FCF_11th_edition_Solutions_Manual

CHAPTER 1 - 2 9. In auction markets like the NYSE, brokers and agents meet at a physical location (the exchange) to match buyers and sellers of assets. Dealer markets like NASDAQ consist of dealers operating at dispersed locales who buy and sell assets themselves, communicating with other dealers either electronically or literally over-the-counter. 10. Such organizations frequently pursue social or political missions, so many different goals are conceivable. One goal that is often cited is revenue minimization; that is, provide whatever goods and services are offered at the lowest possible cost to society. A better approach might be to observe that even a not-for-profit business has equity. Thus, one answer is that the appropriate goal is to maximize the value of the equity. 11. 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. 12. 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 illegal behavior, and the framework of stock valuation explicitly includes these. At the other extreme, we could argue that these are noneconomic phenomena and are best handled through the political process. A classic (and highly relevant) thought question that illustrates this debate goes something like this: “A firm has estimated that the cost of improving the safety of one of its products is $30 million. However, the firm believes that improving the safety of the product will only save $20 million in product liability claims. What should the firm do?” 13. 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. 14. The goal of management should be to maximize the share price for the current shareholders. If management believes that it can improve the profitability of the firm so that the share price will exceed $35, then they should fight the offer from the outside company. If management believes that this bidder or other unidentified bidders will actually pay more than $35 per share to acquire the company, then they should still fight the offer. However, if the current management cannot increase the value of the firm beyond the bid price, and no other higher bids come in, then management is not acting in the interests of the shareholders by fighting the offer. Since current managers often lose their jobs when the corporation is acquired, poorly monitored managers have an incentive to fight corporate takeovers in situations such as this. 15. We would expect agency problems to be less severe in countries with a relatively small percentage of individual ownership. Fewer individual owners should reduce the number of diverse opinions concerning corporate goals. The high percentage of institutional ownership might lead to a higher degree of agreement between owners and managers on decisions concerning risky projects. In addition, institutions may be better able to implement effective monitoring mechanisms on managers than can individual owners, based on the institutions’ deeper resources and experiences with their own management. 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 and a more efficient market for corporate control.

5 SOLUTIONS MANUAL 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, its cash flow to creditors will be negative. 10. The adjustments discussed were purely accounting changes; they had no cash flow or market value consequences unless the new accounting information caused stockholders to revalue the derivatives. 11. Enterprise value is the theoretical takeover price. In the event of a takeover, an acquirer would have to take on the company's debt but would pocket its cash. Enterprise value differs significantly from simple market capitalization in several ways, and it may be a more accurate representation of a firm's value. In a takeover, the value of a firm's debt would need to be paid by the buyer. Thus, enterprise value provides a much more accurate takeover valuation because it includes debt in its value calculation. 12. In general, it appears that investors prefer companies that have a steady earnings stream. If true, this encourages companies to manage earnings. Under GAAP, there are numerous choices for the way a company reports its financial statements. Although not the reason for the choices under GAAP, one outcome is the ability of a company to manage earnings, which is not an ethical decision. Even though earnings and cash flow are often related, earnings management should have little effect on cash flow (except for tax implications). If the market is “fooled” and prefers steady earnings, shareholder wealth can be increased, at least temporarily. However, given the questionable ethics of this practice, the company (and shareholders) will lose value if the practice is discovered. Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. To find owners’ equity, we must construct a balance sheet as follows: Balance Sheet CA $ 5,300 CL $ 4,600 NFA 24,900 LTD 10,300 OE ?? TA $30,200 TL & OE $30,200 We know that total liabilities and owners’ equity (TL & OE) must equal total assets of $30,200. We also know that TL & OE is equal to current liabilities plus long-term debt plus owners’ equity, so owners’ equity is: Owner’s equity = $30,200 – 10,300 – 4,600 = $15,300 NWC = CA – CL = $5,300 – 4,600 = $700

CHAPTER 2 - 8 14. To find the OCF, we first calculate net income. Income Statement Sales $267,000 Costs 148,000 Other expenses 8,200 Depreciation 17,600 EBIT $ 93,200 Interest 12,400 Taxable income $ 80,800 Taxes 32,620 Net income $ 48,180 Dividends $15,500 Additions to RE $32,680 a. OCF = EBIT + Depreciation – Taxes = $93,200 + 17,600 – 32,620 = $78,180 b. CFC = Interest – Net new LTD = $12,400 – (–4,900) = $17,300 Note that the net new long-term debt is negative because the company repaid part of its long- term debt. c. CFS = Dividends – Net new equity = $15,500 – 6,400 = $9,100 d. We know that CFA = CFC + CFS, so: CFA = $17,300 + 9,100 = $26,400 CFA 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 + Depreciation = $25,000 + 17,600 = $42,600 Now we can use: CFA = OCF – Net capital spending – Change in NWC $26,400 = $78,180 – 42,600 – Change in NWC Change in NWC = $9,180 This means that the company increased its NWC by $9,180.

11 SOLUTIONS MANUAL c. Change in NWC = NWC end – NWC beg = (CA end – CL end ) – (CA beg – CL beg ) = ($7,829 – 4,159) – ($6,336 – 3,564) = $3,670 – 2,772 = $898 Net capital spending = NFA end – NFA beg + Depreciation = $22,176 – 18,018 + 5,341 = $9,499 CFA = OCF – Change in NWC – Net capital spending = $8,316 – 898 – 9,499 = –$2,082 The cash flow from assets can be positive or negative, since it represents whether the firm raised funds or distributed funds on a net basis. In this problem, even though net income and OCF are positive, the firm invested heavily in both fixed assets and net working capital; it had to raise a net $2,082 in funds from its stockholders and creditors to make these investments. d. Cash flow to creditors = Interest – Net new LTD = $2,409 – 0 = $2,409 Cash flow to stockholders = Cash flow from assets – Cash flow to creditors = –$2,082 – 2,409 = –$4,491 We can also calculate the cash flow to stockholders as: Cash flow to stockholders = Dividends – Net new equity Solving for net new equity, we get: Net new equity = $1,716 – (–4,491) = $6,207 The firm had positive earnings in an accounting sense (NI > 0) and had positive cash flow from operations. The firm invested $898 in new net working capital and $9,499 in new fixed assets. The firm had to raise $2,082 from its stakeholders to support this new investment. It accomplished this by raising $6,207 in the form of new equity. After paying out $1,716 of this in the form of dividends to shareholders and $2,409 in the form of interest to creditors, $2,082 was left to meet the firm’s cash flow needs for investment. 22. a. Total assets 2014 = $1,005 + 4,144 = $5,149 Total liabilities 2014 = $402 + 2,190= $2,592 Owners’ equity 2014 = $5,149 – 2,592 = $2,557 Total assets 2015 = $1,089 + 4,990 = $6,079 Total liabilities 2015 = $451 + 2,329 = $2,780 Owners’ equity 2015 = $6,079 – 2,780 = $3,299 b. NWC 2014 = CA14 – CL14 = $1,005 – 402 = $603 NWC 2015 = CA15 – CL15 = $1,089 – 451 = $638 Change in NWC = NWC15 – NWC14 = $638 – 603 = $35

CHAPTER 2 - 14 2014 Income Statement 2015 Income Statement Sales $12,730.00 Sales $14,229.00 COGS 4,377.00 COGS 5,178.00 Other expenses 1,041.00 Other expenses 906.00 Depreciation 1,827.00 Depreciation 1,910.00 EBIT $5,485.00 EBIT $6,235.00 Interest 854.00 Interest 1,019.00 EBT $4,631.00 EBT $5,216.00 Taxes (34%) 1,574.54 Taxes (34%) 1,773.44 Net income $3,056.46 Net income $3,442.56 Dividends $1,522.00 Dividends $1,780.00 Additions to RE 1,534.46 Additions to RE 1,662.56 26. OCF = EBIT + Depreciation – Taxes = $6,235 + 1,910 – 1,773.44 = $6,371.56 Change in NWC = NWC end – NWC beg = (CA – CL) end – (CA – CL) beg = ($34,301 – 6,370) – ($31,222 – 6,110) = $2,819 Net capital spending = NFA end – NFA beg + Depreciation = $59,700 – 55,977 + 1,910 = $5,633 Cash flow from assets = OCF – Change in NWC – Net capital spending = $6,371.56 – 2,819 – 5,633 = –$2,080.44 Cash flow to creditors = Interest – Net new LTD Net new LTD = LTD end – LTD beg Cash flow to creditors = $1,019 – ($27,099 – 22,352) = –$3,728 Net new equity = Common stock end – Common stock beg Common stock + Retained earnings = Total owners’ equity Net new equity = (OE – RE) end – (OE – RE) beg = OE end – OE beg + RE beg – RE end RE end = RE beg + Additions to RE15  Net new equity = OE end – OE beg + RE beg – (RE beg + Additions to RE15) = OE end – OE beg – Additions to RE Net new equity = $60,532 – 58,737 – 1,662.56 = $132.44 CFS = Dividends – Net new equity CFS = $1,780 – 132.44 = $1,647.56 As a check, cash flow from assets is –$2,080.44. CFA = Cash flow from creditors + Cash flow to stockholders CFA = –$3,278 + 1,647.56 = –$2,080.44

17 SOLUTIONS MANUAL d. For an online service provider such as Comcast, using a per internet session for costs would allow for comparisons with smaller services. A per subscriber basis would also make sense. e. For a hospital such as Holy Cross, revenues and costs expressed on a per-bed basis would be useful. f. For a college textbook publisher such as McGraw-Hill/Irwin, the leading publisher of finance textbooks for the college market, the obvious standardization would be per book sold. 11. Reporting the sale of Treasury securities as cash flow from operations is an accounting “trick,” and as such, should constitute a possible red flag about the companies accounting practices. For most companies, the gain from a sale of securities should be placed in the financing section. Including the sale of securities in the cash flow from operations would be acceptable for a financial company, such as an investment or commercial bank. 12. Increasing the payables period increases the cash flow from operations. This could be beneficial for the company as it may be a cheap form of financing, but it is basically a one-time change. The payables period cannot be increased indefinitely as it will negatively affect the company’s credit rating if the payables period becomes too long. Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. Using the formula for NWC, we get: NWC = CA – CL CA = CL + NWC CA = $4,380 + 1,920 CA = $6,300 So, the current ratio is: Current ratio = CA / CL Current ratio = $6,300 / $4,380 Current ratio = 1.44 times And the quick ratio is: Quick ratio = (CA – Inventory) / CL Quick ratio = ($6,300 – 3,750) / $4,380 Quick ratio = .58 times

CHAPTER 3 - 20 10. The average time to pay suppliers is the days’ sales in payables, so: Payables turnover = COGS / Accounts payable Payables turnover = $57,382 / $10,432 Payables turnover = 5.50 times Days’ sales in payables = 365 days / Payables turnover Days’ sales in payables = 365 / 5.50 Days’ sales in payables = 66.36 days The company left its bills to suppliers outstanding for 66.36 days on average. A large value for this ratio could imply that either (1) the company is having liquidity problems, making it difficult to pay off its short-term obligations, or (2) that the company has successfully negotiated lenient credit terms from its suppliers. 11. First, we need the enterprise value, which is: Enterprise value = Market capitalization + Debt – Cash Enterprise value = $610,000 + 204,000 – 39,000 Enterprise value = $775,000 And EBITDA is: EBITDA = EBIT + Depreciation & Amortization EBITDA = $96,000 + 143,000 EBITDA = $239,000 So, the enterprise value-EBITDA multiple is: Enterprise value-EBITDA multiple = $775,000 / $239,000 Enterprise value-EBITDA multiple = 3.24 times 12. The equity multiplier is: EM = 1 + D/E EM = 1 + .65 EM = 1.65 One formula to calculate return on equity is: ROE = ROA(EM) ROE = .082(1.65) ROE = .1353, or 13.53% ROE can also be calculated as: ROE = NI / TE

21 SOLUTIONS MANUAL So, net income is: Net income = ROE(TE) Net income = .1353($515,000) Net income = $69,679.50 13. through 15 :   2014 #13   2015 #13 #14 #15 Assets               Current assets               Cash $11,135 2.67% $13,407 2.93% 1.2040 1.0963 Accounts receivable 28,419 6.81% 30,915 6.75% 1.0878 .9905 Inventory 51,163 12.26% 56,295 12.29% 1.1003 1.0018 Total $90,717 21.75% $100,617 21.96% 1.1091 1.0099 Fixed assets Net plant and equipment $326,456 78.25% $357,560 78.04% 1.0953 .9973 Total assets $417,173 100% $458,177 100% 1.0983 1.0000   Liabilities and Owners’ Equity Current liabilities Accounts payable $45,166 10.83% $48,185 10.52% 1.0668 .9714 Notes payable 17,773 4.26% 18,257 3.98% 1.0272 .9353 Total $62,939 15.09% $66,442 14.50% 1.0557 .9612 Long-term debt $44,000 10.55% $39,000 8.51% .8864 .8070 Owners' equity Common stock and paid-in surplus $50,000 11.99% $50,000 10.91% 1.0000 .9105 Accumulated retained earnings 260,234 62.38% 302,735 66.07% 1.1633 1.0592 Total $310,234 74.37% $352,735 76.99% 1.1370 1.0352 Total liabilities and owners' equity $417,173 100% $458,177 100% 1.0983 1.0000 The common-size balance sheet answers are found by dividing each category by total assets. For example, the cash percentage for 2014 is: $11,135 / $417,173 = .0267, or 2.67% This means that cash is 2.67% of total assets. The common-base year answers for Question 14 are found by dividing each category value for 2015 by the same category value for 2014. For example, the cash common-base year number is found by: $13,407 / $11,135 = 1.2040 This means the cash balance in 2015 is 1.2040 times as large as the cash balance in 2014.

CHAPTER 3 - 22 The common-size, common-base year answers for Question 15 are found by dividing the common- size percentage for 2015 by the common-size percentage for 2014. For example, the cash calculation is found by: 2.93% / 2.67% = 1.0963 This tells us that cash, as a percentage of assets, increased by 9.63%. 16.   2014   Sources/Us es     2015 Assets Current assets Cash $11,135 $2,272 U $13,407 Accounts receivable 28,419 2,496 U 30,915 Inventory 51,163 5,132 U 56,295 Total $90,717 $9,900 U $100,617 Fixed assets Net plant and equipment $326,456 $31,104 U $357,560 Total assets $417,173 $41,004 U $458,177   Liabilities and Owners’ Equity Current liabilities Accounts payable $45,166 $3,019 S $48,185 Notes payable 17,773 484 S 18,257 Total $62,939 $3,503 S $66,442 Long-term debt $44,000 –$5,000 U $39,000 Owners' equity Common stock and paid-in surplus $50,000 0 $50,000 Accumulated retained earnings 260,234 42,501 S 302,735 Total $310,234 $42,501 S $352,735 Total liabilities and owners' equity $417,173 $41,004 S $458,177 The firm used $41,004 in cash to acquire new assets. It raised this amount of cash by increasing liabilities and owners’ equity by $41,004. In particular, the needed funds were raised by internal financing (on a net basis), out of the additions to retained earnings, and an increase in current liabilities. 17. a. Current ratio = Current assets / Current liabilities Current ratio 2014 = $90,717 / $62,939 = 1.44 times Current ratio 2015 = $100,617 / $66,442 = 1.51 times b. Quick ratio = (Current assets – Inventory) / Current liabilities Quick ratio 2014 = ($90,717 – 51,163) / $62,939 = .63 times Quick ratio 2015 = ($100,617 – 56,295) / $66,442 = .67 times

23 SOLUTIONS MANUAL c. Cash ratio = Cash / Current liabilities Cash ratio 2014 = $11,135 / $62,939 = .18 times Cash ratio 2015 = $13,407 / $66,442 = .20 times d. NWC ratio = NWC / Total assets NWC ratio 2014 = ($90,717 – 62,939) / $417,173 = .0666, or 6.66% NWC ratio 2015 = ($100,617 – 66,442) / $458,177 = .0746, or 7.46% e. Debt-equity ratio = Total debt / Total equity Debt-equity ratio 2014 = ($62,939 + 44,000) / $310,234 = .34 times Debt-equity ratio 2015 = ($66,442 + 39,000) / $352,735 = .30 times Equity multiplier = 1 + D/E Equity multiplier 2014 = 1 + .34 = 1.34 Equity multiplier 2015 = 1 + .30 = 1.30 f. Total debt ratio = (Total assets – Total equity) / Total assets Total debt ratio 2014 = ($417,173 – 310,234) / $417,173 = .26 times Total debt ratio 2015 = ($458,177 – 352,735) / $458,177 = .23 times Long-term debt ratio = Long-term debt / (Long-term debt + Total equity) Long-term debt ratio 2014 = $44,000 / ($44,000 + 310,234) = .12 times Long-term debt ratio 2015 = $39,000 / ($39,000 + 352,735) = .10 times Intermediate 18. This is a multistep problem involving several ratios. The ratios given are all part of the DuPont Identity. The only DuPont Identity ratio not given is the profit margin. If we know the profit margin, we can find the net income since sales are given. So, we begin with the DuPont Identity: ROE = .11 = (PM)(TAT)(EM) = (PM)(S / TA)(1 + D/E) Solving the DuPont Identity for profit margin, we get: PM = [(ROE)(TA)] / [(1 + D/E)(S)] PM = [(.11)($2,604)] / [(1 + .75)($5,783)] PM = .0283 Now that we have the profit margin, we can use this number and the given sales figure to solve for net income: PM = .0283 = NI / S NI = .0283($5,783) Net income = $163.68

CHAPTER 3 - 24 19. This is a multistep problem involving several ratios. It is often easier to look backward to determine where to start. We need receivables turnover to find days’ sales in receivables. To calculate receivables turnover, we need credit sales, and to find credit sales, we need total sales. Since we are given the profit margin and net income, we can use these to calculate total sales as: PM = .079 = NI / Sales PM = $186,000 / Sales Sales = $2,354,430 Credit sales are 70 percent of total sales, so: Credit sales = .70($2,354,430) Credit sales = $1,648,101 Now we can find receivables turnover by: Receivables turnover = Credit sales / Accounts receivable Receivables turnover = $1,648,101 / $123,840 Receivables turnover = 13.31 times Days’ sales in receivables = 365 days / Receivables turnover Days’ sales in receivables = 365 / 13.31 Days’ sales in receivables = 27.43 days 20. The solution to this problem requires a number of steps. First, remember that Current assets + Net fixed assets = Total assets. So, if we find the CA and the TA, we can solve for NFA. Using the numbers given for the current ratio and the current liabilities, we solve for CA: Current ratio = Current assets / Current liabilities Current assets = Current ratio(Current liabilities) Current assets = 1.25($987) Current assets = $1,233.75 To 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 profit margin: Profit margin = Net income / Sales Net income = Profit margin(Sales) Net income = .086($6,860) Net income = $589.96 We now use the net income figure as an input into ROE to find the total equity: ROE = Net income / Total equity Total equity = Net income / ROE Total equity = $589.96 / .175 Total equity = $3,371.20

25 SOLUTIONS MANUAL Next, we need to find the long-term debt. The long-term debt ratio is: Long-term debt ratio = .45 = LTD / (LTD + Total equity) Inverting both sides gives: 1 / .45 = (LTD + Total equity) / LTD = 1 + (Total equity / LTD) Substituting the total equity into the equation and solving for long-term debt gives the following: 2.222 = 1 + ($3,371.20 / LTD) LTD = $3,371.20 / 1.222 LTD = $2,758.25 Now, we can find the total debt of the company: Total debt = Current liabilities + LTD Total debt = $987 + 2,758.25 Total debt = $3,745.25 And, with the total debt, we can find the TD&E, which is equal to TA: Total assets = Total debt + Total equity Total assets = $3,371.20 + 3,745.25 Total assets = $7,116.45 And finally, we are ready to solve the balance sheet identity as: Net fixed assets = Total assets – Current assets Net fixed assets = $7,116.45 – 1,233.75 Net fixed assets = $5,882.70 21. Child: Profit margin = Net income / Sales Profit margin = $1.50 / $50 Profit margin = .03, or 3% Store: Profit margin = Net income / Sales Profit margin = $10,650,000 / $710,000,000 Profit margin = .015, or 1.50% The advertisement is referring to the store’s profit margin, but a more appropriate earnings measure for the firm’s owners is the return on equity. ROE = NI / TE = NI / (TA – TD) ROE = $10,650,000 / ($390,000,000 – 280,000,000) ROE = .0968, or 9.68%

CHAPTER 3 - 26 22. The solution requires substituting two ratios into a third ratio. Rearranging Debt / Total assets: Firm A Firm B D / TA = .55 D / TA = .40 (TA – E) / TA = .55 (TA – E) / TA = .40 (TA / TA) – (E / TA) = .55 (TA / TA) – (E / TA) = .40 1 – (E / TA) = .55 1 – (E / TA) = .40 E / TA = .45 E / TA = .60 E = .45(TA) E = .60 (TA) Rearranging ROA = Net income / Total assets, we find: NI / TA = .08 NI / TA = .11 NI = .08(TA) NI = .11(TA) Since ROE = Net income / Equity, we can substitute the above equations into the ROE formula, which yields: ROE = .08(TA) / .45(TA) ROE = .11(TA) / .60 (TA) ROE = .08 / .45 ROE = .11 / .60 ROE = .1778, or 17.78% ROE = .1833, or 18.33% 23. This problem requires us to work backward through the income statement. First, recognize that Net income = (1 – T C )EBT. Plugging in the numbers given and solving for EBT, we get: EBT = $17,382 / (1 – .34) EBT = $26,336.36 Now, we can add interest to EBT to get EBIT as follows: EBIT = EBT + Interest EBIT = $26,336.36 + 3,953 EBIT = $30,289.36 To get EBITD (earnings before interest, taxes, and depreciation), the numerator in the cash coverage ratio, add depreciation to EBIT: EBITD = EBIT + Depreciation EBITD = $30,289.36 + 4,283 EBITD = $34,572.36 Now, we can plug the numbers into the cash coverage ratio and calculate: Cash coverage ratio = EBITD / Interest Cash coverage ratio = $34,572.36 / $3,953 Cash coverage ratio = 8.75 times

27 SOLUTIONS MANUAL 24. The only ratio given that includes cost of goods sold is the inventory turnover ratio, so it is the last ratio used. Since current liabilities are given, we start with the current ratio: Current ratio = Current assets / Current liabilities 1.35 = Current assets / $387,000 Current assets = $522,450 Using the quick ratio, we solve for inventory: Quick ratio = (Current assets – Inventory) / Current liabilities Inventory = Current assets – (Quick ratio × Current liabilities) Inventory = $522,450 – (.85 × $387,000) Inventory = $193,500 Inventory turnover = COGS / Inventory 8.40 = COGS / $193,500 COGS = $1,625,400 25. Profit margin = Net income / Sales Profit margin = –£27,835 / £204,350 Profit margin = –.1362, or –13.62% 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 the text are measured in terms of currency. This is one reason why financial ratio analysis is widely used in international finance to compare the business operations of firms and/or divisions across national economic borders. The net income in dollars is: Net income = Profit margin × Sales Net income = –.1362($327,810) Net income = –$44,652 26. Short-term solvency ratios: Current ratio = Current assets / Current liabilities Current ratio 2014 = $68,074 / $61,722 = 1.10 times Current ratio 2015 = $79,974 / $69,426 = 1.15 times Quick ratio = (Current assets – Inventory) / Current liabilities Quick ratio 2014 = ($68,074 – 27,931) / $61,722 = .65 times Quick ratio 2015 = ($79,974 – 32,586) / $69,426 = .68 times Cash ratio = Cash / Current liabilities Cash ratio 2014 = $26,450 / $61,722 = .43 times Cash ratio 2015 = $29,106 / $69,426 = .42 times Asset utilization ratios: Total asset turnover = Sales / Total assets Total asset turnover = $422,045 / $478,319 = .88 times Inventory turnover = Cost of goods sold / Inventory Inventory turnover = $291,090 / $32,586 = 8.93 times

CHAPTER 3 - 28 Receivables turnover = Sales / Accounts receivable Receivables turnover = $422,045 / $18,282 = 23.09 times Long-term solvency ratios: Total debt ratio = (Total assets – Total equity) / Total assets Total debt ratio 2014 = ($425,239 – 268,517) / $425,239 = .37 times Total debt ratio 2015 = ($478,319 – 298,893) / $478,319 = .38 times Debt-equity ratio = Total debt / Total equity Debt-equity ratio 2014 = ($61,722 + 95,000) / $268,517 = .58 times Debt-equity ratio 2015 = ($69,426 + 110,000) / $298,893 = .60 times Equity multiplier = 1 + D/E Equity multiplier 2014 = 1 + .58 = 1.58 times Equity multiplier 2015 = 1 + .60 = 1.60 times Times interest earned = EBIT / Interest Times interest earned = $93,902 / $16,400 = 5.73 times Cash coverage ratio = (EBIT + Depreciation) / Interest Cash coverage ratio = ($93,902 + 37,053) / $16,400 = 7.99 times Profitability ratios: Profit margin = Net income / Sales Profit margin = $50,376 / $422,045 = .1194, or 11.94% Return on assets = Net income / Total assets Return on assets = $50,376 / $478,319 = .1053, or 10.53% Return on equity = Net income / Total equity Return on equity = $50,376 / $298,893 = .1685, or 16.85% 27. The DuPont identity is: ROE = (PM)(TAT)(EM) ROE = (.1194)(.88)(1.60) = .1685, or 16.85%

29 SOLUTIONS MANUAL 28. SMOLIRA GOLF CORP. Statement of Cash Flows For 2015   Cash, beginning of the year $ 26,450                   Operating activities         Net income     $ 50,376   Plus:               Depreciation     $ 37,053     Increase in accounts payable 4,883     Increase in other current liabilities 5,161   Less:               Increase in accounts receivable $ (4,589)     Increase in inventory   (4,655)                   Net cash from operating activities $ 88,229                   Investment activities         Fixed asset acquisition $(78,233)   Net cash from investment activities $(78,233)                   Financing activities         Increase in notes payable $ (2,340)     Dividends paid   (20,000)     Increase in long-term debt 15,000   Net cash from financing activities $(7,340)                   Net increase in cash   $ 2,656                   Cash, end of year   $ 29,106 29. Earnings per share = Net income / Shares Earnings per share = $50,376 / 25,000 = $2.02 per share P/E ratio = Share price / Earnings per share P/E ratio = $58 / $2.02 = 28.78 times Dividends per share = Dividends / Shares Dividends per share = $20,000 / 25,000 = $.80 per share Book value per share = Total equity / Shares Book value per share = $298,893 / 25,000 shares = $11.96 per share

CHAPTER 3 - 30 Market-to-book ratio = Share price / Book value per share Market-to-book ratio = $58 / $11.96 = 4.85 times PEG ratio = P/E ratio / Growth rate PEG ratio = 28.78 / 9 = 3.20 times 30. First, we will find the market value of the company’s equity, which is: Market value of equity = Shares × Share price Market value of equity = 25,000($58) = $1,450,000 The total book value of the company’s debt is: Total debt = Current liabilities + Long-term debt Total debt = $69,426 + 110,000 = $179,426 Now we can calculate Tobin’s Q, which is: Tobin’s Q = (Market value of equity + Book value of debt) / Book value of assets Tobin’s Q = ($1,450,000 + 179,426) / $478,319 Tobin’s Q = 3.41 Using the book value of debt implicitly assumes that the book value of debt is equal to the market value of debt. This will be discussed in more detail in later chapters, but this assumption is generally true. Using the book value of assets assumes that the assets can be replaced at the current value on the balance sheet. There are several reasons this assumption could be flawed. First, inflation during the life of the assets can cause the book value of the assets to understate the market value of the assets. Since assets are recorded at cost when purchased, inflation means that it is more expensive to replace the assets. Second, improvements in technology could mean that the assets could be replaced with more productive, and possibly cheaper, assets. If this is true, the book value can overstate the market value of the assets. Finally, the book value of assets may not accurately represent the market value of the assets because of depreciation. Depreciation is done according to some schedule, generally straight-line or MACRS. Thus, the book value and market value can often diverge.

CHAPTER 4 LONG-TERM FINANCIAL PLANNING AND GROWTH Answers to Concepts Review and Critical Thinking Questions 1. 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 or indirectly support sales. Put differently, a firm’s future need for things like capital assets, employees, inventory, and financing are determined by its future sales level. 2. Two assumptions of the sustainable growth formula are that the company does not want to sell new equity, and that financial policy is fixed. If the company raises outside equity, or increases its debt- equity ratio, it can grow at a higher rate than the sustainable growth rate. Of course the company could also grow faster than its profit margin increases, if it changes its dividend policy by increasing the retention ratio, or its total asset turnover increases. 3. The internal growth rate is greater than 15 percent, because at a 15 percent growth rate the negative EFN indicates that there is excess internal financing. If the internal growth rate is greater than 15 percent, then the sustainable growth rate is certainly greater than 15 percent, because there is additional debt financing used in that case (assuming the firm is not 100 percent equity-financed). As the retention ratio is increased, the firm has more internal sources of funding, so the EFN will decline. Conversely, as the retention ratio is decreased, the EFN will rise. If the firm pays out all its earnings in the form of dividends, then the firm has no internal sources of funding (ignoring the effects of accounts payable); the internal growth rate is zero in this case and the EFN will rise to the change in total assets. 4. The sustainable growth rate is greater than 20 percent, because at a 20 percent growth rate the negative EFN indicates that there is excess financing still available. If the firm is 100 percent equity financed, then the sustainable and internal growth rates are equal and the internal growth rate would be greater than 20 percent. However, when the firm has some debt, the internal growth rate is always less than the sustainable growth rate, so it is ambiguous whether the internal growth rate would be greater than or less than 20 percent. If the retention ratio is increased, the firm will have more internal funding 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 the retention rate is zero, both the internal and sustainable growth rates are zero, and the EFN will rise to the change in total assets. 5. Presumably not, but, of course, if the product had been much less popular, then a similar fate would have awaited due to lack of sales. 6. Since customers did not pay until shipment, receivables rose. The firm’s NWC, but not its cash, increased. At the same time, costs were rising faster than cash revenues, so operating cash flow declined. The firm’s capital spending was also rising. Thus, all three components of cash flow from assets were negatively impacted.

CHAPTER 4 - 32 7. Apparently not! In hindsight, the firm may have underestimated costs and also underestimated the extra demand from the lower price. 8. 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 the need for planning. 9. All three were important, but the lack of cash or, more generally, financial resources ultimately spelled doom. An inadequate cash resource is usually cited as the most common cause of small business failure. 10. Demanding cash up front, increasing prices, subcontracting production, and improving financial resources via new owners or new sources of credit are some of the options. When orders exceed capacity, price increases may be especially beneficial. Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. It is important to remember that equity will not increase by the same percentage as the other assets. If every other item on the income statement and balance sheet increases by 15 percent, the pro forma income statement and balance sheet will look like this: Pro forma income statement Pro forma balance sheet Sales $ 41,400 Assets $30,360 Debt $ 7,245 Costs 34,270 Equity 23,115 Net income $ 7,130 Total $ 30,360 Total $ 30,360 In order for the balance sheet to balance, equity must be: Equity = Total liabilities and equity – Debt Equity = $30,360 – 7,245 Equity = $23,115 Equity increased by: Equity increase = $23,115 – 20,100 Equity increase = $3,015

33 SOLUTIONS MANUAL Net income is $7,130 but equity only increased by $3,015; therefore, a dividend of: Dividend = $7,130 – 3,015 Dividend = $4,115 must have been paid. Dividends paid is the plug variable. 2. Here we are given the dividend amount, so dividends paid is not a plug variable. If the company pays out one-half of its net income as dividends, the pro forma income statement and balance sheet will look like this: Pro forma income statement Pro forma balance sheet Sales $41,400 Assets $30,360 Debt $ 6,300 Costs 34,270 Equity 23,665 Net income $ 7,130 Total $30,360 Total $29,965 Dividends $3,565 Add. to RE $3,565 Note that the balance sheet does not balance. This is due to EFN. The EFN for this company is: EFN = Total assets – Total liabilities and equity EFN = $30,360 – 29,965 EFN = $395 3. An increase of sales to $8,968 is an increase of: Sales increase = ($8,968 – 7,600) / $7,600 Sales increase = .18, or 18% Assuming costs and assets increase proportionally, the pro forma financial statements will look like this: Pro forma income statement Pro forma balance sheet Sales $ 8,968.00 Assets $25,606.00 Debt $ 9,100.00 Costs 6,112.40 Equity 15,455.60 Net income $ 2,855.60 Total $ 25,606.00 Total $24,555.60 If no dividends are paid, the equity account will increase by the net income, so: Equity = $12,600 + 2,855.60 Equity = $15,455.60 So the EFN is: EFN = Total assets – Total liabilities and equity EFN = $25,606 – 24,555.60 EFN = $1,050.40

CHAPTER 4 - 34 4. An increase of sales to $32,085 is an increase of: Sales increase = ($32,085 – 27,900) / $27,900 Sales increase = .15, or 15% Assuming costs and assets increase proportionally, the pro forma financial statements will look like this: Pro forma income statement Pro forma balance sheet Sales $ 32,085 Assets $ 77,050 Debt $ 27,400 Costs 20,815 Equity 43,602 EBIT 11,270 Total $ 77,050 Total $ 71,002 Taxes (40%) 4,508 Net income $ 6,762 The payout ratio is constant, so the dividends paid this year is the payout ratio from last year times net income, or: Dividends = ($2,400 / $5,880)($6,762) Dividends = $2,760 The addition to retained earnings is: Addition to retained earnings = $6,762 – 2,760 Addition to retained earnings = $4,002 And the new equity balance is: Equity = $39,600 + 4,002 Equity = $43,602 So the EFN is: EFN = Total assets – Total liabilities and equity EFN = $77,050 – 71,002 EFN = $6,048 5. Assuming costs, current liabilities, and assets increase proportionally, the pro forma financial statements will look like this: Pro forma income statement Pro forma balance sheet Sales $7,705.00 CA $4,715.00 CL $2,415.00 Costs 5,060.00 FA 9,890.00 LTD 3,700.00 Taxable income $2,645.00 Equity 7,947.42 Taxes (34%) 899.30 TA $14,605.00 Total D&E $14,062.42 Net income $1,745.70

35 SOLUTIONS MANUAL The payout ratio is 40 percent, so dividends will be: Dividends = .40($1,745.70) Dividends = $698.28 The addition to retained earnings is: Addition to retained earnings = $1,745.70 – 698.28 Addition to retained earnings = $1,047.42 So the EFN is: EFN = Total assets – Total liabilities and equity EFN = $14,605 – 14,062.42 EFN = $542.58 6. To calculate the internal growth rate, we first need to calculate the ROA, which is: ROA = NI / TA ROA = $3,480 / $36,600 ROA = .0951, or 9.51% The plowback ratio, b , is one minus the payout ratio, so: b = 1 – .30 b = .70 Now we can use the internal growth rate equation to get: Internal growth rate = (ROA × b ) / [1 – (ROA × b )] Internal growth rate = [.0951(.70)] / [1 – .0951(.70)] Internal growth rate = .0713, or 7.13% 7. To calculate the sustainable growth rate, we first need to calculate the ROE, which is: ROE = NI / TE ROE = $3,480 / $20,200 ROE = .1723, or 17.23% The plowback ratio, b , is one minus the payout ratio, so: b = 1 – .30 b = .70 Now we can use the sustainable growth rate equation to get: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Sustainable growth rate = [.1723(.70)] / [1 – .1723(.70)] Sustainable growth rate = .1371, or 13.71%

CHAPTER 4 - 36 8. The maximum percentage sales increase is the sustainable growth rate. To calculate the sustainable growth rate, we first need to calculate the ROE, which is: ROE = NI / TE ROE = $7,722 / $48,000 ROE = .1609, or 16.09% The plowback ratio, b , is one minus the payout ratio, so: b = 1 – .30 b = .70 Now we can use the sustainable growth rate equation to get: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Sustainable growth rate = [.1609(.70)] / [1 – .1609(.70)] Sustainable growth rate = .1269, or 12.69% So, the maximum dollar increase in sales is: Maximum increase in sales = $53,500(.1269) Maximum increase in sales = $6,789.33 9. Assuming costs vary with sales and a 20 percent increase in sales, the pro forma income statement will look like this: HEIR JORDAN CORPORATION Pro Forma Income Statement Sales $55,200 Costs 45,120 Taxable income $10,080 Taxes (35%) 3,528 Net income $6,552 The payout ratio is constant, so the dividends paid this year is the payout ratio from last year times net income, or: Dividends = ($2,400 / $5,460)($6,552) Dividends = $2,880 And the addition to retained earnings will be: Addition to retained earnings = $6,552 – 2,880 Addition to retained earnings = $3,672

37 SOLUTIONS MANUAL 10. Below is the balance sheet with the percentage of sales for each account on the balance sheet. Notes payable, total current liabilities, long-term debt, and all equity accounts do not vary directly with sales. HEIR JORDAN CORPORATION Balance Sheet ($) (%) ($) (%) Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 2,950 6.41 Accounts payable $ 2,400 5.22 Accounts receivable 4,100 8.91 Notes payable 5,400 n/a Inventory 6,400 13.91 Total $ 7,800 n/a Total $ 13,450 29.24 Long-term debt 28,000 n/a Fixed assets Owners’ equity Net plant and Common stock and equipment 41,300 89.78 paid-in surplus $15,000 n/a Retained earnings 3,950 n/a Total $ 18,950 n/a Total liabilities and owners’ Total assets $ 54,750 119.02 equity $ 54,750 n/a 11. Assuming costs vary with sales and a 15 percent increase in sales, the pro forma income statement will look like this: HEIR JORDAN CORPORATION Pro Forma Income Statement Sales $52,900 Costs 43,240 Taxable income $9,660 Taxes (34%) 3,381 Net income $6,279 The payout ratio is constant, so the dividends paid this year is the payout ratio from last year times net income, or: Dividends = ($2,400 / $5,460)($6,279) Dividends = $2,760 And the addition to retained earnings will be: Addition to retained earnings = $6,279 – 2,760 Addition to retained earnings = $3,519 The new accumulated retained earnings on the pro forma balance sheet will be: New accumulated retained earnings = $3,950 + 3,519 New accumulated retained earnings = $7,469

CHAPTER 4 - 38 The pro forma balance sheet will look like this: HEIR JORDAN CORPORATION Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 3,392.50 Accounts payable $ 2,760.00 Accounts receivable 4,715.00 Notes payable 5,400.00 Inventory 7,360.00 Total $ 8,160.00 Total $ 15,467.50 Long-term debt 28,000.00 Fixed assets Net plant and Owners’ equity equipment 47,495.00 Common stock and paid-in surplus $ 15,000.00 Retained earnings 7,469.00 Total $ 22,469.00 Total liabilities and owners’ Total assets $ 62,962.50 equity $ 58,629.00 So the EFN is: EFN = Total assets – Total liabilities and equity EFN = $62,962.50 – 58,629.00 EFN = $4,333.50 12. We need to calculate the retention ratio to calculate the internal growth rate. The retention ratio is: b = 1 – .30 b = .70 Now we can use the internal growth rate equation to get: Internal growth rate = (ROA × b ) / [1 – (ROA × b )] Internal growth rate = [.08(.70)] / [1 – .08(.70)] Internal growth rate = .0593, or 5.93% 13. We need to calculate the retention ratio to calculate the sustainable growth rate. The retention ratio is: b = 1 – .25 b = .75 Now we can use the sustainable growth rate equation to get: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Sustainable growth rate = [.15(.75)] / [1 – .15(.75)] Sustainable growth rate = .1268, or 12.68%

39 SOLUTIONS MANUAL 14. We first must calculate the ROE to calculate the sustainable growth rate. To do this we must realize two other relationships. The total asset turnover is the inverse of the capital intensity ratio, and the equity multiplier is 1 + D/E. Using these relationships, we get: ROE = (PM)(TAT)(EM) ROE = (.075)(1 / .65)(1 + .60) ROE = .1846, or 18.46% The plowback ratio is one minus the dividend payout ratio, so: b = 1 – ($16,000 / $67,000) b = .7612 Now we can use the sustainable growth rate equation to get: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Sustainable growth rate = [.1846(.7612)] / [1 – .1846(.7612)] Sustainable growth rate = .1635, or 16.35% 15. We must first calculate the ROE using the DuPont ratio to calculate the sustainable growth rate. The ROE is: ROE = (PM)(TAT)(EM) ROE = (.052)(3.4)(1.30) ROE = .2298, or 22.98% The plowback ratio is one minus the dividend payout ratio, so: b = 1 – .35 b = .65 Now we can use the sustainable growth rate equation to get: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Sustainable growth rate = [.2298(.65)] / [1 – .2298(.65)] Sustainable growth rate = .1756, or 17.56% Intermediate 16. To determine full capacity sales, we divide the current sales by the capacity the company is currently using, so: Full capacity sales = $690,000 / .92 Full capacity sales = $750,000 The maximum sales growth is the full capacity sales divided by the current sales, so: Maximum sales growth = ($750,000 / $690,000) – 1 Maximum sales growth = .0870, or 8.70%

CHAPTER 4 - 40 17. To find the new level of fixed assets, we need to find the current percentage of fixed assets to full capacity sales. Doing so, we find: Fixed assets / Full capacity sales = $520,000 / $750,000 Fixed assets / Full capacity sales = .6933 Next, we calculate the total dollar amount of fixed assets needed at the new sales figure. Total fixed assets = .6933($790,000) Total fixed assets = $547,733.33 The new fixed assets necessary is the total fixed assets at the new sales figure minus the current level of fixed assts. New fixed assets = $547,733.33 – 520,000 New fixed assets = $27,733.33 18. We have all the variables to calculate ROE using the DuPont identity except the profit margin. If we find ROE, we can solve the DuPont identity for profit margin. We can calculate ROE from the sustainable growth rate equation. For this equation we need the retention ratio, so: b = 1 – .30 b = .70 Using the sustainable growth rate equation and solving for ROE, we get: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] .12 = [ROE(.70)] / [1 – ROE(.70)] ROE = .1531, or 15.31% Now we can use the DuPont identity to find the profit margin as: ROE = PM(TAT)(EM) .1531 = PM(1 / .95)(1 + .85) PM = (.1531) / [(1 / .95)(1.85)] PM = .0786, or 7.86% 19. We are given the profit margin. Remember that: ROA = PM(TAT) We can calculate the ROA from the internal growth rate formula, and then use the ROA in this equation to find the total asset turnover. The retention ratio is: b = 1 – .25 b = .75

41 SOLUTIONS MANUAL Using the internal growth rate equation to find the ROA, we get: Internal growth rate = (ROA × b ) / [1 – (ROA × b )] .065 = [ROA(.75)] / [1 – ROA(.75)] ROA = .0814, or 8.14% Plugging ROA and PM into the equation we began with and solving for TAT, we get: ROA = (PM)(TAT) .0814 = .07(TAT) TAT = .0814 / .07 TAT = 1.16 times 20. We should begin by calculating the D/E ratio. We calculate the D/E ratio as follows: Total debt ratio = .35 = TD / TA Inverting both sides we get: 1 / .35 = TA / TD Next, we need to recognize that TA / TD = 1 + TE / TD Substituting this into the previous equation, we get: 1 / .35 = 1 + TE /TD Subtract 1 (one) from both sides and inverting again, we get: D/E = 1 / [(1 / .35) – 1] D/E = .54 With the D/E ratio, we can calculate the EM and solve for ROE using the DuPont identity: ROE = (PM)(TAT)(EM) ROE = (.059)(1.90)(1 + .54) ROE = .1725, or 17.25% Now we can calculate the retention ratio as: b = 1 – .30 b = .70 Finally, putting all the numbers we have calculated into the sustainable growth rate equation, we get: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Sustainable growth rate = [.1725(.70)] / [1 – .1725(.70)] Sustainable growth rate = .1373, or 13.73%

CHAPTER 4 - 42 21. To calculate the sustainable growth rate, we first must calculate the retention ratio and ROE. The retention ratio is: b = 1 – $9,400 / $16,200 b = .4198 And the ROE is: ROE = $16,200 / $55,000 ROE = .2945, or 29.45% So, the sustainable growth rate is: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Sustainable growth rate = [.2945(.4198)] / [1 – .2945(.4198)] Sustainable growth rate = .1411, or 14.11% If the company grows at the sustainable growth rate, the new level of total assets is: New TA = 1.1411($73,000 + 55,000) = $146,058.09 To find the new level of debt in the company’s balance sheet, we take the percentage of debt in the capital structure times the new level of total assets. The additional borrowing will be the new level of debt minus the current level of debt. So: New TD = [D / (D + E)](TA) New TD = [$73,000 / ($73,000 + 55,000)]($146,058.09) New TD = $83,298.76 And the additional borrowing will be: Additional borrowing = $83,298.76 – 73,000 Additional borrowing = $10,298.76 The growth rate that can be supported with no outside financing is the internal growth rate. To calculate the internal growth rate, we first need the ROA, which is: ROA = $16,200 / ($73,000 + 55,000) ROA = .1266, or 12.66% This means the internal growth rate is: Internal growth rate = (ROA × b ) / [1 – (ROA × b )] Internal growth rate = [.1266(.4198)] / [1 – .1266(.4198)] Internal growth rate = .0561, or 5.61%

43 SOLUTIONS MANUAL 22. Since the company issued no new equity, shareholders’ equity increased by retained earnings. Retained earnings for the year were: Retained earnings = NI – Dividends Retained earnings = $29,000 – 6,200 Retained earnings = $22,800 So, the equity at the end of the year was: Ending equity = $161,000 + 22,800 Ending equity = $183,800 The ROE based on the end of period equity is: ROE = $29,000 / $183,800 ROE = .1578, or 15.78% The plowback ratio is: Plowback ratio = Addition to retained earnings / NI Plowback ratio = $22,800 / $29,000 Plowback ratio = .7862, or 78.62% 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 = [.1578(.7862)] / [1 – .1578(.7862)] Sustainable growth rate = .1416, or 14.16% The ROE based on the beginning of period equity is ROE = $29,000 / $161,000 ROE = .1801, or 18.01% Using the shortened equation for the sustainable growth rate and the beginning of period ROE, we get: Sustainable growth rate = ROE × b Sustainable growth rate = .1801 × .7862 Sustainable growth rate = .1416, or 14.16 Using the shortened equation for the sustainable growth rate and the end of period ROE, we get: Sustainable growth rate = ROE × b Sustainable growth rate = .1578 × .7862 Sustainable growth rate = .1240, or 12.40%

CHAPTER 4 - 44 Using the end of period ROE in the shortened sustainable growth rate equation results in a growth rate that is too low. This will always occur whenever the equity increases. If equity increases, the ROE based on end of period equity is lower than the ROE based on the beginning of period equity. The ROE (and sustainable growth rate) in the abbreviated equation is based on equity that did not exist when the net income was earned. 23. The ROA using end of period assets is: ROA = $29,000 / $305,000 ROA = .0951, or 9.51% The beginning of period assets had to have been the ending assets minus the addition to retained earnings, so: Beginning assets = Ending assets – Addition to retained earnings Beginning assets = $305,000 – 22,800 Beginning assets = $282,200 And the ROA using beginning of period assets is: ROA = $29,000 / $282,200 ROA = .1028, or 10.28% Using the internal growth rate equation presented in the text, we get: Internal growth rate = (ROA × b ) / [1 – (ROA × b )] Internal growth rate = [.0951(.7862)] / [1 – .0951(.7862)] Internal growth rate = .0808, or 8.08% Using the formula ROA × b , and beginning of period assets: Internal growth rate = .1028 × .7862 Internal growth rate = .0808, or 8.08% Using the formula ROA × b , and end of period assets: Internal growth rate = .0951 × .7862 Internal growth rate = .0748, or 7.48% Using the end of period ROA in the shortened internal growth rate equation results in a growth rate that is too low. This will always occur whenever the assets increase. If assets increase, the ROA based on end of period assets is lower than the ROA based on the beginning of period assets. The ROA (and internal growth rate) in the abbreviated equation is based on assets that did not exist when the net income was earned.

45 SOLUTIONS MANUAL 24. Assuming costs vary with sales and a 20 percent increase in sales, the pro forma income statement will look like this: FLEURY INC. Pro Forma Income Statement Sales $ 1,069,920 Costs 832,320 Other expenses 21,888 EBIT $ 215,712 Interest 13,400 Taxable income $ 202,312 Taxes(35%) 70,809 Net income $ 131,503 The payout ratio is constant, so the dividends paid this year is the payout ratio from last year times net income, or: Dividends = ($35,684 / $108,134)($131,503) Dividends = $43,396 And the addition to retained earnings will be: Addition to retained earnings = $131,503 – 43,396 Addition to retained earnings = $88,107 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $174,730 + 88,107 New retained earnings = $262,837 The pro forma balance sheet will look like this: FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 29,136 Accounts payable $ 78,240 Accounts receivable 44,484 Notes payable 16,320 Inventory 100,080 Total $ 94,560 Total $ 173,700 Long-term debt 155,000 Fixed assets Net plant and Owners’ equity equipment 475,800 Common stock and paid-in surplus $ 130,000 Retained earnings 262,837 Total $ 392,837 Total liabilities and owners’ Total assets $ 649,500 equity $ 642,397

CHAPTER 4 - 46 So the EFN is: EFN = Total assets – Total liabilities and equity EFN = $649,500 – 642,397 EFN = $7,103 25. First, we need to calculate full capacity sales, which is: Full capacity sales = $891,600 / .80 Full capacity sales = $1,114,500 The full capacity ratio at full capacity sales is: Full capacity ratio = Fixed assets / Full capacity sales Full capacity ratio = $396,500 / $1,114,500 Full capacity ratio = .35576 The fixed assets required at the projected sales figure is the full capacity ratio times the projected sales level: Total fixed assets = .35576($1,069,920) = $380,640 So, EFN is: EFN = ($173,700 + 380,640) – $642,397 = –$88,057 Note that this solution assumes that fixed assets are decreased (sold) so the company has a 100 percent fixed asset utilization. If we assume fixed assets are not sold, the answer becomes: EFN = ($173,700 + 396,500) – $642,397 = –$72,197 26. The D/E ratio of the company is: D/E = ($81,520 + 155,000) / $304,730 D/E = .7762 So the new total debt amount will be: New total debt = .7762($392,837) New total debt = $304,905 This is the new total debt for the company. Given that our calculation for EFN is the amount that must be raised externally and does not increase spontaneously with sales, we need to subtract the spontaneous increase in accounts payable. The new level of accounts payable, which is the current accounts payable times the sales growth, will be: Spontaneous increase in accounts payable = $65,200(.20) Spontaneous increase in accounts payable = $13,040

47 SOLUTIONS MANUAL This means that $13,040 of the new total debt is not raised externally. So, the debt raised externally, which will be the EFN is: EFN = New total debt – (Beginning LTD + Beginning CL + Spontaneous increase in AP) EFN = $304,905 – ($155,000 + 78,240 + 16,320) = $55,345 The pro forma balance sheet with the new long-term debt will be: FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 29,136 Accounts payable $ 78,240 Accounts receivable 44,484 Notes payable 16,320 Inventory 100,080 Total $ 94,560 Total $ 173,700 Long-term debt 210,345 Fixed assets Net plant and Owners’ equity equipment 475,800 Common stock and paid-in surplus $ 130,000 Retained earnings 262,837 Total $ 392,837 Total liabilities and owners’ Total assets $ 649,500 equity $ 697,743 The funds raised by the debt issue can be put into an excess cash account to make the balance sheet balance. The excess debt will be: Excess debt = $697,743 – 649,500 = $48,243 To make the balance sheet balance, the company will have to increase its assets. We will put this amount in an account called excess cash, which will give us the following balance sheet:

CHAPTER 4 - 48 FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 29,136 Accounts payable $ 78,240 Excess cash 48,243 Accounts receivable 44,484 Notes payable 16,320 Inventory 100,080 Total $ 94,560 Total $ 221,943 Long-term debt 210,345 Fixed assets Net plant and Owners’ equity equipment 475,800 Common stock and paid-in surplus $ 130,000 Retained earnings 262,837 Total $ 392,837 Total liabilities and owners’ Total assets $ 697,743 equity $ 697,743 The excess cash has an opportunity cost that we discussed earlier. Increasing fixed assets would also not be a good idea since the company already has enough fixed assets. A likely scenario would be the repurchase of debt and equity in its current capital structure weights. The company’s debt-assets and equity-assets are: Debt-assets = .7762 / (1 + .7762) = .44 Equity-assets = 1 / (1 + .7762) = .56 So, the amount of debt and equity needed will be: Total debt needed = .44($649,500) = $283,824 Equity needed = .56($649,500) = $365,676 So, the repurchases of debt and equity will be: Debt repurchase = ($94,560 + 210,345) – 283,824 = $21,081 Equity repurchase = $392,837 – 365,676 = $27,161 Assuming all of the debt repurchase is from long-term debt, and the equity repurchase is entirely from the retained earnings, the final pro forma balance sheet will be:

49 SOLUTIONS MANUAL FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 29,136 Accounts payable $ 78,240 Accounts receivable 44,484 Notes payable 16,320 Inventory 100,080 Total $ 94,560 Total $ 173,700 Long-term debt 189,264 Fixed assets Net plant and Owners’ equity equipment 475,800 Common stock and paid-in surplus $ 130,000 Retained earnings 235,676 Total $ 365,676 Total liabilities and owners’ Total assets $ 649,500 equity $ 649,500 Challenge 27. The pro forma income statements for all three growth rates will be: FLEURY INC. Pro Forma Income Statement 15 % Sales Growth 20% Sales Growth 25% Sales Growth Sales $1,025,340 $1,069,920 $1,114,500 Costs 797,640 832,320 867,000 Other expenses 20,976 21,888 22,800 EBIT $206,724 $215,712 $224,700 Interest 13,400 13,400 13,400 Taxable income $193,324 $202,312 $211,300 Taxes (35%) 67,663 70,809 73,955 Net income $125,661 $131,503 $137,345 Dividends $41,468 $43,396 $45,324 Add to RE 84,193 88,107 92,021 We will calculate the EFN for the 15 percent growth rate first. Assuming the payout ratio is constant, the dividends paid will be: Dividends = ($35,684 / $108,134)($125,661) Dividends = $41,468 And the addition to retained earnings will be: Addition to retained earnings = $125,661 – 41,468

CHAPTER 4 - 50 Addition to retained earnings = $84,193 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $174,730 + 84,193 New retained earnings = $258,923 The pro forma balance sheet will look like this: 15% Sales Growth : FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 27,922 Accounts payable $ 74,980 Accounts receivable 42,631 Notes payable 16,320 Inventory 95,910 Total $ 91,300 Total $ 166,463 Long-term debt $ 155,000 Fixed assets Net plant and Owners’ equity equipment 455,975 Common stock and paid-in surplus $ 130,000 Retained earnings 258,923 Total $ 388,923 Total liabilities and owners’ Total assets $ 622,438 equity $ 635,223 So the EFN is: EFN = Total assets – Total liabilities and equity EFN = $622,438 – 635,223 EFN = –$12,785 At a 20 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be: Dividends = ($35,684 / $108,134)($131,503) Dividends = $43,396 And the addition to retained earnings will be: Addition to retained earnings = $131,503 – 43,396 Addition to retained earnings = $88,107 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $174,730 + 88,107 New retained earnings = $262,837

51 SOLUTIONS MANUAL The pro forma balance sheet will look like this: 20% Sales Growth : FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 29,136 Accounts payable $ 78,240 Accounts receivable 44,484 Notes payable 16,320 Inventory 100,080 Total $ 94,560 Total $ 173,700 Long-term debt 155,000 Fixed assets Net plant and Owners’ equity equipment 475,800 Common stock and paid-in surplus $ 130,000 Retained earnings 262,837 Total $ 392,837 Total liabilities and owners’ Total assets $ 649,500 equity $ 642,397 So the EFN is: EFN = Total assets – Total liabilities and equity EFN = $649,500 – 642,397 EFN = $7,103 At a 25 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be: Dividends = ($35,684 / $108,134)($137,345) Dividends = $45,324 And the addition to retained earnings will be: Addition to retained earnings = $137,345 – 45,324 Addition to retained earnings = $92,021 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $174,730 + 92,021 New retained earnings = $266,751

CHAPTER 4 - 52 The pro forma balance sheet will look like this: 25% Sales Growth : FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 30,350 Accounts payable $ 81,500 Accounts receivable 46,338 Notes payable 16,320 Inventory 104,250 Total $ 97,820 Total $ 180,938 Long-term debt $ 155,000 Fixed assets Net plant and Owners’ equity equipment 495,625 Common stock and paid-in surplus $ 130,000 Retained earnings 266,751 Total $ 396,751 Total liabilities and owners’ Total assets $ 676,563 equity $ 649,571 So the EFN is: EFN = Total assets – Total liabilities and equity EFN = $676,563 – 649,571 EFN = $26,991 28. The pro forma income statements for all three growth rates will be: FLEURY INC. Pro Forma Income Statement 20% Sales Growth 30% Sales Growth 35% Sales Growth Sales $1,069,920 $1,159,080 $1,203,660 Costs 832,320 901,680 936,360 Other expenses 21,888 23,712 24,624 EBIT $215,712 $233,688 $242,676 Interest 13,400 13,400 13,400 Taxable income $202,312 $220,288 $229,276 Taxes (35%) 70,809 77,101 80,247 Net income $131,503 $143,187 $149,029 Dividends $43,396 $47,251 $49,179 Add to RE 88,107 95,936 99,850 At a 30 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be: Dividends = ($35,684 / $108,134)($143,187) Dividends = $47,251

53 SOLUTIONS MANUAL And the addition to retained earnings will be: Addition to retained earnings = $143,187 – 47,251 Addition to retained earnings = $95,936 The new addition to retained earnings on the pro forma balance sheet will be: New addition to retained earnings = $174,730 + 95,936 New addition to retained earnings = $270,666 The new total debt will be: New total debt = .7762($400,666) New total debt = $310,982 So, the new long-term debt will be the new total debt minus the new short-term debt, or: New long-term debt = $310,982 – 101,080 New long-term debt = $209,902 The pro forma balance sheet will look like this: Sales growth rate = 30% and debt/equity ratio = .7762: FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 31,564 Accounts payable $ 84,760 Accounts receivable 48,191 Notes payable 16,320 Inventory 108,420 Total $ 101,080 Total $ 188,175 Long-term debt 209,902 Fixed assets Net plant and Owners’ equity equipment 515,450 Common stock and paid-in surplus $ 130,000 Retained earnings 270,666 Total $ 400,666 Total liabilities and owners’ Total assets $ 703,625 equity $ 711,647 So the excess debt raised is: Excess debt = $711,647 – 703,625 Excess debt = $8,022

CHAPTER 4 - 54 And the EFN is: EFN = $310,982 – 101,080 – 155,000 EFN = $54,902 At a 35 percent growth rate, and assuming the payout ratio is constant, the dividends paid will be: Dividends = ($35,684 / $108,134)($149,029) Dividends = $49,179 And the addition to retained earnings will be: Addition to retained earnings = $149,029 – 49,179 Addition to retained earnings = $99,850 The new retained earnings on the pro forma balance sheet will be: New retained earnings = $174,730 + 99,850 New retained earnings = $274,580 The new total debt will be: New total debt = .7762($404,580) New total debt = $314,020 So, the new long-term debt will be the new total debt minus the new short-term debt, or: New long-term debt = $314,020 – 104,340 New long-term debt = $209,680

55 SOLUTIONS MANUAL Sales growth rate = 35% and debt/equity ratio = .7762: FLEURY INC. Pro Forma Balance Sheet Assets Liabilities and Owners’ Equity Current assets Current liabilities Cash $ 32,778 Accounts payable $ 88,020 Accounts receivable 50,045 Notes payable 16,320 Inventory 112,590 Total $ 104,340 Total $ 195,413 Long-term debt $ 209,680 Fixed assets Net plant and Owners’ equity equipment 535,275 Common stock and paid-in surplus $ 130,000 Retained earnings 274,580 Total $ 404,580 Total liabilities and owners’ Total assets $ 730,688 equity $ 718,600 So the excess debt raised is: Excess debt = $718,600 – 730,688 Excess debt = –$12,088 At a 35 percent growth rate, the firm will need funds in the amount of $12,088 in addition to the external debt already raised. So, the EFN will be: EFN = $54,680 + 12,088 EFN = $66,767 29. We need the ROE to calculate the sustainable growth rate. The ROE is: ROE = (PM)(TAT)(EM) ROE = (.061)(1 / 1.80)(1 + .35) ROE = .0458, or 4.58% Now we can use the sustainable growth rate equation to find the retention ratio as: Sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Sustainable growth rate = .12 = [.0458( b )] / [1 – .0458( b ) b = 2.34 This implies the payout ratio is: Payout ratio = 1 – b Payout ratio = 1 – 2.34 Payout ratio = –1.34

CHAPTER 4 - 56 This is a dividend payout ratio of negative 134 percent, which is impossible. The growth rate is not consistent with the other constraints. The lowest possible payout rate is 0, which corresponds to a retention ratio of 1, or total earnings retention. The maximum sustainable growth rate for this company is: Maximum sustainable growth rate = (ROE × b ) / [1 – (ROE × b )] Maximum sustainable growth rate = [.0458(1)] / [1 – .0458(1)] Maximum sustainable growth rate = .0479, or 4.79% 30. We know that EFN is: EFN = Increase in assets – Addition to retained earnings The increase in assets is the beginning assets times the growth rate, so: Increase in assets = A  g The addition to retained earnings next year is the current net income times the retention ratio, times one plus the growth rate, so: Addition to retained earnings = (NI  b )(1 + g ) And rearranging the profit margin to solve for net income, we get: NI = PM(S) 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 ] g EFN = – PM(S) b + [A – PM(S) b ] g 31. We start with the EFN equation we derived in Problem 30 and set it equal to zero: EFN = 0 = – PM(S) b + [A – PM(S) b ] g Substituting the rearranged profit margin equation into the internal growth rate equation, we have: Internal growth rate = [PM(S) b ] / [A – PM(S) b ] Since: ROA = NI / A ROA = PM(S) / A We can substitute this into the internal growth rate equation and divide both the numerator and denominator by A. This gives: Internal growth rate = {[PM(S) b ] / A} / {[A – PM(S) b ] / A} Internal growth rate = b (ROA) / [1 – b (ROA)]

57 SOLUTIONS MANUAL To derive the sustainable growth rate, we must realize that to maintain a constant D/E ratio with no external equity financing, EFN must equal the addition to retained earnings times the D/E ratio: EFN = (D/E)[PM(S)b(1 + g )] EFN = A( g ) – PM(S)b(1 + g ) Solving for g and then dividing numerator and denominator by A: Sustainable growth rate = PM(S) b (1 + D/E) / [A – PM(S) b (1 + D/E )] Sustainable growth rate = [ROA(1 + D/E ) b ] / [1 – ROA(1 + D/E )b] Sustainable growth rate = b (ROE) / [1 – b (ROE)] 32. 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 numerator is equal to beginning of period equity, that is: (TE E – NI × b ) = TE B Substituting this into the previous equation, we get: Sustainable rate = (NI × b ) / TE B Which is equivalent to: Sustainable rate = (NI / TE B ) × b Since ROE B = NI / TE B The sustainable growth rate equation is: Sustainable growth rate = ROE B × b For 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 )

CHAPTER 4 - 58 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 numerator is equal to beginning of period assets, that is: (TA E – NI × b ) = TA B Substituting this into the previous equation, we get: Internal growth rate = (NI × b ) / TA B Which is equivalent to: Internal growth rate = (NI / TA B ) × b Since ROA B = NI / TA B The internal growth rate equation is: Internal growth rate = ROA B × b

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount rate ( r ), and the life of the investment ( t ). 2. Compounding refers to the growth of a dollar amount through time via reinvestment of interest earned. It is also the process of determining the future value of an investment. Discounting is the process of determining the value today of an amount to be received in the future. 3. Future values grow (assuming a positive rate of return); present values shrink. 4. The future value rises (assuming it’s positive); the present value falls. 5. It would appear to be both deceptive and unethical to run such an ad without a disclaimer or explanation. 6. It’s a reflection of the time value of money. TMCC gets to use the $24,099. If TMCC uses it wisely, it will be worth more than $100,000 in 30 years. 7. This will probably make the security less desirable. TMCC will only repurchase the security prior to maturity if it is to its advantage, i.e., interest rates decline. Given the drop in interest rates needed to make this viable for TMCC, it is unlikely the company will repurchase the security. This is an example of a “call” feature. Such features are discussed at length in a later chapter. 8. The key considerations would be: (1) Is the rate of return implicit in the offer attractive relative to other, similar risk investments? and (2) How risky is the investment; i.e., how certain are we that we will actually get the $100,000? 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 $100,000. This rise is just a reflection of the time value of money. As time passes, the time until receipt of the $100,000 grows shorter, and the present value rises. In 2019, the price will probably be higher for the same reason. We cannot be sure, however, because interest rates could be much higher, or TMCC’s financial position could deteriorate. Either event would tend to depress the security’s price.

CHAPTER 5 - 60 Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. The time line for the cash flows is: 0 7 $9,000 FV The simple interest per year is: $9,000 × .08 = $720 So after 7 years you will have: $720 × 7 = $5,040 in interest. The total balance will be $9,000 + 5,040 = $14,040 With compound interest we use the future value formula: FV = PV(1 + r ) t FV = $9,000(1.08) 7 = $15,424.42 The difference is: $15,424.42 – 14,040 = $1,384.42 2. To find the FV of a lump sum, we use: FV = PV(1 + r ) t 0 11 $1,975 FV FV = $1,975(1.13) 11 = $7,575.83 0 7 $6,734 FV FV = $6,734(1.09) 7 = $12,310.02

61 SOLUTIONS MANUAL 0 14 $81,346 FV FV = $81,346(1.12) 14 = $397,547.04 0 8 $192,050 FV FV = $192,050(1.06) 8 = $306,098.52 3. To find the PV of a lump sum, we use: PV = FV / (1 + r ) t 0 13 PV $15,451 PV = $15,451 / (1.09) 13 = $5,039.79 0 4 PV $51,557 PV = $51,557 / (1.07) 4 = $39,332.59 0 29 PV $886,073 PV = $886,073 / (1.24) 29 = $1,730.78 0 40 PV $550,164 PV = $550,164 / (1.35) 40 = $3.37 4. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 0 4 –$181 $297 FV = $297 = $181(1 + r ) 4 ; r = ($297 / $181) 1/4 – 1 = .1318, or 13.18%

CHAPTER 5 - 62 0 18 –$335 $1,080 FV = $1,080 = $335(1 + r ) 18 ; r = ($1,080 / $335) 1/18 – 1 = .0672, or 6.72% 0 19 –$48,000 $185,382 FV = $185,382 = $48,000(1 + r ) 19 ; r = ($185,382 / $48,000) 1/19 – 1 = .0737, or 7.37% 0 25 –$40,353 $531,618 FV = $531,618 = $40,353(1 + r ) 25 ; r = ($531,618 / $40,353) 1/25 – 1 = .1086, or 10.86% 5. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for t , we get: t = ln(FV / PV) / ln(1 + r ) 0 ? –$560 $1,389 FV = $1,389 = $560(1.06) t ; t = ln($1,389 / $560) / ln 1.06 = 15.59 years 0 ? –$810 $1,821 FV = $1,821 = $810(1.09) t ; t = ln($1,821 / $810) / ln 1.09 = 9.40 years 0 ? –$18,400 $289,715 FV = $289,715 = $18,400(1.11) t ; t = ln($289,715 / $18,400) / ln 1.11 = 26.41 years 0 ? –$21,500 $430,258 FV = $430,258 = $21,500(1.13) t ; t = ln($430,258 / $21,500) / ln 1.13 = 24.52 years

63 SOLUTIONS MANUAL 6. The time line is: 0 18 –$67,000 $320,000 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 r = ($320,000 / $67,000) 1/18 – 1 r = .0908, or 9.08% 7. To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for t , we get: t = ln(FV / PV) / ln(1 + r ) The length of time to double your money is: 0 ? –$1 $2 FV = $2 = $1(1.073) t t = ln 2 / ln 1.073 = 9.84 years The length of time to quadruple your money is: 0 ? –$1 $4 FV = $4 = $1(1.073) t t = ln 4 / ln 1.073 = 19.68 years Notice that the length of time to quadruple your money is twice as long as the time needed to double your money (any difference in these answers is due to rounding). This is an important concept of time value of money.

CHAPTER 5 - 64 8. The time line is: 0 13 –$200,300 $306,900 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 r = ($306,900 / $200,300) 1/13 – 1 r = .0334, or 3.34% 9. The time line is: 0 ? –$45,000 $225,000 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for t , we get: t = ln(FV / PV) / ln(1 + r ) t = ln ($225,000 / $45,000) / ln 1.048 = 34.33 years 10. The time line is: 0 20 PV $475,000,000 To find the PV of a lump sum, we use: PV = FV / (1 + r) t PV = $475,000,000 / (1.061) 20 = $145,340,259.61

65 SOLUTIONS MANUAL 11. The time line is: 0 80 PV $2,000,000 To find the PV of a lump sum, we use: PV = FV / (1 + r) t PV = $2,000,000 / (1.075) 80 = $6,142.61 12. The time line is: 0 111 $50 FV To find the FV of a lump sum, we use: FV = PV(1 + r ) t FV = $50(1.043) 111 = $5,352.15 13. The time line is: 0 119 –$150 $1,620,000 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 r = ($1,620,000 / $150) 1/119 – 1 = .0812, or 8.12% To find the FV of the first prize in 2040, we use: 0 26 $1,620,000 FV FV = PV(1 + r ) t FV = $1,620,000(1.0812) 26 = $12,324,441.95

CHAPTER 5 - 66 14. The time line is: 0 4 –$12,377,500 $10,311,500 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 r = ($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. Intermediate 15. The time line from minting to the first sale is: 0 192 –$15 $430,000 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 r = ($430,000 / $15) 1/192 – 1 = .0549, or 5.49% The time line from the first sale to the second sale is: 0 35 –$430,000 $4,582,500 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 r = ($4,582,500 / $430,000) 1/35 – 1 = .0699, or 6.99%

67 SOLUTIONS MANUAL The time line from minting to the second sale is: 0 227 –$15 $4,582,500 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 r = ($4,582,500 / $15) 1/227 – 1 = .0572, or 5.72% 16. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for r , we get: r = (FV / PV) 1 / t – 1 a. The time line is: 0 20 –$50 $100 r = (FV / PV) 1 / t – 1 r = ($100 / $50) 1/20 – 1 r = .0353, or 3.53% b. The time line is: 0 10 –$50 FV FV = PV(1 + r ) t FV = $50(1 + .001) 10 FV = $50.50

CHAPTER 5 - 68 c. The time line is: 0 10 –$50.50 $100 r = (FV / PV) 1 / t – 1 r = ($100 / $50.50) 1/10 – 1 r = .0707, or 7.07% 17. The time line is: 0 9 PV $225,000 To find the PV of a lump sum, we use: PV = FV / (1 + r) t PV = $225,000 / (1.12) 9 = $81,137.26 18. To find the FV of a lump sum, we use: FV = PV(1 + r ) t 0 45 $5,000 FV FV = $5,000(1.10) 45 = $364,452.42 0 35 $5,000 FV FV = $5,000(1.10) 35 = $140,512.18 Better start early! 19. The time line is: 0 2 8 $20,000 FV We need to find the FV of a lump sum. However, the money will only be invested for six years, so the number of periods is six. FV = PV(1 + r ) t FV = $20,000(1.073) 6 = $30,523.08

69 SOLUTIONS MANUAL 20. The time line is: 0 2 ? –$15,000 $75,000 To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r ) t Solving for t , we get: t = ln(FV / PV) / ln(1 + r ) t = ln($75,000 / $15,000) / ln(1.09) = 18.68 So, the money must be invested for 18.68 years. However, you will not receive the money for another two years. From now, you’ll wait: 2 years + 18.68 years = 20.68 years

CHAPTER 5 - 70 Calculator Solutions 1. Enter 7 8% $9,000 N I/Y PV PMT FV Solve for $15,424.42 $15,424.42 – 14,040 = $1,384.42 2. Enter 11 13% $1,975 N I/Y PV PMT FV Solve for $7,575.83 Enter 7 9% $6,734 N I/Y PV PMT FV Solve for $12,310.02 Enter 14 12% $81,346 N I/Y PV PMT FV Solve for $397,547.04 Enter 8 6% $192,050 N I/Y PV PMT FV Solve for $306,098.52 3. Enter 13 9% $15,451 N I/Y PV PMT FV Solve for $5,039.79 Enter 4 7% $51,557 N I/Y PV PMT FV Solve for $39,332.59 Enter 29 24% $886,073 N I/Y PV PMT FV Solve for $1,730.78 Enter 40 35% $550,164 N I/Y PV PMT FV Solve for $3.37

71 SOLUTIONS MANUAL 4. Enter 4 $181  $297 N I/Y PV PMT FV Solve for 13.18% Enter 18 $335  $1,080 N I/Y PV PMT FV Solve for 6.72% Enter 19 $48,000  $185,382 N I/Y PV PMT FV Solve for 7.37% Enter 25 $40,353  $531,618 N I/Y PV PMT FV Solve for 10.86% 5. Enter 6% $560  $1,389 N I/Y PV PMT FV Solve for 15.59 Enter 9% $810  $1,821 N I/Y PV PMT FV Solve for 9.40 Enter 11% $18,400  $289,715 N I/Y PV PMT FV Solve for 26.41 Enter 13% $21,500  $430,258 N I/Y PV PMT FV Solve for 24.52 6. Enter 18 $67,000  $320,000 N I/Y PV PMT FV Solve for 9.08% 7. Enter 7.3% $1  $2 N I/Y PV PMT FV Solve for 9.84

CHAPTER 5 - 72 Enter 7.3% $1  $4 N I/Y PV PMT FV Solve for 19.68 8. Enter 13 $200,300  $306,900 N I/Y PV PMT FV Solve for 3.34% 9. Enter 4.80% $45,000  $225,000 N I/Y PV PMT FV Solve for 34.33 10. Enter 20 6.1% $475,000,000 N I/Y PV PMT FV Solve for $145,340,259.61 11. Enter 80 7.5% $2,000,000 N I/Y PV PMT FV Solve for $6,142.61 12. Enter 111 4.30% $50 N I/Y PV PMT FV Solve for $5,352.15 13. Enter 119  $150 $1,620,000 N I/Y PV PMT FV Solve for 8.12% Enter 26 8.12% $1,620,000 N I/Y PV PMT FV Solve for $12,324,441.95 14. Enter 4  $12,377,500 $10,311,500 N I/Y PV PMT FV Solve for –4.46% 15. Enter 192  $15 $430,000 N I/Y PV PMT FV Solve for 5.49% Enter 35  $430,000 $4,582,500

73 SOLUTIONS MANUAL N I/Y PV PMT FV Solve for 6.99% Enter 227  $15 $4,582,500 N I/Y PV PMT FV Solve for 5.72% 16. a. Enter 20  $50 $100 N I/Y PV PMT FV Solve for 3.53% b. Enter 10 .10%  $50 N I/Y PV PMT FV Solve for $50.50 c. Enter 10  $50.50 $100 N I/Y PV PMT FV Solve for 7.07% 17. Enter 9 12% $225,000 N I/Y PV PMT FV Solve for $81,137.26 18. Enter 45 10% $5,000 N I/Y PV PMT FV Solve for $364,452.42 Enter 35 10% $5,000 N I/Y PV PMT FV Solve for $140,512.18 19. Enter 6 7.30% $20,000 N I/Y PV PMT FV Solve for $30,523.08

CHAPTER 5 - 74 20. Enter 9%  $15,000 $75,000 N I/Y PV PMT FV Solve for 18.68 From now, you’ll wait 2 + 18.68 = 20.68 years

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow ( C ), the discount rate ( r ), and the number of payments, or the life of the annuity, t . 2. Assuming positive cash flows, both the present and the future values will rise. 3. Assuming positive cash flows, the present value will fall and the future value will rise. 4. It’s deceptive, but very common. The basic concept of time value of money is that a dollar today is not worth the same as a dollar tomorrow. The deception is particularly irritating given that such lotteries are usually government sponsored! 5. If the total money is fixed, you want as much as possible as soon as possible. The team (or, more accurately, the team owner) wants just the opposite. 6. The better deal is the one with equal installments. 7. Yes, they should. APRs generally don’t provide the relevant rate. The only advantage is that they are easier to compute, but with modern computing equipment, that advantage is not very important. 8. A freshman does. The reason is that the freshman gets to use the money for much longer before interest starts to accrue. The subsidy is the present value (on the day the loan is made) of the interest that would have accrued up until the time it actually begins to accrue. 9. The problem is that the subsidy makes it easier to repay the loan, not obtain it. However, ability to repay the loan depends on future employment, not current need. For example, consider a student who is currently needy, but is preparing for a career in a high-paying area (such as corporate finance!). Should this student receive the subsidy? How about a student who is currently not needy, but is preparing for a relatively low-paying job (such as becoming a college professor)?

CHAPTER 6 - 76 10. In general, viatical settlements are ethical. In the case of a viatical settlement, it is simply an exchange of cash today for payment in the future, although the payment depends on the death of the seller. The purchaser of the life insurance policy is bearing the risk that the insured individual will live longer than expected. Although viatical settlements are ethical, they may not be the best choice for an individual. In a Businessweek article (October 31, 2005), options were examined for a 72- year-old male with a life expectancy of eight years and a $1 million dollar life insurance policy with an annual premium of $37,000. The four options were: (1) Cash the policy today for $100,000. (2) Sell the policy in a viatical settlement for $275,000. (3) Reduce the death benefit to $375,000, which would keep the policy in force for 12 years without premium payments. (4) Stop paying premiums and don’t reduce the death benefit. This will run the cash value of the policy to zero in five years, but the viatical settlement would be worth $475,000 at that time. If he died within five years, the beneficiaries would receive $1 million. Ultimately, the decision rests with the individual on what they perceive as best for themselves. The values that will affect the value of the viatical settlement are the discount rate, the face value of the policy, and the health of the individual selling the policy. 11. This is a trick question. The future value of a perpetuity is undefined since the payments are perpetual. Finding the future value at any particular point automatically ignores all cash flows beyond that point. 12. The ethical issues surrounding payday loans are more complex than they might first appear. On the one hand, the interest rates are astronomical, and the people paying those rates are typically among the worst off financially to begin with. On the other hand, and unfortunately, payday lenders are essentially the lenders of last resort for some. And the fact is that paying $15 for a two-week loan of $100 might be a bargain compared to the alternatives such as having utilities disconnected or paying bank overdraft fees. Restricting or banning payday lending also has the effect of encouraging loan sharking, where rates are even higher and collection practices much less consumer friendly (no payday loan company has ever demanded a pound of flesh nearest the heart as did Shylock in The Merchant of Venice ). As a final note, such loans are by definition extremely risky, with a higher likelihood of default. As we will discuss later, higher-risk investments necessarily demand a higher return. Solutions to Questions and Problems NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. Basic 1. The time line is: 0 1 2 3 4 PV $680 $810 $940 $1,150 To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we use: PV = FV / (1 + r ) t

77 SOLUTIONS MANUAL PV@10% = $680 / 1.10 + $810 / 1.10 2 + $940 / 1.10 3 + $1,150 / 1.10 4 = $2,779.30 PV@18% = $680 / 1.18 + $810 / 1.18 2 + $940 / 1.18 3 + $1,150 / 1.18 4 = $2,323.27 PV@24% = $680 / 1.24 + $810 / 1.24 2 + $940 / 1.24 3 + $1,150 / 1.24 4 = $2,054.62 2. The times lines are: 0 1 2 3 4 5 6 7 8 PV $4,700 $4,700 $4,700 $4,700 $4,700 $4,700 $4,700 $4,700 0 1 2 3 4 5 PV $6,700 $6,700 $6,700 $6,700 $6,700 To find the PVA, we use the equation: PVA = C ({1 – [1/(1 + r ) t ] } / r ) At a 5 percent interest rate: X@5%: PVA = $4,700{[1 – (1/1.05) 8 ] / .05 } = $30,377.10 Y@5%: PVA = $6,700{[1 – (1/1.05) 5 ] / .05 } = $29,007.49 And at a 15 percent interest rate: X@15%: PVA = $4,700{[1 – (1/1.15) 8 ] / .15 } = $21,090.41 Y@15%: PVA = $6,700{[1 – (1/1.15) 5 ] / .15 } = $22,459.44 Notice that the PV of cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a 15 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more valuable since it has larger cash flows. At the higher interest rate, these bigger cash flows early are more important since the cost of waiting (the interest rate) is so much greater. 3. The time line is: 0 1 2 3 4 $1,225 $1,345 $1,460 $1,590 To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a lump sum, we use: FV = PV(1 + r ) t FV@8% = $1,225(1.08) 3 + $1,345(1.08) 2 + $1,460(1.08) + $1,590 = $6,278.76

CHAPTER 6 - 78 FV@11% = $1,225(1.11) 3 + $1,345(1.11) 2 + $1,460(1.11) + $1,590 = $6,543.12 FV@24% = $1,225(1.24) 3 + $1,345(1.24) 2 + $1,460(1.24) + $1,590 = $7,804.09 Notice, since we are finding the value at Year 4, the cash flow at Year 4 is simply added to the FV of the other cash flows. In other words, we do not need to compound this cash flow. 4. To find the PVA, we use the equation: PVA = C ({1 – [1/(1 + r ) t ] } / r ) 0 1 … 15 PV $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 PVA@15 yrs: PVA = $5,500{[1 – (1/1.06) 15 ] / .06} = $53,417.37 0 1 … 40 PV $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 PVA@40 yrs: PVA = $5,500{[1 – (1/1.06) 40 ] / .06} = $82,754.63 0 1 … 75 PV $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 PVA@75 yrs: PVA = $5,500{[1 – (1/1.06) 75 ] / .06} = $90,507.16 To find the PV of a perpetuity, we use the equation: PV = C / r 0 1 … ∞ PV $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 PV = $5,500 / .06 = $91,666.67 Notice that as the length of the annuity payments increases, the present value of the annuity approaches the present value of the perpetuity. The present value of the 75-year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years is only $1,159.50. 5. The time line is: 0 1 … 15 $38,000 C C C C C C C C C

79 SOLUTIONS MANUAL Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the PVA equation: PVA = C ({1 – [1 / (1 + r ) t ] } / r ) PVA = $38,000 = $ C {[1 – (1 / 1.058 15 ) ] / .058} We can now solve this equation for the annuity payment. Doing so, we get: C = $38,000 / 9.840437 = $3,861.62 6. The time line is: 0 1 2 3 4 5 6 7 $57,000 C C C C C C C To find the PVA, we use the equation: PVA = C ({1 – [1 / (1 + r ) t ] } / r ) PVA = $57,000{[1 – (1 / 1.078 7 ) ] / .078} = $298,803.87 7. Here we need to find the FVA. The equation to find the FVA is: FVA = C {[(1 + r ) t – 1] / r } 0 1 … 20 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 FVA for 20 years = $4,000[(1.097 20 – 1) / .097] = $221,439.14 0 1 … 40 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 FVA for 40 years = $4,000[(1.097 40 – 1) / .097] = $1,631,984.09 Notice that because of exponential growth, doubling the number of periods does not merely double the FVA. 8. The time line is: 0 1 … 12 $50,000 C C C C C C C C C Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the FVA equation: FVA = C {[(1 + r ) t – 1] / r }

CHAPTER 6 - 80 $50,000 = $ C [(1.062 12 – 1) / .062] We can now solve this equation for the annuity payment. Doing so, we get: C = $50,000 / 17.06825 = $2,929.42 9. The time line is: 0 1 2 3 4 5 $50,000 C C C C C Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the PVA equation: PVA = C ({1 – [1/(1 + r ) t ] } / r ) $50,000 = C {[1 – (1/1.075 5 ) ] / .075} We can now solve this equation for the annuity payment. Doing so, we get: C = $50,000 / 4.04588 = $12,358.24 10. The time line is: 0 1 … ∞ PV $40,00 0 $40,000 $40,000 $40,000 $40,000 $40,000 $40,00 0 $40,000 $40,000 This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation: PV = C / r PV = $40,000 / .051 = $784,313.73 11. The time line is: 0 1 … ∞ –$650,000 $40,000 $40,000 $40,00 0 $40,000 $40,000 $40,000 $40,000 $40,000 $40,000 Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using the PV of a perpetuity equation: PV = C / r $650,000 = $40,000 / r We can now solve for the interest rate as follows: r = $40,000 / $650,000 = .0615, or 6.15%

81 SOLUTIONS MANUAL 12. For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m )] m – 1 EAR = [1 + (.07 / 4)] 4 – 1 = .0719, or 7.19% EAR = [1 + (.17 / 12)] 12 – 1 = .1839, or 18.39% EAR = [1 + (.13 / 365)] 365 – 1 = .1388, or 13.88% To find the EAR with continuous compounding, we use the equation: EAR = e q – 1 EAR = e .10 – 1 = .1052, or 10.52% 13. Here we are given the EAR and need to find the APR. Using the equation for discrete compounding: EAR = [1 + (APR / m )] m – 1 We can now solve for the APR. Doing so, we get: APR = m [(1 + EAR) 1/ m – 1] EAR = .1240 = [1 + (APR / 2)] 2 – 1 APR = 2[(1.1240) 1/2 – 1] = .1204, or 12.04% EAR = .1170 = [1 + (APR / 12)] 12 – 1 APR = 12[(1.1170) 1/12 – 1] = .1112, or 11.12% EAR = .0950 = [1 + (APR / 52)] 52 – 1 APR = 52[(1.0950) 1/52 – 1] = .0950, or 9.50% Solving the continuous compounding EAR equation: EAR = e q – 1 We get: APR = ln(1 + EAR) APR = ln(1 + .0790) APR = .0760, or 7.60% 14. For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m )] m – 1 So, for each bank, the EAR is: First National: EAR = [1 + (.1240 / 12)] 12 – 1 = .1313, or 13.13% First United: EAR = [1 + (.1270 / 2)] 2 – 1 = .1310, or 13.10% Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding periods within a year will also affect the EAR.

CHAPTER 6 - 82 15. The reported rate is the APR, so we need to convert the EAR to an APR as follows: EAR = [1 + (APR / m )] m – 1 APR = m [(1 + EAR) 1/ m – 1] APR = 365[(1.165) 1/365 – 1] = .1528, or 15.28% This is deceptive because the borrower is actually paying annualized interest of 16.5 percent per year, not the 15.28 percent reported on the loan contract. 16. The time line is: 0 1 … 34 $2,400 FV For this problem, we need to find the FV of a lump sum using the equation: FV = PV(1 + r ) t It is important to note that compounding occurs semiannually. To account for this, we will divide the interest rate by two (the number of compounding periods in a year), and multiply the number of periods by two. Doing so, we get: FV = $2,400[1 + (.079 / 2)] 17(2) = $8,958.68 17. For this problem, we need to find the FV of a lump sum using the equation: FV = PV(1 + r ) t It is important to note that compounding occurs daily. To account for this, we will divide the interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by 365. Doing so, we get: 0 1 … 5(365) $7,000 FV FV in 5 years = $7,000[1 + (.067 / 365)] 5(365) = $9,785.28 0 1 … 10(365) $7,000 FV FV in 10 years = $7,000[1 + (.067 / 365)] 10(365) = $13,678.82 0 1 … 20(365) $7,000 FV FV in 20 years = $7,000[1 + (.067 / 365)] 20(365) = $26,730.02

83 SOLUTIONS MANUAL 18. The time line is: 0 1 … 10(365) PV $65,000 For this problem, we need to find the PV of a lump sum using the equation: PV = FV / (1 + r ) t It is important to note that compounding occurs daily. To account for this, we will divide the interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by 365. Doing so, we get: PV = $65,000 / [(1 + .07 / 365) 10(365) ] = $32,280.21 19. The APR is simply the interest rate per period times the number of periods in a year. In this case, the interest rate is 32 percent per month, and there are 12 months in a year, so we get: APR = 12(32%) = 384% To find the EAR, we use the EAR formula: EAR = [1 + (APR / m )] m – 1 EAR = (1 + .32) 12 – 1 = 2,698.25% Notice that we didn’t need to divide the APR by the number of compounding periods per year. We do this division to get the interest rate per period, but in this problem we are already given the interest rate per period. 20. The time line is: 0 1 … 60 $79,500 C C C C C C C C C We first need to find the annuity payment. We have the PVA, the length of the annuity, and the interest rate. Using the PVA equation: PVA = C ({1 – [1 / (1 + r ) t ] } / r ) $79,500 = $ C [1 – {1 / [1 + (.058 / 12)] 60 } / (.058 / 12)] Solving for the payment, we get: C = $79,500 / 51.97521 = $1,529.58 To find the EAR, we use the EAR equation: EAR = [1 + (APR / m )] m – 1

CHAPTER 6 - 84 EAR = [1 + (.058 / 12)] 12 – 1 = .0596, or 5.96% 21. The time line is: 0 1 … ? –$18,000 $500 $500 $500 $500 $500 $500 $500 $500 $500 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 ) $18,000 = $500{[1 – (1 / 1.015) t ] / .015} Now we solve for t : 1 / 1.015 t = 1 – {[($18,000) / ($500)](.015)} 1 / 1.015 t = .46 1.015 t = 1 / .46 = 2.174 t = ln 2.174 / ln 1.015 = 52.16 months 22. The time line is: 0 1 –$3 $4 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 week The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks in a year, so: APR = (52)33.33% = 1,733.33% And using the equation to find the EAR: EAR = [1 + (APR / m )] m – 1 EAR = [1 + .3333] 52 – 1 = 313,916,515.69% 23. The time line is: 0 1 … ∞ –$215,000 $1,400 $1,400 $1,400 $1,400 $1,400 $1,400 $1,400 $1,400 $1,400

85 SOLUTIONS MANUAL Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using the PV of a perpetuity equation: PV = C / r $215,000 = $1,400 / r We can now solve for the interest rate as follows: r = $1,400 / $215,000 = .0065, or .65% per month The interest rate is .65% per month. To find the APR, we multiply this rate by the number of months in a year, so: APR = (12).65% = 7.81% And using the equation to find an EAR: EAR = [1 + (APR / m )] m – 1 EAR = [1 + .0065] 12 – 1 = 8.10% 24. The time line is: 0 1 … 360 $450 $450 $450 $450 $450 $450 $450 $450 $450 This problem requires us to find the FVA. The equation to find the FVA is: FVA = C {[(1 + r ) t – 1] / r } FVA = $450[{[1 + (.10 / 12) ] 360 – 1} / (.10 / 12)] FVA = $1,017,219.57 25. The time line is: 0 1 … 30 $5,400 $5,400 $5,400 $5,400 $5,400 $5,400 $5,400 $5,400 $5,400 In the previous problem, the cash flows are monthly and the compounding period is monthly. This compounding periods are still monthly, but since the cash flows are annual, we need to use the EAR to calculate the future value of annual cash flows. It is important to remember that you have to make sure the compounding periods of the interest rate are the same as the timing of the cash flows. In this case, we have annual cash flows, so we need the EAR since it is the true annual interest rate you will earn. So, finding the EAR: EAR = [1 + (APR / m )] m – 1 EAR = [1 + (.10 / 12)] 12 – 1 = .1047, or 10.47%

CHAPTER 6 - 86 Using the FVA equation, we get: FVA = C {[(1 + r ) t – 1] / r } FVA = $5,400[(1.1047 30 – 1) / .1047] FVA = $971,435.17 26. The time line is: 0 1 … 16 PV $2,200 $2,200 $2,200 $2,200 $2,200 $2,200 $2,200 $2,200 $2,200 The cash flows are simply an annuity with four payments per year for four years, or 16 payments. We can use the PVA equation: PVA = C ({1 – [1 / (1 + r ) t ] } / r ) PVA = $2,200{[1 – (1 / 1.0043) 16 ] / .0043} PVA = $33,945.97 27. The time line is: 0 1 2 3 4 PV $790 $860 $0 $1,340 The cash flows are annual and the compounding period is quarterly, so we need to calculate the EAR to make the interest rate comparable with the timing of the cash flows. Using the equation for the EAR, we get: EAR = [1 + (APR / m )] m – 1 EAR = [1 + (.09 / 4)] 4 – 1 = .0931, or 9.31% And now we use the EAR to find the PV of each cash flow as a lump sum and add them together: PV = $790 / 1.0931 + $860 / 1.0931 2 + $1,340 / 1.0931 4 PV = $2,381.12 28. The time line is: 0 1 2 3 4 PV $2,480 $0 $3,920 $2,170 Here the cash flows are annual and the given interest rate is annual, so we can use the interest rate given. We can find the PV of each cash flow and add them together. PV = $2,480 / 1.0618 + $3,920 / 1.0618 3 + $2,170 / 1.0618 4 = $7,317.47

87 SOLUTIONS MANUAL Intermediate 29. 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: .075(10) = .75 First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor of $1 minus the initial investment of $1, or: (1 + r ) 10 – 1 Setting the two equal, we get: (.075)(10) = (1 + r ) 10 – 1 r = 1.75 1/10 – 1 = .0576, or 5.76% 30. Here we need to convert an EAR into interest rates for different compounding periods. Using the equation for the EAR, we get: EAR = [1 + (APR / m )] m – 1 EAR = .125 = (1 + r ) 2 – 1; r = (1.125) 1/2 – 1 = .0607, or 6.07% per six months EAR = .125 = (1 + r ) 4 – 1; r = (1.125) 1/4 – 1 = .0299, or 2.99% per quarter EAR = .125 = (1 + r ) 12 – 1; r = (1.125) 1/12 – 1 = .0099, or .99% per month Notice that the effective six-month rate is not twice the effective quarterly rate because of the effect of compounding. 31. Here we need to find the FV of a lump sum, with a changing interest rate. We must do this problem in two parts. After the first six months, the balance will be: FV = $7,000 [1 + (.005 / 12)] 6 = $7,017.52 This is the balance in six months. The FV in another six months will be: FV = $7,017.52[1 + (.185 / 12)] 6 = $7,692.18 The problem asks for the interest accrued, so, to find the interest, we subtract the beginning balance from the FV. The interest accrued is: Interest = $7,692.18 – 7,000 = $692.18

CHAPTER 6 - 88 32. Although the stock and bond accounts have different interest rates, we can draw one time line, but we need to remember to apply different interest rates. The time line is: 0 1 ... 360 … 660 Stock $850 $850 $850 $850 $850 C C C Bond $350 $350 $350 $350 $350 We need to find the annuity payment in retirement. Our retirement savings ends and the retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the retirement savings. So, we find the FV of the stock account and the FV of the bond account and add the two FVs. Stock account: FVA = $850[{[1 + (.10 / 12) ] 360 – 1} / (.10 / 12)] = $1,921,414.74 Bond account: FVA = $350[{[1 + (.06 / 12) ] 360 – 1} / (.06 / 12)] = $351,580.26 So, the total amount saved at retirement is: $1,921,414.74 + 351,580.26 = $2,272,995.00 Solving for the withdrawal amount in retirement using the PVA equation gives us: PVA = $2,272,995.00 = $ C [1 – {1 / [1 + (.05 / 12)] 300 } / (.05 / 12)] C = $2,272,995.00 / 171.0600 = $13,287.70 withdrawal per month 33. We need to find the FV of a lump sum in one year and two years. It is important that we use the number of months in compounding since interest is compounded monthly in this case. So: 0 … 12 $1 FV in one year = $1(1.0074) 12 = $1.09 0 … 24 $1 FV in two years = $1(1.0074) 24 = $1.19 There is also another common alternative solution. We could find the EAR, and use the number of years as our compounding periods. So we will find the EAR first: EAR = (1 + .0074) 12 – 1 = .0925, or 9.25% Using the EAR and the number of years to find the FV, we get: 0 1 $1 FV

89 SOLUTIONS MANUAL FV in one year = $1(1.0925) 1 = $1.09 0 1 2 $1 FV FV in two years = $1(1.0925) 2 = $1.19 Either method is correct and acceptable. We have simply made sure that the interest compounding period is the same as the number of periods we use to calculate the FV. 34. Here we are finding the annuity payment necessary to achieve the same FV. The interest rate given is an APR of 10.5 percent, with monthly deposits. We must make sure to use the number of months in the equation. So, using the FVA equation: Starting today: 0 1 … 480 $1,000,000 C C C C C C C C C FVA = C [{[1 + (.105/12) ] 480 – 1} / (.105/12)] C = $1,000,000 / 7,369.04 = $135.70 Starting in 10 years: 0 1 … 120 121 … 480 $1,000,000 C C C C C C C FVA = C [{[1 + (.105/12) ] 360 – 1} / (.105/12)] C = $1,000,000 / 2,516.40 = $397.39 Starting in 20 years: 0 1 … 240 241 … 480 $1,000,000 C C C C C FVA = C [{[1 + (.105/12) ] 240 – 1} / (.105/12)] C = $1,000,000 / 810.50 = $1,233.80 Notice that a deposit for half the length of time, i.e., 20 years versus 40 years, does not mean that the annuity payment is doubled. In this example, by reducing the savings period by one-half, the deposit necessary to achieve the same ending value is about 10 times as large.