1 + .06) 15 = $59.275. We need to find the required return **of** the stock. Using the constant growth model, we can solve theequation for R. Doing so, we find:R = (D 1 / P 0 ) + g = ($3.10 / $48.00) + .05 = 11.46%6. Using the constant growth model, we find the price **of** the stock today is:P 0 = D 1 / (R – g) = $3.60 / (.13 – .045) = $42.357. We know the stock has a required return **of** 12 percent, and the dividend and capital gains yield areequal, so:Dividend yield = 1/2(.12) = .06 = Capital gains yieldNow we know both the dividend yield and capital gains yield. The dividend is simply the stock pricetimes the dividend yield, so:D 1 = .06($70) = $4.20This is the dividend next year. The question asks for the dividend this year. Using the relationshipbetween the dividend this year and the dividend next year:D 1 = D 0 (1 + g)We can solve for the dividend that was just paid:$4.20 = D 0 (1 + .06)D 0 = $4.20 / 1.06 = $3.968. The price **of** any financial instrument is the PV **of** the future cash flows. The future dividends **of** thisstock are an annuity for eight years, so the price **of** the stock is the PVA, which will be:P 0 = $12.00(PVIFA 10%,8 ) = $64.02

B-104 SOLUTIONS9. The growth rate **of** earnings is the return on equity times the retention ratio, so:g = ROE × bg = .14(.60)g = .084 or 8.40%To find next year’s earnings, we simply multiply the current earnings times one plus the growth rate,so:Next year’s earnings = Current earnings(1 + g)Next year’s earnings = $20,000,000(1 + .084)Next year’s earnings = $21,680,000Intermediate10. Here we are finding the YTM **of** semiannual coupon bonds for various maturity lengths. The bondprice equation is:P = C(PVIFA R%,t ) + $1,000(PVIF R%,t )Miller Corporation bond:P 0 = $40(PVIFA 3%,26 ) + $1,000(PVIF 3%,26 ) = $1,178.77P 1 = $40(PVIFA 3%,24 ) + $1,000(PVIF 3%,24 ) = $1,169.36P 3 = $40(PVIFA 3%,20 ) + $1,000(PVIF 3%,20 ) = $1,148.77P 8 = $40(PVIFA 3%,10 ) + $1,000(PVIF 3%,10 ) = $1,085.30P 12 = $40(PVIFA 3%,2 ) + $1,000(PVIF 3%,2 ) = $1,019.13P 13 = $1,000Modigliani Company bond:Y: P 0 = $30(PVIFA 4%,26 ) + $1,000(PVIF 4%,26 ) = $840.17P 1 = $30(PVIFA 4%,24 ) + $1,000(PVIF 4%,24 ) = $847.53P 3 = $30(PVIFA 4%,20 ) + $1,000(PVIF 4%,20 ) = $864.10P 8 = $30(PVIFA 4%,10 ) + $1,000(PVIF 4%,10 ) = $918.89P 12 = $30(PVIFA 4%,2 ) + $1,000(PVIF 4%,2 ) = $981.14P 13 = $1,000All else held equal, the premium over par value for a premium bond declines as maturity approaches,and the discount from par value for a discount bond declines as maturity approaches. This is called“pull to par.” In both cases, the largest percentage price changes occur at the shortest maturitylengths.Also, notice that the price **of** each bond when no time is left to maturity is the par value, even thoughthe purchaser would receive the par value plus the coupon payment immediately. This is because wecalculate the clean price **of** the bond.

CHAPTER 5 B- 10511. The bond price equation for this bond is:P 0 = $1,040 = $42(PVIFA R%,18 ) + $1,000(PVIF R%,18 )Using a spreadsheet, financial calculator, or trial and error we find:R = 3.887%This is the semiannual interest rate, so the YTM is:YTM = 2 3.887% = 7.77%The current yield is:Current yield = Annual coupon payment / Price = $84 / $1,040 = 8.08%The effective annual yield is the same as the EAR, so using the EAR equation from the previouschapter:Effective annual yield = (1 + 0.03887) 2 – 1 = 7.92%12. The company should set the coupon rate on its new bonds equal to the required return. The requiredreturn can be observed in the market by finding the YTM on outstanding bonds **of** the company. So,the YTM on the bonds currently sold in the market is:P = $1,095 = $40(PVIFA R%,40 ) + $1,000(PVIF R%,40 )Using a spreadsheet, financial calculator, or trial and error we find:R = 3.55%This is the semiannual interest rate, so the YTM is:YTM = 2 3.55% = 7.10%13. This stock has a constant growth rate **of** dividends, but the required return changes twice. To find thevalue **of** the stock today, we will begin by finding the price **of** the stock at Year 6, when both thedividend growth rate and the required return are stable forever. The price **of** the stock in Year 6 willbe the dividend in Year 7, divided by the required return minus the growth rate in dividends. So:P 6 = D 6 (1 + g) / (R – g) = D 0 (1 + g) 7 / (R – g) = $3.00 (1.05) 7 / (.11 – .05) = $70.36Now we can find the price **of** the stock in Year 3. We need to find the price here since the requiredreturn changes at that time. The price **of** the 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 in Year 3 is:P 3 = $3.00(1.05) 4 / 1.14 + $3.00(1.05) 5 / 1.14 2 + $3.00(1.05) 6 / 1.14 3 + $70.36 / 1.14 3P 3 = $56.35

B-106 SOLUTIONSFinally, we can find the price **of** the stock today. The price today will be the PV **of** the dividends inYears 1, 2, and 3, plus the PV **of** the stock in Year 3. The price **of** the stock today is:P 0 = $3.00(1.05) / 1.16 + $3.00(1.05) 2 / (1.16) 2 + $3.00(1.05) 3 / (1.16) 3 + $56.35 / (1.16) 3= $43.5014. Here we have a stock that pays no dividends for 10 years. Once the stock begins paying dividends, itwill have a constant growth rate **of** dividends. We can use the constant growth model at that point. Itis important to remember that general form **of** the constant dividend growth formula is:P t = [D t × (1 + g)] / (R – g)This means that since we will use the dividend in Year 10, we will be finding the stock price in Year9. The dividend growth model is similar to the PVA and the PV **of** a perpetuity: The equation givesyou the PV one period before the first payment. So, the price **of** the stock in Year 9 will be:P 9 = D 10 / (R – g) = $8.00 / (.13 – .06) = $114.29The price **of** the stock today is simply the PV **of** the stock price in the future. We simply discount thefuture stock price at the required return. The price **of** the stock today will be:P 0 = $114.29 / 1.13 9 = $38.0415. The price **of** a stock is the PV **of** the future dividends. This stock is paying four dividends, so theprice **of** the stock is the PV **of** these dividends using the required return. The price **of** the stock is:P 0 = $12 / 1.11 + $15 / 1.11 2 + $18 / 1.11 3 + $21 / 1.11 4 = $49.9816. With supernormal dividends, we find the price **of** the stock when the dividends level **of**f at a constantgrowth rate, and then find the PV **of** the future stock price, plus the PV **of** all dividends during thesupernormal growth period. The stock begins constant growth in Year 5, so we can find the price **of**the stock in Year 4, one year before the constant dividend growth begins, as:P 4 = D 4 (1 + g) / (R – g) = $2.00(1.05) / (.13 – .05) = $26.25The price **of** the stock today is the PV **of** the first four dividends, plus the PV **of** the Year 4 stockprice. So, the price **of** the stock today will be:P 0 = $8.00 / 1.13 + $6.00 / 1.13 2 + $3.00 / 1.13 3 + $2.00 / 1.13 4 + $26.25 / 1.13 4 = $31.1817. With supernormal dividends, we find the price **of** the stock when the dividends level **of**f at a constantgrowth rate, and then find the PV **of** the 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 3, one year before the constant dividend growth begins as:P 3 = D 3 (1 + g) / (R – g) = D 0 (1 + g 1 ) 3 (1 + g 2 ) / (R – g 2 ) = $2.80(1.25) 3 (1.07) / (.13 – .07) = $97.53

CHAPTER 5 B- 107The price **of** the stock today is the PV **of** the first three dividends, plus the PV **of** the Year 3 stockprice. The price **of** the stock today will be:P 0 = 2.80(1.25) / 1.13 + $2.80(1.25) 2 / 1.13 2 + $2.80(1.25) 3 / 1.13 3 + $97.53 / 1.13 3P 0 = $77.9018. Here we need to find the dividend next year for a stock experiencing supernormal growth. We knowthe stock price, the dividend growth rates, and the required return, but not the dividend. First, weneed to realize that the dividend in Year 3 is the current dividend times the FVIF. The dividend inYear 3 will be:D 3 = D 0 (1.30) 3And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate, or:D 4 = D 0 (1.30) 3 (1.18)The stock begins constant growth in Year 4, so we can find the price **of** the stock in Year 4 as thedividend in Year 5, divided by the required return minus the growth rate. The equation for the price**of** the stock in Year 4 is:P 4 = D 4 (1 + g) / (R – g)Now we can substitute the previous dividend in Year 4 into this equation as follows:P 4 = D 0 (1 + g 1 ) 3 (1 + g 2 ) (1 + g 3 ) / (R – g 3 )P 4 = D 0 (1.30) 3 (1.18) (1.08) / (.14 – .08) = 46.66D 0When we solve this equation, we find that the stock price in Year 4 is 46.66 times as large as thedividend today. Now we need to find the equation for the stock price today. The stock price today isthe PV **of** the dividends in Years 1, 2, 3, and 4, plus the PV **of** the Year 4 price. So:P 0 = D 0 (1.30)/1.14 + D 0 (1.30) 2 /1.14 2 + D 0 (1.30) 3 /1.14 3 + D 0 (1.30) 3 (1.18)/1.14 4 + 46.66D 0 /1.14 4We can factor out D 0 in the equation, and combine the last two terms. Doing so, we get:P 0 = $70.00 = D 0 {1.30/1.14 + 1.30 2 /1.14 2 + 1.30 3 /1.14 3 + [(1.30) 3 (1.18) + 46.66] / 1.14 4 }Reducing the equation even further by solving all **of** the terms in the braces, we get:$70 = $33.04D 0D 0 = $70.00 / $33.04 = $2.12This is the dividend today, so the projected dividend for the next year will be:D 1 = $2.12(1.30) = $2.75

B-108 SOLUTIONS19. We are given the stock price, the dividend growth rate, and the required return, and are asked to findthe dividend. Using the constant dividend growth model, we get:P 0 = $50 = D 0 (1 + g) / (R – g)Solving this equation for the dividend gives us:D 0 = $50(.14 – .08) / (1.08) = $2.7820. The price **of** a share **of** preferred stock is the dividend payment divided by the required return. Weknow the dividend payment in Year 6, so we can find the price **of** the stock in Year 5, one yearbefore the first dividend payment. Doing so, we get:P 5 = $9.00 / .07 = $128.57The price **of** the stock today is the PV **of** the stock price in the future, so the price today will be:P 0 = $128.57 / (1.07) 5 = $91.6721. If the company’s earnings are declining at a constant rate, the dividends will decline at the same ratesince the dividends are assumed to be a constant percentage **of** income. The dividend next year willbe less than this year’s dividend, soP 0 = D 0 (1 + g) / (R – g) = $5.00(1 – .10) / [(.14 – (–.10)] = $18.7522. Here we have a stock paying a constant dividend for a fixed period, and an increasing dividendthereafter. We need to find the present value **of** the two different cash flows using the appropriatequarterly interest rate. The constant dividend is an annuity, so the present value **of** these dividends is:PVA = C(PVIFA R,t )PVA = $1(PVIFA 2.5%,12 )PVA = $10.26Now we can find the present value **of** the dividends beyond the constant dividend phase. Using thepresent value **of** a growing annuity equation, we find:P 12 = D 13 / (R – g)P 12 = $1(1 + .005) / (.025 – .005)P 12 = $50.25This is the price **of** the stock immediately after it has paid the last constant dividend. So, the presentvalue **of** the future price is:PV = $50.25 / (1 + .025) 12PV = $37.36The price today is the sum **of** the present value **of** the two cash flows, so:P 0 = $10.26 + 37.36P 0 = $47.62

CHAPTER 5 B- 10923. We can find the price **of** the stock in Year 4 when it begins a constant increase in dividends using thegrowing perpetuity equation. So, the price **of** the stock in Year 4, immediately after the dividendpayment, is:P 4 = D 4 (1 + g) / (R – g)P 4 = $2(1 + .06) / (.16 – .06)P 4 = $21.20The stock price today is the sum **of** the present value **of** the two fixed dividends plus the presentvalue **of** the future price, so:P 0 = $2 / (1 + .16) 3 + $2 / (1 + .16) 4 + $21.20 / (1 + .16) 4P 0 = $14.0924. Here we need to find the dividend next year for a stock with nonconstant growth. We know the stockprice, the dividend growth rates, and the required return, but not the dividend. First, we need torealize that the dividend in Year 3 is the constant dividend times the FVIF. The dividend in Year 3will be:D 3 = D(1.04)The equation for the stock price will be the present value **of** the constant dividends, plus the presentvalue **of** the future stock price, or:P 0 = D / 1.12 + D /1.12 2 + D(1.04)/(.12 – .04)/1.12 2$30 = D / 1.12 + D /1.12 2 + D(1.04)/(.12 – .04)/1.12 2We can factor out D 0 in the equation, and combine the last two terms. Doing so, we get:$30 = D{1/1.12 + 1/1.12 2 + [(1.04)/(.12 – .04)] / 1.12 2 }Reducing the equation even further by solving all **of** the terms in the braces, we get:$30 = D(12.0536)D = $30 / 12.0536 = $2.4925. The required return **of** a stock consists **of** two components, the capital gains yield and the dividendyield. In the constant dividend growth model (growing perpetuity equation), the capital gains yield isthe same as the dividend growth rate, or algebraically:R = D 1 /P 0 + g

B-110 SOLUTIONSWe can find the dividend growth rate by the growth rate equation, or:g = ROE × bg = .11 × .75g = .0825 or 8.25%This is also the growth rate in dividends. To find the current dividend, we can use the informationprovided about the net income, shares outstanding, and payout ratio. The total dividends paid is thenet income times the payout ratio. To find the dividend per share, we can divide the total dividendspaid by the number **of** shares outstanding. So:Dividend per share = (Net income × Payout ratio) / Shares outstandingDividend per share = ($10,000,000 × .25) / 1,250,000Dividend per share = $2.00Now we can use the initial equation for the required return. We must remember that the equationuses the dividend in one year, so:R = D 1 /P 0 + gR = $2(1 + .0825)/$40 + .0825R = .1366 or 13.66%26. First, we need to find the annual dividend growth rate over the past four years. To do this, we canuse the future value **of** a lump sum equation, and solve for the interest rate. Doing so, we find thedividend growth rate over the past four years was:FV = PV(1 + R) t$1.66 = $0.90(1 + R) 4R = ($1.66 / $0.90) 1/4 – 1R = .1654 or 16.54%We know the dividend will grow at this rate for five years before slowing to a constant rateindefinitely. So, the dividend amount in seven years will be:D 7 = D 0 (1 + g 1 ) 5 (1 + g 2 ) 2D 7 = $1.66(1 + .1654) 5 (1 + .08) 2D 7 = $4.1627. a. We can find the price **of** the all the outstanding company stock by using the dividends the sameway we would value an individual share. Since earnings are equal to dividends, and there is nogrowth, the value **of** the company’s stock today is the present value **of** a perpetuity, so:P = D / RP = $800,000 / .15P = $5,333,333.33

CHAPTER 5 B- 111The price-earnings ratio is the stock price divided by the current earnings, so the price-earningsratio **of** each company with no growth is:P/E = Price / EarningsP/E = $5,333,333.33 / $800,000P/E = 6.67 timesb. Since the earnings have increased, the price **of** the stock will increase. The new price **of** the allthe outstanding company stock is:P = D / RP = ($800,000 + 100,000) / .15P = $6,000,000.00The price-earnings ratio is the stock price divided by the current earnings, so the price-earningswith the increased earnings is:P/E = Price / EarningsP/E = $6,000,000 / $800,000P/E = 7.50 timesc. Since the earnings have increased, the price **of** the stock will increase. The new price **of** the allthe outstanding company stock is:P = D / RP = ($800,000 + 200,000) / .15P = $6,666,666.67The price-earnings ratio is the stock price divided by the current earnings, so the price-earningswith the increased earnings is:P/E = Price / EarningsP/E = $6,666,666.67 / $800,000P/E = 8.33 times28. a. If the company does not make any new investments, the stock price will be the present value **of**the constant perpetual dividends. In this case, all earnings are paid dividends, so, applying theperpetuity equation, we get:P = Dividend / RP = $7 / .12P = $58.33b. The investment is a one-time investment that creates an increase in EPS for two years. Tocalculate the new stock price, we need the cash cow price plus the NPVGO. In this case, theNPVGO is simply the present value **of** the investment plus the present value **of** the increases inEPS. SO, the NPVGO will be:NPVGO = C 1 / (1 + R) + C 2 / (1 + R) 2 + C 3 / (1 + R) 3NPVGO = –$1.75 / 1.12 + $1.90 / 1.12 2 + $2.10 / 1.12 3NPVGO = $1.45

B-112 SOLUTIONSSo, the price **of** the stock if the company undertakes the investment opportunity will be:P = $58.33 + 1.45P = $59.78c. After the project is over, and the earnings increase no longer exists, the price **of** the stock willrevert back to $58.33, the value

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