CH-4-5 Measuring Corporate Performance – Fundamentals of Corporate Finance : Book Guide

This Document Contains Chapters 4 to 5 Brealey 5CE Solutions to Chapter 4 1. NOTE: there is more than one definition for some of these ratios. That is why providing the ratio definition is important. One big issue is when calculating ratios that combine information from the income statement with the statement of financial position. Either the value of the asset at the start of the year, the end of the year or the average of the beginning and end of year values are compared to the income statement number. This occurs because the income statement shows values measured over the year and a statement of financial position is relevant for a either the start or the end of the year. Most of the ratios used here are the definitions from Table 4.6. a. LT debt ratio = LT debt/(LT debt +equity) = 7,018/(7,018 + 9,724) = .42 b. Total debt ratio = Current liab. + LT debt + Other LT liab./Total assets = (4,794 + 7,018 + 6,178)/27,714 = .65 c. Times interest earned = EBIT/Interest expenses = 2,566/685 = 3.75 d. Cash coverage ratio = (EBIT+ Depreciation)/Interest expenses = (2,566+ 2,518)/685 = 7.42 e. Current ratio =Current assets/Current liabilities = 3,525/4,794 = .74 f. Quick ratio = (Cash and marketable securities + Receivables)/Current liab. = (89 + 2382)/4794 = .52 g. Operating profit margin = NOPAT/Sales NOPAT =net operating profit after tax=Net income + after-tax interest expense Need to estimate tax rate: Estimated tax rate = taxes/income before tax = 570/1,881 = .303 After-tax interest expense = (1 – tax rate) x interest expense = (1-.303) x 685 = 477.4 NOPAT = Net income + after-tax interest expense = 1,311 + 477.4 = 1,788.4 Operating profit margin = NOPAT/Sales = 1,788.4/13,193 = .135, 13.5% h. Inventory turnover = cost of sales/average inventories = 4060 (238+187) /2 = 4060 212.5 = 19.11 times i. Days in inventory = average inventories/ daily cost of sales = (238+187)/2 4060/365 = 212.5 4060/365 = 19.10 days 4-1 This Document Contains Chapters 4 to 5 j. Average collection period =average receivables/daily sales = (2,490+2,382)/2 13,193/365 = 67.39 days k. Return on Equity ROE= Net income/equity at start of year = 1,311 9,121 = .144 = 14.4% ROE = Net income/average equity = 1,311 (9,121+ 9,724) / 2 = .139 = 13.9% l. Return on assets ROA = NOPAT/(assets at start of year) =1,788.4/27,503 = .0650 = 6.5% ROA = NOPAT/(Average assets) = 1,788.4 (27,503 + 27,714) / 2 = .0648 = 6.48% m. Payout ratio = Dividends/net income = 856/1312 = .65 2. Market capitalization = stock price x shares outstanding = $84 x 205 million = $17,220 million Market value added = Market capitalization – book value of equity = $17,220 million - $9,724 million = $7,496 million Market-to-book ratio = Market capitalization/ book value of equity = $17,220 million / $9,724 million = 1.77 Earnings per share = $84 - $47.43 = $36.57 Note: Book value stock price = $9,724 million / 205 million = $47.43 3. Estimated tax rate = taxes/income before tax = 570/1,881 = .303 = 30.3% Economic Value Add EVA = NOPAT- Cost of capital x average total capitalization NOPAT = net operating profit after tax=Net income + after-tax interest expense After-tax interest expense = (1 – tax rate) x interest expense = (1-.303) x 685 = 477.4 NOPAT = Net income + after-tax interest expense = 1,311 + 477.4 = 1,788.4 4-2 Total Capitalization at start of year = long-term debt and leases + equity = 6,833+9,121 = 15,954 Total Capitalization at end of year = long-term debt and leases + equity = 7,018+9,724 = 16,742 Average total capitalization = (15,954 + 16,742)/2 = 16,348 EVA = NOPAT- Cost of capital x average total capitalization = 1,788.4 - 8.3% x 16,348 = 431.516 4. a) Since EVA is NOPAT – cost of capital x total capitalization, an increase in cost of capital from 6.7% to 8% would decrease EVA. Equation 4.1 shows that AstraZeneca’s EVA at the 6.7% cost of capital was $6,121 million. Using an 8% cost of capital and the values for AstraZeneca’s EVA in equation 4.1 the new EVA would be lower : EVA = net income + after-tax interest expense – cost of capital x capitalization = $8,081 + $383 - .08 x $34,970 = $8,464 - $2,797.6 = $5,666.4 million b) Accounting profits (net income) would not be impacted as it is calculated by sales (revenue) minus expenses, and does not take into account cost of capital. c) EVA seems to be the better measure of a firm’s performance as it not only accounts for expenses like net income, but also cost of capital. 5. a) Market Value Added, MVA = market value of shares – book value of equity Market value of shares = stock price x number of shares. Market Value Added will decrease as the market stock price declines. Using the information for AstraZeneca on page 99 the MVA of AstraZeneca when the stock price was US$46.10 and there were 1,438 million shares was $46.19 x1,438 = $66,421.22 million. If stock price fell by 5 percent it would be (1-.05)x US$46.19 = $43.8805 and new would be MVA = $43.8805 x 1,438 =$63,100.16 million 4-3 b) If the fall is due to general stock market uncertainty (as it seems to be the case, both fell by the same percentage), and is out of the control of AstraZeneca managers, then this cannot be contributed to their performance. c) In this case, the stock price decline appears to be from risks and events within the managers’ control, and not of the market. Thus, this can be attributed to management performance. 6. a) AstraZeneca’s sustainable growth rate in Table 4.6 was calculated as: sustainable growth rate = plowback ratio x ROE = .57 x .388 = .221, 22.1% Note that plowback ratio = 1 – payout ratio. So, if the payout ratio increases, the plowback ratio decreases and thus the sustainable growth rate is lower. If AstraZeneca’s payout ratio is 70 percent its sustainable growth rate would be: sustainable growth rate = (1-payout ratio) x ROE= (1-.7) x .388 = .1164, 11.64%, which is a lower sustainable growth rate. b) If the ROE decreases to 30%, the sustainable growth rate would decrease. Assuming that the plowback ratio is still 57%, the new sustainable growth rate with the lower ROE would be: sustainable growth rate = .57 x .30 = .171, 17.1%, which is a lower sustainable growth rate. 7. ROA = NOPAT/(Average assets) = 1,788.4 (27,503 + 27,714) / 2 = .0648 = 6.48% Asset turnover = Revenue Average assets = (27,50313,193 + 27,714) / 2 = .4779 Operating profit margin = NOPATSales = 1,788.4 13,193 = .1356 Asset turnover × operating profit margin = .4779 × . 1355 = .0648 = ROA 4-4 8. Using this definition of ROE: ROE = Net income/average equity = 1,311 (9,121+ 9,724) / 2 = .139 = 13.9% Average equity = (9,121 + 9,724)/2 = 9,422.5 Using the DuPont formula for ROE (incorporating average equity): ROE = leverage ratio x asset turnover ratio x operating profit margin x debt burden = Assets Average Equity × Sales Assets × NOPAT Sales x Net Income NOPAT NOPAT = Net income + after-tax interest expense = 1,311 + 477.4 = 1,788.4 Leverage ratio x asset turnover ratio x operating profit margin x debt burden = Assets Average Equity × Sales Assets × NOPAT Sales x Net Income NOPAT = 27,714 9,422.5 × 13,193 27,714 × 1,788.4 13,193 × 1,311 1,788.4 = .139, or 13.9% We confirm that the DuPont formula gives ROE. 9. a. High asset turnover means high sales relative to assets. The consulting firm will have relatively few assets. The major “asset” is the know-how of its employees, which is not measured on its balance sheet. It will have the higher asset turnover ratio. AstraZeneca, by contrast, has significant balance sheet assets and will have a lower asset turnover. It takes a lot of PP&E and intangible assets to manufacture its products. b. Catalog shopping will generate far more sales relative to assets than a traditional retailer like Mark’s Work Wearhouse. The online retailer does not have to sell goods from stores, reducing its fixed assets relative to bricks-and- mortar stores. Catalog stores probably can also maintain relatively lower inventories. Its asset turnover ratio will be higher. c. Standard supermarkets has a far higher ratio of sales to assets. The supermarket itself is a simple building and the store sells a high volume of goods with relatively low mark-ups (profit margins). Its asset turnover will be the highest of all the companies in this question. An electric utility produces electricity and requires large fixed assets, including power generating facilities and transmission lines. 10. Return on Capital, ROC, can be calculate using total capitalization or average capitalization. Also EVA can be calculated with total capital or average total capital. We will use total capitalization in each formula. 4-5 ROC = NOPAT/ total capitalization So: NOPAT = ROC x total capitalization EVA= NOPAT- Cost of capital x total capitalization Replace NOPAT in EVA with this equation NOPAT= ROC x total capitalization. So: EVA = ROC x total capitalization - Cost of capital x total capitalization = total capitalization x (ROC - Cost of capital) Thus, EVA is positive if ROC exceeds the cost of capital. EVA is about value created for investors, and cost of capital is the return investors give up by investing in a company rather than on their own. If the ROC is higher than the cost of capital, it means that they are better off investing in a company and thus the EVA would be positive. 11. a. Debt-equity ratio = Long-term debt Equity , a leverage measure which signifies how much debt for each dollar of equity a firm has. b. Return on equity = Net income Equity , a profitability measure which shows how much income is earned per dollar of equity c. Operating profit margin = NOPAT Sales , a profitability measure which indicates how much profit is generated per dollar of revenue d. Inventory turnover = cost of sales average inventories , an efficiency measure of inventories that uses of the cost of sales rather than sales. e. Current ratio = Current assets Current liabilities , a liquidity measure which compares a firm’s liquid assets against its liquid liabilities 4-6 f. Average collection period = Average Trade Receivables Daily Revenue , an efficiency measure which signifies on average, how fast customers pay g. Quick ratio = Cash+Marketable Securities+Trade Receivables Current Liabilities , a liquidity measure which compares only a firm’s most liquid assets (excluding inventory unlike the current ratio) against its current liabilities 12. If AstraZeneca borrows $300 million from its unused line of credit and invests the funds in marketable securities, both current assets and current liabilities will increase. Use the data in Table 4.2 a. Liquidity ratios Before the new borrowing AstraZeneca’s liquidity ratios were: Net working capital Total assets = 25,131-16,787 0.1487 56,127 = Current ratio = current assets current liabilities = 25,131 16,787 = 1.497 Quick ratio = cash and cash equivalent + current other investments +trade receivables current liabilities = 11,068 1,491 5,864 1.0975 16,787 + + = Cash ratio = cash and cash equivalent current liabilities = 11,068 0.6593 16,787 = Then it if used the line of credit to borrow $300 million and invest it in marketable securities the net working capital to total asset ratio would be: Net working capital Total assets = Although net working capital does not change, total assets are increased, and the ratio of net working capital to total assets decreases from .1487 to .1479, indicating that the company is less liquid. 4-7 Current ratio= current assets current liabilities = = 1.488 Current ratio decreases from 1.497 Quick ratio = cash and cash equivalent + current other investments +trade receivables current liabilities = 11,068 300 1,491 5,864 16,787 300 + + + + = 1.0957 Quick ratio decreases from 1.0975 Cash ratio = cash and cash equivalent current liabilities = 11,068 300 0.6653 16,787 300 + = + Cash ratio increases from 0.6593 The transaction would result in decrease in two of the four liquidity ratios but an increase in 2 of the liquidity ratios. So that the company might have appeared to be less liquid or more liquid. However, is the company really more liquid? It could sell its marketable securities and clear its borrowings from the line of credit. However, a financial analyst would be very unlikely to conclude that the company was more liquid if it knew the details of the transaction. b. Leverage ratios The debt-equity and long-term debt ratios would be unaffected since they are calculated using only long-term debt. The total debt ratio will increase, however: Total liabilities Total assets = = .585 The total liabilities/total asset ratio before the use of the line of credit 32,717/ 56,127 = .583 The slight increase in the total debt ratio indicates that the company would appear to be very slightly more leveraged. However, a financial analyst would conclude that the company is actually no more leveraged than prior to the transaction. 4-8 13. a. Current ratio will be unaffected. Inventories are replaced with either cash or accounts receivable, but total current assets are unchanged. b. Current ratio will be unaffected. Due trade payables are replaced with the bank loan, but total current liabilities are unchanged. c. Current ratio will be unaffected. Receivables are replaced with cash, but total current assets are unchanged. d. Current ratio will be unaffected. Inventories replace cash, but total current assets are unchanged. 14. The current ratio will be unaffected. Inventories replace cash, and total current assets are unchanged. The quick ratio falls, however, since inventories are not included in the most liquid assets. 15. Average collection period equals average receivables divided by daily sales: Average collection period = average trade receivables daily sales = 6333 9800/365 = 236 days 16. Days in inventory = inventories daily cost of sales = 400 73,000/365 = 2 days 17. Days in inventory = Annual Inventory cost of goods sold/365 Annual cost of goods sold = Inventory x 365/Days in inventory = $10,000 × 365/30 = $121,667 Inventory turnover = 10,000 121,667 = 12.167 times per year. 18. a. Interest expense = .08 × $10 million = $800,000 Times interest earned = EBIT/Interest expense= $1,000,000/$800,000 = 1.25 b. Cash coverage ratio = interest expense EBIT + Depreciation = 800,000 1,000,000 + 200,000 = 1.5 4-9 c. Fixed-payment cash-coverage = E800,000 + 300,000 E =A 1.09 19. a. ROA = ANOPAT Assets E Aand using the Du Pont System ROA can be decomposed to: ROA = A Sales Assets E ×AA NOPAT Sales E =A Asset turnover × Net profit margin ROA = 3 × .05 = .15 = 15% b. ROE = ANet Income Assets E Aand using Du Pont System ROE can be decomposed to: ROE = AAssets Equity E A × A Sales Assets E ×A EBIT + depreciation interest expense + debt repayment = 1,000,000 + 200,000 ANOPAT Sales E ×A A Net Income NOPAT E A Note that A Sales Assets E ×A A NOPAT Sales E =A ROA And ANet Income NOPAT E Ais called Debt Burden So: ROE = AAssets Equity E A × ROA × debt burden If debt/equity = 1, then debt = equity, so total assets are twice equity and AAssets Equity E =A 2/1 = 2 To calculate the debt burden both Net Income and NOPAT need to be calculated. EBIT, earnings before interest expense and taxes, is $20,000 Interest expense = $8,000 Taxes = $8,000 Net Income = EBIT – Interest – Taxes = 20,000 – 8,000 – 8,000 = 12,000 NOPAT, net operating profit after tax=Net income + after-tax interest expense Need to estimate tax rate: Estimated tax rate = taxes/income before tax Income before taxes, EBT = EBIT – Interest = 20,000 – 8,000 = 12,000 Estimated tax rate = 8,000/12,000 = .667 After-tax interest expense = (1 – tax rate) x interest expense = (1 - .667) x 8,000 = 2,664 4-10 NOPAT = Net income + after-tax interest expense =4,000+2,664=6,664 Debt Burden = ANet Income NOPAT E =A A 4,000 E6,664 E =A .60 ROE = AAssets Equity E ×A ROA × Debt Burden = 2 × .15 × .60 = .18 = 18% 20. Average collection period, ACP = Average receivables/ (Annual Sales/365) Rearrange the equation to calculate sales: (Annual Sales/365 days) = Average receivables/ Average collection period Annual Sales = (Average receivables/ Average collection period) × 365 Average Receivables = $3,000 and Average collection period = 20 days Annual sales = ($3,000/20 days) × 365 days = $54,750 Asset turnover = Sales/average total assets = $54,750/$75,000 = .73 ROA = Asset turnover × operating profit margin = .73 × .05 = .0365 = 3.65% 21. Debt-equity ratio = LAong-term debt Equity E =A .4 Rearrange the equation to calculate Long-term debt: Long-term debt = Debt-equity ratio × Equity Long-term debt = .4 × $1,000,000 = $400,000 Current ratio = A Current assets Current liabilities E =A 2.0 and Current assets = $200,000. So Current liabilities = ACurrent assets Current ratio E A = $200,000/2 = $100,000 Since the only current liabilities are notes payable, Notes payable = $100,000 Total liabilities = current liabilities + long-term debt =$100,000+$400,000 = $500,000 Total assets = total liabilities + equity = $500,000 + $1,000,000 = $1,500,000 Total debt ratio = $500,000/$1,500,000 = .33 4-11 22. A Book Debt Book Equity E =A .5 AMarket Equity Book Equity E =A 2 The ratio of Book Debt/Book Equity divided by Market Equity/Book Equity cancels out Book Equity and give the value of the Book Debt/Market Equity: A Book Debt Book Equity E A ÷ A Market Equity Book Equity E =A A Book Debt Book Equity E ×A A Book Equity Market Equity E A = A Book Debt Market Equity E So : A Book Debt Market Equity E =A A Book Debt Book Equity E A ÷ A Market Equity Book Equity E =A A .5 2 E =A .25 23. EBIT = Revenues – COGS – Depreciation = $3,000,000 – $2,500,000 – $200,000 = $300,000 Interest expense = interest rate × debt face value = 8% × 1,000,000 = $80,000 Times interest earned = EBIT/Interest expense = 300,000/80,000 = 3.75 24. CFA Corp. has less debt relative to equity than the industry average but its ratio of EBIT plus depreciation to interest expense is lower. Perhaps the firm has a lower ROA than its competitors, and is therefore generating less EBIT per dollar of assets. Perhaps the firm pays a higher interest rate on its debt. Or perhaps its depreciation charges are lower because it uses less capital or older capital. 25. The decline in market interest rates will increase the market value of the fixed-rate debt and thus increase the market-value debt-equity ratio. The value of the floating rate debt will not change with the change in interest rates. By this measure, leverage will increase. The decline in interest rates will also reduce the firm’s interest payments on the floating rate debt, which will increase the times-interest-earned ratio. By this measure, leverage will decrease. The impact of the lower rates on “leverage” is thus ambiguous. The firm has higher indebtedness relative to assets, but greater ability to cover its cash flow obligations. 26. a. The shipping company, which has more tangible assets, will tend to have the higher debt-equity ratio. (It is discussed later in the textbook that firms holding 4-12 tangible assets with active secondary markets tend to maintain higher debt- equity ratios.) b. Food Producers Inc. is in a more mature industry and probably has fewer favourable opportunities for reinvesting income. We would expect it to have the higher payout ratio. c. The paper mill will have higher sales per dollar of assets. It is less capital intensive (that is, has less capital per dollar of sales) than the integrated pulp and paper manufacturer. d. The discount outlet sells many of its goods for cash. The power company bills monthly and usually gives customers a month to pay bills. It will have the longer collection period. 27. Leverage ratios will be of interest to banks or other investors lending money to the firm. They want to be assured that the firm is not borrowing more than it can reasonably be expected to be able to repay. Liquidity ratios are also of interest to creditors who prefer that a firm’s current assets are well in excess of its current liabilities. Liquidity ratios are especially important to those who lend to the firm for short periods, for example, by extending trade credit. If a firm buys goods on credit, the seller wants to know that when the bill comes due, the firm will have enough cash on hand to pay it. Efficiency ratios might be of interest to stock market analysts who want to know how well the firm is being run. They are also of great concern to the firm’s own management, which needs to know if it is running as tight a ship as its competitors. Profitability ratios will be of interest to security market investors who want to know the return on their investment and be able to compare the company to other companies. Also they are of great concern to the firm’s own management, who needs to know how well the company is doing compared to its competitors. Market value ratios are important to security market investors who want to know how the firm is being valued relative to its earnings power as well as relative to other firms in the industry. 4-13 28. ABC Corp Income Statement for This Year UMillions of dollars Net sales $170.00 Cost of goods sold 129.60 Selling, general & administrative expenses 10.00 Depreciation U 20.00 EBIT 10.40 Interest expense U 1.60 Income before tax 8.80 Tax 2U.641 Net income $ 6.159 ABC Corp Balance Sheet UMillions of dollars UThis Year Last Year Assets Cash and marketable securities $ 11 $ 20 Receivables 44 34 Inventories U 22U U26U Total current assets 77 80 Net property, plant, equipment U 38U U25U Total assets $115 $105 Liabilities & Shareholders’ Equity Accounts payable $ 25 $20 Notes payable U 30U U35U Total current liabilities 55 55 Long-term debt 24 20 Shareholders’ equity U 36U U30U Total liabilities & Shareholders’ equity $115 $105 4-14 USolution Procedure 1. Total assets = Total liabilities and Shareholders’ equity So: From the Balance Sheet: This Year Total assets = 115 2. From the Balance Sheet: This Year Total current liabilities = Accounts payable + Notes Payable = 25 + 30 = 55 3. Long-term debt ratio = Long-term debt/(Long-term debt + Equity) = 0.4 So: Long-term debt = (Long-term debt + Equity) × Long-term debt ratio To solve for (Long-term debt + Equity) use this equation: Total Assets = Total liabilities and Shareholders’ equity = Current liabilities + Long-term debt + Equity So: (Long-term debt + Equity) = Total Assets - Current liabilities This Year (Long-term debt + Equity) = 115 – 55 = 60 This Year Long-term debt = (Long-term debt + Equity) × Long-term debt ratio = 60 × .4 = 24 4. Total liabilities and Shareholders’ equity (which also equals Total Assets) = Current liabilities + Long-term debt + Equity So: Equity = Total Assets - Current liabilities - Long-term debt This Year Equity = 115 – 55 – 24 = 36 5. Current ratio = Current assets/Current liabilities = 1.4 So: Total current assets = Total current liabilities × Current ratio This Year Total current assets = 55 × 1.4 = 77 6. Cash ratio = Cash and marketable securities/Current liabilities = 0.2 So: Cash and marketable securities = Current liabilities × Cash ratio This Year Cash and marketable securities = 55 × 0.2 = 11 7. Quick ratio = (Cash and marketable securities+Trade receivables)/ Current liabilities=1 So: Cash and marketable securities+Trade receivables = Current liabilities × Quick ratio This Year Cash and marketable securities+Trade receivables = 55 × 1 = 55 Since This Year Cash and marketable securities = 11 So: This Year Trade receivables = (Cash and marketable securities+Trade receivables) – Cash = 55-11 = 44 8. Inventories = Current Assets - Cash and marketable securities - Trade receivables So: This Year Inventories = 77 – 11 – 44 = 22 9. Net Property, plant, equipment = Total assets – Current assets So: This Year Property, plant, equipment = 115 – 77 = 38 10. Average collection period, ACP = Average trade receivables/(Annual Sales/365) = 83.735 So: Annual sales = (Average trade receivables/ACP) × 365 Average trade receivables = (Last Year Trade Receivables + This Year Trade Receivable)/2 4-15 = (34 +44)/2 = 39 So: This Year Annual Sales = (39/83.735) × 365 = 170.00 11. Inventory turnover = Cost of goods sold/ Average inventories = 5.4 So: Cost of goods sold = Average inventories × Inventory turnover Average inventories = (Last Year Inventories + This Year Inventories)/2 = (26 +22)/2 = 24 So: This Year Cost of goods sold = 24 × 5.4 = 129.6 12. EBIT = Net Sales - Cost of goods sold - SGA - Depreciation By the way SGA is Selling, general, and administrative expenses So: This Year EBIT = 170 – 129.6 – 10 – 20 = 10.4 13. Times interest earned = EBIT/interest expense = 6.5 So: Interest expense = EBIT/ Times interest earned So: This Year Interest Expense = 10.4/6.5 = 1.6 14. Income before tax = EBIT – Interest Expense So: This Year Income before tax = 10.4 – 1.6 = 8.8 15. Return on Equity, ROE = Net Income/equity at start of year = .2053 So : Net income = equity at start of year × ROE So: This Year Net Income = 30 × .2053 = 6.159 16. Net income = Income before tax – Tax So: Tax = Income before tax – Net income So: This Year Tax = 8.8 - 6.159 = 2.641 4-16 29. a. See table and graph below Net Profit Margin (%) Asset Turnover Food Products 5.18 1.22 Transportation 12.72 1.33 Automobiles and Components 5.27 1.43 Chemicals 13.09 0.79 Metals and Mining 38.43 0.4 Retailers 4.59 1.96 Media 12.37 0.59 Biotechnology -29.95 0.42 Capital Goods 1.33 1.26 Gold Mines 17.9 0.34 4-17 You can see that the Retail industry with the highest asset turnover (1.96) had low profit margin (4.59%). In comparison, the Metals and Mining industry had highest profit margin (38.43%) but one of the lower asset turnovers (0.4). The scatter diagram shows that asset turnover generally declines as profit margin increases. This relationship makes sense as firms with low profit margins need to generate move volume of sales. That is, if margins are low, each dollar of total assets must work harder to produce the same amount of total profit. b. See table and graph below Current Ratio Quick Ratio Food Products 1.86 0.6 Transportation 1.01 0.83 Automobiles and Components 1.82 1.12 Chemicals 1.77 1.04 Metals and Mining 3.96 3.17 Retailers 1.47 0.64 Media 1.51 1.2 Biotechnology 8.45 7.34 Capital Goods 2.21 1.24 Gold Mines 3.84 2.83 These two measures of liquidity appear to move together. Higher quick ratios are associated with higher current ratios. You may conclude that once you one of these ratios 4-18 there is little to be gained by calculating the other ratio. However, analysts should use caution as some firms may have very high current ratios but the result may be due to hight level of illiquid assets, such as old inventory. 30. As of September 9P th P, 2011, there was no information on interim financial information available on finance.yahoo.com so this is obtained from the AstraZeneca website’s second quarter 2011 statement. Since this is a semi-annual F/S, figures have been multiplied by 2 as necessary (used as a crude estimate). Most measures are very similar to those calculated for the 2010 year-end. The most major difference is in performance measures – the market value added and market- to-book ratio dropped, this may be due to general stock market uncertainty and perhaps also from various lawsuits launched against in the first two quarters. Profitability measures all increased slightly. Inventory turnover and receivables turnover both declined, which signifies that AZN is tying up more assets and customers are now taking longer to pay. Leverage barely changed. AZN is slightly more liquid, and the payout rate and sustainable growth rate improved as well. As a debtor, I would be interested in leverage measures and liquidity measures (how much debt AZN has taken on and how much liquid assets it has to make interest payments). As both have improved on a minor scale, I would be better off at 2011 second quarter than I was at the 2010 year end. Selected financial measures for AstraZenecaCo (2011 Second Quarter and 2010) 2010 2011 Second Quarter 0TPerformance Measures Market value added ($ millions) 43,208 60,390 – 23,815 = 36,575 Market-to-book ratio 2.86 60,390 / 23,815 = 2.54 0TProfitability Measures Return on equity (ROE) 38.8% 5,038 x 2 / 23,815 = 42.3% Return on assets (ROA) 15.4% (5,038+493) x 2 / 53,406 = 20.7% Return on capital (ROC) 24.2% (5,038+493) x 2 / (372+9,210+3,034+23,815) = 30.4% EVA ($ US millions) 6,245 (5,038+493) x 2 - (0.067 x 36,431) = 8,622 Operating profit margin 25.4% (5,038+493) / 16,722 = 33.1% 0TEfficiency Measures Asset turnover 0.60 16,722 x 2 / 53,406 = 0.63 Inventory turnover 3.72 (2,821 x 2) / 2,021 = 2.79 Days in inventory 98 2,021 / (2,821 x 2 / 365) = 131 Receivables turnover 5.5 16,722 x 2 / 8,320 = 4.02 Average collection period (days) 66 8,320 / (16,722 x 2 / 365) = 91 0TLeverage Measures Long-term debt ratio 28% 9,210 / (9,210+23,815) = 27.9% Long-term debt-equity ratio 39% 9,210 / 23,815 = 38.7% Debt-to-asset ratio 58% 29,591 / 53,406 = 55.4% Times interest earned 11.1 6,366 / 493 = 12.9 Cash coverage ratio 13.8 (6,366+1,037) / 493 = 15 4-19 0TLiquidity Measures Net working capital to total assets 0.14 (22,691-13,838) / 53,406 = 0.17 Current ratio 1.5 22,691 / 13,838 = 1.64 Quick ratio 1.12 (9,613+679+8,320) / 13,838 = 1.34 Cash ratio 0.66 9,613 / 13,838 = 0.69 0TGrowth Measures Payout ratio .43 2,646 / 5,038 = .53 Sustainable growth rate 21.1% 0.53 x 42.3% = 22.4% SScatter 4-20 Solution for Mini-Case for Chapter 4 Problems for HH are apparent in the areas of debt and assets. Leverage ratios improved between 2006 and 2010 (all of various leverage measures have decreased from 2006 to 2010) but have increased in 2011. Table 4.12 shows that both long-term and short-term debt (debt due for repayment) has increased significantly in 2011 relative to 2010. Liquidity ratios began to deteriorate in 2007, at the same time that the number of employees increased substantially. Further deterioration in liquidity ratios occurred in 2008, when inventories more than doubled and current liabilities increased by more than 85%. At the same time, sales remained virtually unchanged from 2007. Table 4.10 shows that Hobby Horse (HH) sales have been increasing each year, since 2006. Also the number of stores and the number of employees have increased each year. But it has negative EBIT and negative net income in 2011, despite the fact that both have been increasing from 2006 to 2010. To better understand what is going on it useful to look at financial ratios. Key financial ratios calculated with the financial highlights are reported in the table below and below that are additional ratios for 2011 using the 2011 financial statements. Sales growth rate is variable but always positive. The company has increased the number of stores and its employees but sales per store and sales per employee are not staying constant or increasing at a steady rate. The 2011 leverage and liquidity ratios show that Hobby Horse is currently experiencing some difficulty. Although long-term debt leverage ratios improved between 2006 and 2010 (all of various leverage measures have decreased from 2006 to 2010) but have increased in 2011. The liquidity measures (times interest earned, current ratio and net working capital to total assets) for 2006 to 2010 do not indicate liquidity crisis. In those years the times interest earned was generally rising, and around 4 to near 5, which are strong numbers. Also current ratios are bigger than 1 which is indicates liquidity. But in 2010 current ratio and net working capital to total asset ratios fell although time interest earned increased. This suggests that the firm current assets or current liabilities changed in 2010. These liquidity measures are the worst in 2011. The fact that time interest earned is negative and current ratio is less than one indicates liquidity issues for HH. The negative ratio of net working capital to assets indicates that current liabilities in 2011 are greater than current assets, another indicator of poor liquidity. The 2011 quick and cash ratios in the second table below are both less than 1, other indicators of current liquidity issues for HH. Table 4.12 shows that both long-term and short-term debt (debt due for repayment) has increased significantly in 2011 relative to 2010. These results could be due to a single poor Christmas season coming at a time of too-rapid growth, rather than being indicative of a long-term unsustainable business situation. In this regard, it is interesting to compare 4-21 Hobby Horse's financial ratios with those of other retailers (Table 4.7). Comparing HH’s 2011 performance with the ratios in Table 4.7, reveals that HH is more highly levered, much less profitable and less liquid than other retailers. However, if you use 2009’s performance, HH looks much more like the average, although it still seems to more levered but has higher operating return on assets and ROE than average. Over time HH’s asset turnover has increased but it fell in 2011. You can see in Table 4.12 that inventories increase in 2011 relative to 2010. This is likely due to the poor sales at Christmas. The company had purchased the inventories for the Christmas season but they did not get sold. Now looking at the various measures of profitability shows the poor financial profit in the 2011. Notice that ROA, operating return on assets and ROC (return on capital) were generally rising from 2006 to 2010. ROA’s “DuPont factors,” operating profit margin and asset turnover, also generally rose from 2006 to 2008. Both the level and the trend of ROA would seem to indicate a successful underlying business. Consistent with the apparent success of the firm, management has been steadily expanding the business, with slow but steady growth of assets from $959 million in 2006 to $1,249 million in 2010 (6.8% compounded annual growth). The 72.5% growth in assets from 2010 to 2011 is far more aggressive. But this is likely due to excessive inventory due to less sales than planned for. Ratios based on the most recent year’s statements have deteriorated. Losses have apparently created the need for additional borrowing: long-term debt and the debt-equity ratio have both increased. Asset turnover is down, presumably due to the rapid expansion of assets in the face of flat sales. Of particular concern is the rapid growth of inventories, from $203 million to $479 million. Days of products sitting in inventory is currently 88 days. This may reflect in part a particularly bad Christmas season. An obvious question is whether the unsold inventory has any substantial value. If the extra inventory is just an unanticipated build up of goods and can be sold in the next few months, then it is not overly worrisome. But if it will never be sold and is effectively worthless, then the firm’s debt-equity ratio is much higher than it appears. The firm is encountering difficulties, but in the long term it seems like a situation that can be salvaged. The underlying business ratios seem stable, and the recent problems seem due to too-rapid expansion. The firm has already acted to reverse the situation by selling 15 of its stores, which will bring total stores almost back to the 2010 level. If the recent disappointing Christmas season is indeed a blip, rather than a signal of long-lived downturn in general economic conditions, the firm should be fully able to return to its formerly strong economic position. As the firm sells off assets, the bank should demand as a condition of renewing the loan that the revenue from the asset sales be used to improve the firm’s liquidity ratio. With $484 million of debt due for repayment (see SFP in Table 4.12), the company clearly has a refinancing problem. Selling off 15 stores probably won't help much here. It would be important to ask the company about its plans to deal with the debt repayment. 4-22 NOTE: Different definitions of various ratios are reported below. The big issue is when comparing a value from an income statement with a value from the statement of financial position (balance sheet). Do you use the statement of financial position at the beginning of year, the end of the year or the average of the beginning and end of the year values. You will see that the trends are generally similar regardless of which ratio definition is used. NOTE: Sales growth rate calculation formula Let g be the annual sales growth rate: So 2011 Net Sales = (1+g) 2010 Net Sale Solve for the 2011 annual sales growth rate: g = (2011 Net Sales/2010 Net Sales) -1 Key Financial Ratios 2006 - 2011 U2011 U2010 U2009 U2008 U2007 U2006 Annual sales growth rate 0.011 0.165 0.018 0.122 0.154 Annual asset growth rate 0.275 0.004 0.116 0.113 0.045 Number of employees per store =Employees/Number of stores 54.404 53.552 46.493 53.207 53.382 49.841 Sales per store = Net Sales/number of stores 13.963 14.995 13.483 15.196 14.665 13.758 Sales per employee = Net Sales/employees 0.257 0.280 0.290 0.286 0.275 0.276 Long-term debt ratio=Long-term debt/(Long-term debt+equity) 0.259 0.180 0.331 0.383 0.439 0.493 Long-term Debt-Equity ratio =Long-term debt/Equity 0.349 0.219 0.496 0.620 0.784 0.972 Debt-to-asset ratio=Total liabilities/Total assets (Total Liabilities = Current liabilities + Long-term Debt) 0.575 0.420 0.518 0.550 0.594 0.662 Times Interest Earned =EBIT/Interest -0.243 4.952 3.938 4.190 4.417 3.391 Current ratio =Current assets/Current liabilities 0.984 1.285 1.411 1.440 1.420 1.322 NWC/Total assets=(Current assets - Current liabilities)/total assets -0.007 0.083 0.115 0.119 0.116 0.107 Asset Turnover =Net Sales/average assets 2.359 2.659 2.412 2.641 2.543 Asset Turnover =Net Sales/end of year assets 2.105 2.653 2.287 2.508 2.488 2.252 Asset Turnover =Net Sales/start of year assets 2.683 2.664 2.552 2.790 2.600 Tax rate 0.240 0.240 0.240 0.240 0.240 0.240 NOPAT =Net income +(1-tax rate) x interest - 20.880 236.88 194.40 186.08 161.48 110.96 Operating profit margin =NOPAT/Sales -0.006 0.071 0.068 0.067 0.065 0.051 Operating profit margin =(NI + interest)/Sales -0.004 0.076 0.074 0.072 0.069 0.056 4-23 Return on assets, ROA = NOPAT/assets at start of year -0.017 0.190 0.174 0.186 0.168 ROA = NOPAT/(assets at end of year) -0.013 0.190 0.156 0.167 0.161 0.116 ROA = NOPAT/(average total assets) -0.015 0.190 0.165 0.176 0.165 Total Capital =long-term debt + equity 912.00 884.00 896.00 813.00 726.00 639.00 Return on capital, ROC = NOPAT/(start of year total capital) -0.024 0.264 0.239 0.256 0.253 Return on capital, ROC = NOPAT/(end of year total capital) -0.023 0.268 0.217 0.229 0.222 0.174 Return on capital, ROC = NOPAT/(average total capital) -0.023 0.266 0.228 0.242 0.237 ROE =Net income/equity at start of year -0.068 0.316 0.289 0.349 0.386 ROE =Net income/equity at end of year -0.072 0.261 0.242 0.283 0.307 0.235 ROE = Net income/average equity -0.070 0.285 0.263 0.312 0.342 Additional Ratios for 2011 Ratios Cash coverage ratio =(EBIT + depreciation)/interest 4.054 Quick ratio=(cash and cash equivalents+trade receivables)/current liabilities 0.279 Cash ratio=(cash and cash equivalents)/current liabilities 0.021 Inventory turnover=COGS/(average inventories) 5.836 Inventory turnover=COGS/(2011 inventories) 4.154 Days in inventory=(average inventories)/(COGS/365) 62.545 Days in inventory =(2011 inventories)/(COGS/365) 87.857 Receivables turnover = sales/(average trade receivables) 18.114 Receivables turnover = sales/(2011 trade receivables) 19.040 Collection period (days)=(average trade receivables)/sales/365 20.151 Collection period (days)=(2011 trade receivables)/(sales/365) 19.170 Trade payables turnover = COGS/(average trade payables) 42.340 Trade payables turnover = COGS/(2011 trade payables) 21.170 Payment period(days)=(average trade payables)/(COGS/365) 13.940 Payment period(days)=(2011 trade payables)/(COGS/365) 17.241 Payout ratio=Dividends/net income 0 4-24 Brealey 5CE Solutions to Chapter 5 Note: Unless otherwise stated, assume that cash flows occur at the end of each year. 1. a. 100/(1.08)10 = $46.32 b. 100/(1.08)20 = $21.45 c. 100/(1.04)10 = $67.56 d. 100/(1.04)20 = $45.64 2. a. 100 × (1.08)10 = $215.89 b. 100 × (1.08)20 = $466.10 c. 100 × (1.04)10 = $148.02 d. 100 × (1.04)20 = $219.11 3. With simple interest, you earn 4% of $1000, or $40 each year. There is no interest on interest. After 10 years, you earn total interest of $400, and your account accumulates to $1400. With compound interest, your account grows to 1000 × (1.04)10 = $1480. Therefore $80 is interest on interest. 4. FV = 700 PV = 700/(1.05)5 = $548.47 5. Present Value Years Future Value Interest Rate* a. $400 11 $684 5% = (684 400 ) 1/11 – 1 b. $183 4 $249 8% = (249 183 ) 1/4 – 1 c. $300 7 $300 0% = (300 300 ) 1/7 – 1 To find the interest rate, we rearrange the equation FV = PV × (1 + r)n to conclude that r = (FV PV ) 1/n - 1 To use a financial calculator for (a) enter PV= (-)400, FV = 684, PMT = 0, n = 11 and compute the interest rate. 5-1 6. You should compare the present values of the two annuities. Discount Present Value of Present Value of Rate 10-year, $1000 annuity 15-year, $800 annuity a. 5% 7721.73 8303.73 b. 20% 4192.47 3740.38 When the interest rate is low, as in part (a), the longer (i.e., 15-year) but smaller annuity is more valuable because the impact of discounting on the present value of future payments is less severe. When the interest rate is high, as in part (b), the shorter but higher annuity is more valuable. In this case, with the 20 percent interest rate, the present value of more distant payments is substantially reduced, making it better to take the shorter but higher annuity. 7. PV = 200/1.05 + 400/1.052 + 300/1.053 = 190.48 + 362.81 + 259.15 = $812.44 8. In these problems, you can either solve the equation provided directly, or you can use your financial calculator setting PV = (−)400, FV = 1000, PMT = 0, i as specified by the problem. Then compute n on the calculator. a. 400 × (1 + .04)t = 1,000 t = 23.36 periods b. 400 × (1 + .08)t = 1,000 t = 11.91 periods c. 400 × (1 + .16)t = 1,000 t = 6.17 periods Note: To solve directly, use the natural log function, ln. For example, for (a), ln[ (1.04)t ] = ln[1000/400] t × ln[1.04] = 0.91629 t = 0.91629/.03922 = 23.36 period. Using the calculator: PV = (-)400, FV = 1000, i = 4, compute n to get n = 23.36. 9. a. PV = 100 × PVIFA(.08,10) = 100 × 6.7101 = 671.01 b. PV = 100 × PVIFA(.08,20) = 100 × 9.8181 = 981.81 c. PV = 100 × PVIFA(.04,10) = 100 × 8.1109 = 811.09 d. PV = 100 × PVIFA(.04,20) = 100 × 13.5903 = 1,359.03 10. a. FV = 100 × FVIFA(.08,10) = 100 × 14.4866 = 1,448.66 b. FV = 100 × FVIFA(.08,20) = 100 × 45.7620 = 4,576.20 c. FV = 100 × FVIFA(.04,10) = 100 × 12.0061 = 1,200.61 d. FV = 100 × FVIFA(.04,20) = 100 × 29.7781 = 2,977.81 5-2 11. APR Compounding Period Per Period Rate, APR/m Effective annual rate a. 12% 1 month (m = 12/yr) .12/12 =.01 1.0112− 1 = .1268 = 12.68% b. 8% 3 months (m = 4/yr) .08/4 = .02 1.024 − 1 = .0824 = 8.24% c. 10% 6 months (m = 2/yr) .10/2 = .05 1.052 − 1 = .1025 = 10.25% 12. Effective Annual Rate, EAR Compounding Period Number of Periods per year, m Per period rate, (1+EAR)1/m -1 APR, m × per period rate a. 10.0% 1 month 12 1.11/12 – 1 = .008 12×.008 = .096 = 9.6% b. 6.09% 6 months 2 1.06091/2 − 1 = .03 2×.03 = .06 = 6% c. 8.24% 3 months 4 1.08241/4 − 1 = .02 4×.02 = .08 = 8% 13. We need to find the value of n for which 1.08n = 2. You can solve to find that n = 9.01 years. On a financial calculator you would enter PV = (−)1, FV = 2, PMT = 0, i = 8 and then compute n. 14. Semiannual compounding means that the 8.5 percent loan really carries interest of 4.25 percent per half year. Similarly, the 8.4 percent loan has a monthly rate of .7 percent. APR Period m Effective annual rate = (1 + per period rate)m – 1 8.5% 6 months 2 (1.0425)2 − 1 = .0868 = 8.68% 8.4% 1 month 12 (1.007)12 − 1 = .0873 = 8.73% Choose the 8.5 percent loan for its slightly lower effective rate. 5-3 15. APR = 1% × 52 = 52% EAR = (1 + .01)52 − 1 = .6777 = 67.77% 16. Our answer assumes that the investment was made at the beginning of 1900 and now it is the end of 2011. Thus the investment was for 112 years (2011 – 1900 + 1). a. 1000 × (1.05)112 = $236,157.37 b. PV × (1.05)112 = 1,000,000 implies that PV = $4,234.46 17. $1000 × 1.05 = $1050.00 First-year interest = $50 $1050 × 1.05 = $1102.50 Second-year interest = $1102.50 − $1050 = $52.50 After 9 years, your account has grown to 1000 × (1.05)9 = $1551.33 After 10 years, your account has grown to 1000 × (1.05)10 = $1628.89 Interest earned in tenth year = $1628.89 − $1551.33 = $77.56 18. Method 1: If you earned simple interest (without compounding) then the total growth in your account after 25 years would be 4% per year × 25 years = 100%, and your money would double. With compound interest, your money would grow faster, and therefore would require less than 25 years to double. Method 2: Another quick way to answer the question is with the Rule of 72. Dividing 72 by 4 gives 18 years, which is less than 25. The exact answer is 17.673 years, found by solving 2000 = 1000 × (1.04)n. [On your calculator, input PV = (-) 1000, FV = 2000, i = 4, PMT = 0, and compute the number of periods.] 19. We solve 422.21 × (1 + r)10 = 1000. This implies that r = 9%. [On your calculator, input PV = (-)422.21, FV = 1000, n = 10, PMT = 0, and compute the interest rate.] 20. The number of payment periods: n = 12 × 4 = 48. If the payment is denoted PMT, then 5-4 PMT × annuity factor( 10 12 %, 48 periods) = 8,000 PMT = $202.90. The monthly interest rate is 10/12 = .8333 percent. Therefore, the effective annual interest rate on the loan is (1.008333)12 − 1 = .1047 = 10.47 percent. 21. a. PV = 100 × annuity factor(6%, 3 periods) = 100 ×  1  .06 − 1 .06(1.06)3 = $267.30 b. If the payment stream is deferred by an extra year, each payment will be discounted by an additional factor of 1.06. Therefore, the present value is reduced by a factor of 1.06 to 267.30/1.06 = $252.17. 22. a. This is an annuity problem with PV = (-)80,000, PMT = 600, FV = 0, n = 20 × 12 = 240 months. Use a financial calculator to solve for i, the monthly rate on this annuity: i = .5479%. EAR = (1 +.005479)12 − 1 = .06776 = 6.776% APR = 12 × monthly interest rate = 12 × .5479% = 6.5748%, compounded monthly b. Again use a financial calculator and enter n = 240, i = .5%, FV = 0, PV = (−)80,000 and compute PMT = $573.14 23. a. With PV = 9,000 and FV = 10,000, the annual interest rate is defined by 9,000 × (1 + r) = 10,000, which implies that r = 11.11%. b. Your present value is 10,000 (1 − d), and the future value you pay back is 10,000. Therefore, the annual interest rate is determined by: PV × (1 + r) = FV [10,000 (1 – d)] × (1 + r) = 10,000 5-5 1 + r = 1 1- d ⇒ r = 1 1- d − 1 = 1 d- d > d Since 0 < d < 1, then 1 – d d. So r must be greater than d. c. With a discount interest loan, the discount is calculated as a fraction of the future value of the loan. In fact, the proper way to compute the interest rate is as a fraction of the funds borrowed. Since PV is less than FV, the interest payment is a smaller fraction of the future value of the loan than it is of the present value. Thus, the true interest rate exceeds the stated discount factor of the loan. 24. If we assume cash flows come at the end of each period (ordinary annuity) when in fact they actually come at the beginning (annuity due), we discount each cash flow by one period too many. Therefore we can obtain the PV of an annuity due by multiplying the PV of an ordinary annuity by (1 + r). Similarly, the FV of an annuity due also equals the FV of an ordinary annuity times (1 + r). Because each cash flow comes at the beginning of the period, it has an extra period to earn interest compared to an ordinary annuity. 25. a. Solve for i in the following equation: 10,000 = 275 × PVIFA(i, 48) Using the calculator, set PV = -10,000, PMT = 275, FV = 0, n = 48 and solve for i i= 1.19544% per month APR = 12 × 1.19544% = 14.3453% EAR = (1 + .0119544)12 – 1 = .153271, or 15.3271% b. Annual payment = 12 × 275 = 3,300 Repeat the steps in (a) to find the EAR of this car loan to see which loan is charging the lower interest rate: Solve for i in the following equation: 10,000 = 3,330 × PVIFA(i, 4) Using the calculator, set PV = -10,000, PMT = 3,300, FV = 0, n = 4 and solve for i i= 12.11% per year Little Bank's loan interest rate of 12.11% is less than the EAR of 15.53% on Big Bank's loan. With a lower interest rate, Little Bank's loan is better. c. Find the annual loan payment, P, such that 10,000 = X × PVIFA(15.3271%, 4) Using the calculator, set PV = -10,000, FV = 0, n = 4, i = 15.3271 and solve for PMT = $3,525.86. By comparison, 12 times $275 per month is $3,300. The annual payment on a 4-year loan equivalent to $275 per month for 48 months is greater than 12 times the monthly payment of $275 because of the benefit of delaying payment to the end of each year. The borrower gets to delay payment and therefore is better off. If Little Bank doesn't charge at least $3,525.86 annually, it earns less on its loan than Big Bank earns on its loan. 5-6 26. a. Compare the present value of the lease to cost of buying the truck. PV lease = 8,000 × PVIFA(7%, 6) = -$38,132.32 It is cheaper to lease than buy because by leasing the truck will cost only $38,132.32, rather than $40,000. Of course, the crucial assumption here is that the truck is worthless after 6 years. If you buy the truck, you can still operate it after 6 years. If you lease it, you must return the truck and replace it. b. If the lease payments are payable at the start of each year, then the present value of the lease payments are: PV annuity due lease = 8,000 + 8,000 × PVIFA(7%, 5) = 8,000 + 32,801.58 = $40,801.58. Note too that PV of an annuity due = PV of ordinary annuity × (1 + r). Therefore, with immediate payment, the value of the lease payments increases from its value in the previous problem to $38,132 × 1.07 = $40,801 which is greater than $40,000 (the cost of buying a truck). Therefore, if the first payment on the lease is due immediately, it is cheaper to buy the truck than to lease it. 27. a. Compare the PV of the payments. Assume the product sells for $100. Installment plan: Down payment = .25 × 100 = 25 Three installments of .25 × 100 = 25 PV = 25 + 25 × annuity factor(6%, 3 years) = $91.83. Pay in full: Payment net of discount = $90. Choose this payment plan for its lower present value of payments. Note: The pay-in-full payment plan will have the lowest present value of payments, regardless of the chosen product price. b. Installment plan: PV = 25 × annuity factor(6%, 4 years) = $86.63. Now the installment plan offers the lower present value of payments. 28. a. PMT × annuity factor(12%, 5 years) = 1000 PMT × 3.6048 = 1000 PMT = $277.41 b. If the first payment is made immediately instead of in a year, the annuity factor will be greater by a factor of 1.12. Therefore PMT × (3.6048 × 1.12) = 1000. PMT = $247.69. 5-7 29. This problem can be approached in two steps. First, find the PV of the $10,000, 10-year annuity as of year 3, when the first payment is exactly one year away (and is therefore an ordinary annuity). Then discount the value back to today. Using a financial calculator, 1) PMT = 10,000; FV = 0; n = 10; i = 6%. Compute PV3 = $73,600.87 2) PV0 = PV3 (1 + r)3 = 1.063 73,600.87 = $61,796.71 A second way to solve the problem is the take the difference between a 13-year annuity and a 3-year annuity, valued as of the end of year 0: PV of delayed annuity = 10,000 × PVIFA(6%,13) – 10,000 × PVIFA(6%,3) = 10,000 × (8.852683 – 2.673012) = 10,000 × 6.179671 = $61,796.71 30. Note: Assume that this is a Canadian mortgage. The monthly payment is based on a $175,000 loan with a 300-month (12 × 25 years) amortization. The posted interest rate of 6 percent has a 6-month compounding period. Its EAR is (1 + .06/2)2 – 1 = .0609, or 6.09%. The monthly interest rate equivalent to 6.09% annual is (1.0609)1/12 – 1 = .004939, or 0.4939%. PMT × annuity factor(.4939%, 300) = 175,000 PMT = $1,119.71. When the mortgage expires in 5 years, there will be 20 years remaining in the amortization period, or 240 months. The loan balance in five years will be the present value of the 240 payments: Loan balance in 5 years = $1,119.71 × Annuity factor (.4939%, 240 periods) = $157,215. 31. The EAR of the posted 7% rate is (1 + .07/2)2 – 1 = .071225. The monthly interest rate equivalent is (1.071225)1/12 – 1 = .00575, or 0.575%. The payment on the mortgage is computed as follows: PMT × annuity factor (.575%, 300 periods) = 350,000 PMT = $2,451.44 per month. If you pay the monthly mortgage payment in two equal installments, you will pay $2,451.44/2, or $1,225.72 every two weeks. Thus each year you make 26 payments. The bi-weekly equivalent of the 7% posted interest rate is (1.071225)1/26 – 1 = 5-8 .002649, or .2649% every two weeks. Now calculate the number of periods it will take to pay off the mortgage: $1,225.72 × Annuity factor (.2649%, n periods) = $350,000 Using the calculator: PMT = 1,225.49, PV = (-)350,000, i = .2649 and compute n = 533.84. This is the number of bi-weekly periods. Divide by 26 to get the number of years: 533.84/26 = 20.5 years. If you pay bi-weekly, the mortgage is paid off 5.5 years sooner than if you pay monthly. 32. a. Input PV = (-)1,000, FV = 0, i = 8%, n = 4, compute PMT which equals $301.92 b. Loan Year End Interest Year End Amortization Time Balance Due on Balance Payment of Loan 0 $1,000.00 $ 80.00 $301.92 $221.92 1 $778.08 $62.25 $301.92 $239.67 2 $538.41 $43.07 $301.92 $258.85 3 $279.56 $22.37 $301.92 $279.56 4 0 0 — — c. 301.92 × annuity factor (8%, 3 years) = 301.92 × 2.5771 = $778.08, which equals the loan balance after one year. 33. The loan repayment is an annuity with present value $4248.68. Payments are made monthly, and the monthly interest rate is 1%. We need to equate this expression to the amount borrowed, $4248.68, and solve for the number of months, n. [On your calculator, input PV = (−) 4248.68, FV = 0, i = 1%, PMT = 200, and compute n.] The solution is n = 24 months, or 2 years. The effective annual rate on the loan is (1.01)12 − 1 = .1268 = 12.68% 34. The present value of the $2 million, 20-year annuity, discounted at 8%, is $19,636,295. If the payment comes one year earlier, the PV increases by a factor of 1.08 to $21,207,198. 35. The real rate is zero. With a zero real rate, we simply divide her savings by the years of retirement: $450,000/30 = $15,000 per year. 5-9 36. Per month interest = 6%/12 = .5% per month FV in 1 year (12 months) = 1000 × (1.005)12 = $1,061.68 FV in 1.5 years (18 months) = 1000 × (1.005)18 = $1,093.93 37. You are repaying the loan with an annuity of payments. The PV of those payments must equal $100,000. Therefore, 804.62 × annuity factor(r, 360 months) = 100,000 which implies that the interest rate is .750% per month. [On your calculator, input PV = (−)100,000, FV = 0, n = 360, PMT = 804.62, and compute the interest rate.] The effective annual rate is (1.00750)12 − 1 = .0938 = 9.38%. If the lender is a Canadian financial institution, the quoted rate will be the APR for a 6-month compounding period: (1 + APR 2 )2 – 1 = .0938 APR 2 = (1.0938)1/2 -1 = .04585 APR = 2 × [(1.0938)1/2 -1] = .0917 or 9.17%, which is lower than the effective annual rate. Note: A simpler APR calculation is .750% × 12 = 9%. However, this is not how Canadian mortgage lenders calculate their APRs. 38. EAR = e.04 -1 = 1.0408 -1 = .0408 = 4.08% 39. The PV of the payments under option (a) is 11,000, assuming the $1,000 rebate is paid immediately. The PV of the payments under option (b) is $250 × annuity factor(1%, 48 months) = $9,493.49 Option (b) is the better deal. 40. 100 × e.10×6 = $182.21 100 × e.06×10 = $182.21 5-10 41. Your savings goal is 30,000 = FV. You currently have in the bank 20,000 = PV. The PMT = (-) 100 and r = .5%. Solve for n to find n = 44.74 months. Note: You may have to solve this by trial-and-error if your calculator cannot handle these numbers. 42. The present value of your payments to the bank equals: $100 × annuity factor(8%, 10 years) = $671.01 The present value of your receipts is the value of a $100 perpetuity deferred for 10 years: 100 .08 × 1 (1.08)10 = $578.99 This is a bad deal if you can earn 8% on your other investments. 43. If you live forever, you will receive a $100 perpetuity which has present value 100/r. Therefore, 100/r = 2500, which implies that r = 4 percent 44. r = 10,000/125,000 = .08 = 8 percent. 45. Suppose the purchase price is $1. If you pay today, you get the discount and pay only $.97. If you wait a month, you must pay $1. Thus, you can view the deferred payment as saving a cash flow of $.97 today, but paying $1 in a month. The monthly rate is therefore .03/.97 = .0309, or 3.09%. The effective annual rate is (1.0309)12 − 1 = .4408 = 44.08%. 46. You borrow $1000 and repay the loan by making 12 monthly payments of $100. We find that r = 2.923% by solving: 100 × annuity factor(r, 12 months) = 1000 [On your calculator, input PV = (−)1,000, FV = 0, n = 12, PMT = 100, and compute the interest rate.] The APR is therefore 2.923% × 12 = 35.08% and the effective annual rate is (1.02923)12 − 1 = .4130 = 41.30% How do we know that the true rates must be greater than 20%? If you borrow 5-11 $1000 and repay $1200 in one year, the rate of interest on the loan is 20%. Here, with add-on interest, you make the $1200 repayment sooner. Because of the time value of money, the effective interest rate must be higher than 20%. 47. You will have to pay back the original $1000 plus 3 × 20% = 60% of the loan amount, or $1600 over the three years. This implies monthly payments of $1600/36 = $44.44 The monthly interest rate is obtained by solving: 44.44 × annuity factor(r, 36) = 1000 On your calculator, input PV = (−)1,000, FV = 0, n = 36, PMT = 44.44, and compute the interest rate as 2.799% per month. The APR is therefore 2.799% × 12 = 33.59%, and the effective annual rate is (1.02799)12 − 1 = .3927 = 39.27% 48. For every $1000 you borrow, your present value is 1000 (1 − d), and the future value you pay back is 1000. Therefore, the annual interest rate is determined by: PV × (1 + r) = FV [10,000 (1 – d)] × (1 + r) = 10,000 1 + r = 1 −1 d ⇒ r = 1 −1 d − 1 = 1 −d d > d If d = 20%, then the effective annual interest rate is .2/.8 = .25 = 25%. 49. The semi-annual interest rate paid at First National is 0.062/2 = .031, or 3.1% every six months. After one year, each dollar invested will grow to: $1 × (1.031)2 = $1.06296 and the EAR is 6.296% The monthly interest rate paid at Second National is .06/12 = .005, or 0.5% every month. After one year, each dollar invested will grow to: $1 × (1.005)12 = $1.06168 and the EAR is 6.168%. First National pays the higher effective annual rate. 5-12 50. Since the $20 origination fee is taken out of the initial proceeds of the loan, the amount actually borrowed is $1000 − $20 = $980. The monthly rate is found by solving the following equation for r: 90 × annuity factor(r, 12) = 980 r = 1.527% per month. The effective rate is (1.01527)12 -1 = .1994 = 19.94%. 51. Monthly interest rate = (1.08)1/12 – 1 = .006434 or .6434% Football Quarterback: Total salary paid over contract = 5 years × 3 million/year = $15 million Monthly salary = 3 million/12 months per year = $250,000 at the end of each month for 12 ×5, or 60 months PV = 250,000 PVIFA(.6434%,60 months) = $12,411,236.45 Hockey Player Total salary paid over contract = $4 million + 5 × $2.1 million = $14.5 million Monthly salary = $2.1 million/12 months per year = $175,000 at the end of each month PV = 4 million + 175,000 PVIFA(.6434%,60 months) = $12,687,865.52 The quarterback is wrong. The hockey player’s contract has a higher present value. 52. a. Per month interest rate = 7%/12 = .005833333 48-month loan: PV = 400 × PVIFA(7%/12,48) = $16,704.08 60-month loan: PV = 400 × PVIFA(7%/12,60) = $20,200.80 Bill will buy a car for $16,704.08 if he arranges a 48-month loan and will buy a car for $20,200.80 if he arranges a 60-month loan. b. To fairly compare the two loans, both time periods must be the same. We assume that Bill will keep the car for 5 years, regardless of which loan he takes. Bill’s wealth at the end of 5 years will depend on the value of the car and the balance in his bank account. Wealth in five years if take the 48-month loan and buy the $16,704.08 car: (1) Value of car at the end of 5 years: starting value × (1 – depreciation rate)5 = $16,704.08 × (1 - .18)5 = $6,192.87 (2) Savings = $400 invested each month for one year: FV = 400 × FVIFA(5%/12,12) = $4,911.54 (3) Total wealth = $6,192.87 + $4,911.54 = $11,104.41 Wealth in five years if take the 60-month loan and buy the $20,200.80 car: (1) Value of car at the end of 5 years: starting value × (1 – depreciation rate)5 5-13 = $20,200.80 × (1 - .18)5 = $7,489.24 (2) Savings = none, spent all spare cash on car payments (3) Total wealth = $7,489.24 We haven’t compared Bill’s happiness from owning the more expensive car to his happiness from owning the less expensive car. On the other hand, we’ve not compare the cost of fuel or maintenance either. Whether the more expensive car is worth it has not been established. 53. a. The present value of the ultimate sales price is 4 million/(1.08)5 = $2.722 million. b. The present value of the sales price is less than the cost of the property, so this would not be an attractive opportunity. c. The value of the total cash flows from the property is now PV = .2 × annuity factor(8%, 5 years) + 4/(1.08)5 = .80 + 2.72 = $3.52 million To solve with a calculator, enter: PMT = .2, FV = 4, i = 8%, n = 5 and compute PV. Therefore, the property is an attractive investment if you can buy it for $3 million. 54. PV of cash inflows = [120,000/1.12 + 180,000/1.122 + 300,000/1.123] = $464,172 This exceeds the cost of the factory, so the investment is attractive. 55. a. The present value of the future payoff is 2000/(1.05)10 = $1227.83. This is a good deal: PV exceeds the initial investment. You can solve this also by looking at the future value of investing $1,000 at the opportunity cost of 5% for 10 years: FV = 1.0510 × 1000 = $1628.9. This is less than the $2000 payoff expected from the investment. The investment is a good deal. Another way to answer this question is to figure out the interest rate that the investment is offering: Find r such that (1 + r)10 × 1000 = 2000. Either 5-14 using the calculator or solving directly, gives r = .07177, or 7.177%. Clearly it is better to invest at 7.177% than at 5%. b. The PV is now only 2000/(1.10)10 = $771.09, which is less than the initial investment. Therefore, this is a bad deal. Another way to look at the investment: Since we know that the investment offers a 7.177% return, it makes no sense to undertake the investment if we can invest elsewhere and earn 10%. 56. The future value of the payments into your savings fund must accumulate to $500,000. We assume that payments are made at the end of the year. We choose the payment so that PMT × future value of an annuity = $500,000. On your calculator, enter: n = 40; i = 5; PV = 0; FV = 500,000. Compute PMT to be $4,139.08. 57. If you invest the $100,000 received in year 10 until your retirement in year 40, it will grow to $100,000 × (1.05)30 = $432,194. Therefore, your savings plan would need to generate a future value of only $500,000 – $432,194 = $67,806. This would require a savings stream of only $561.31. 58. By the time you retire you will need $40,000 × future value annuity factor(5%, 20 periods) = $498,488.41. The future value of the payments into your savings fund must accumulate to $498,488.41. We choose the payment so that PMT × future value of an annuity = $498,488.41. On your calculator, enter: n = 40; i = 5; PV = 0; FV = 498,488.41. Compute PMT to be $4,126.57. 59. After 30 years the couple will have accumulated the future value of a $3,000 annuity, plus the present value of the $10,000 gift. The sum of the savings from these sources is: $3,000 × future value annuity factor (30,8%) = $339,849.63 $10,000 × 1.0825 = 68,484.75 $408,334.38 If they wish to accumulate $800,000 by retirement, they need to save an additional amount per year to provide additional accumulations of $391,665.62. This requires additional annual savings of $3,457.40. [On your calculator, input i = 8; n = 30; PV = 0; FV = 391,665.62 and compute PMT.] 5-15 60. a. The present value of the planned consumption stream as of the retirement date will be $30,000 × annuity factor(25,8%) = $320,243.29. Therefore, they need to have accumulated this level of savings by the time they retire. So their savings plan must provide a future value of $320,243.29. With 50 years to save at 8%, the savings annuity must be $558.14 Another way to think about this is to recognize that the present value of the savings stream must equal the present value of the consumption stream. The PV of consumption as of today is = $320,243.29/(1.08)50 = $6,827.98 Therefore, we set the present value of savings equal to this value, and solve for the required savings stream. Using the calculator: enter PV = (-)6,827.98 n = 50, i = 8, FV = 0 and solve for PMT. b. The couple needs to accumulate additional savings with a present value of $60,000/(1.08)20 = $12,872.89. The total PV of savings is now $12,872.89 + $6,827.98 = $19,700.77. Now we solve for the required savings stream as follows: n = 50; i = 8; PV = (−)19,700.77; FV = 0; and solve for PMT as $1,610.40. They need to save $1,610.40 each year for the next 50 years. 61. Note: Ignore taxes. Monthly interest rate = .08/12 = .00666666 Borrow and buy the copier Monthly loan payments : 20,000 = PMT × PVIFA(8%/12, 60) PMT = $405.53 Cash flows if borrow/buy: Month Cash flows 0 1 2 3 … 58 59 60 Loan +20,000 -405.53 -405.53 -405.53 -405.53 -405.53 -405.53 Salvage +5,000 PV(borrow/buy cash flows) = +20,000 – 405.53 × PVIFA(.08/12,60) + 5,000 × PVIF(.08/12,60) = + 3,356.05 Cash flows if lease: Month Cash flows 0 1 2 3 … 58 59 60 Lease +20,000 - X -X - X - X - X - X 0 5-16 Salvage 0 PV(Lease cash flows) = +20,000 – X – X × PVIFA(.08/12, 59) Find X such that PV(borrow/buy cash flows) = PV(lease cash flows): 3,356.05 = +20,000 – X – X × PVIFA(.08/12, 59) 16,643.95 = X + X × PVIFA(.08/12, 59) = X × annuity paid at the beginning of the period (.08/12, 60) This can be solved using a financial calculator. Set the calculator for payments at the beginning of the period. (With a BAII Plus: enter 2nd PMT (which is BEG), followed by 2nd Enter (which is SET). You should see the word END on the screen change to the word BEG.) PV = (-) 16,643.95, i = .08/12, n = 60, FV = 0, solve for PMT = 335.24 The lease payment equivalent to borrowing and buying is $335.24 per month, paid at the start of each month. 62. a. $60,000/8.2 = $7,317.07. Her real income increased from $6,000 to $7,317.07. b. Years to retirement = 2011 – 1950 = 61 years Salary inflation rate, s: (1 + s)61 × $6,000 = $60,000 s = (60,000/6,000)1/61 – 1 = .0385, or 3.85% Cost of goods inflation rate, c: (1 + c)61 × 1 = 8.2 c = 8.21/61 – 1 = .0351, or 3.51% 63. 1 + nominal rate = (1 + real rate) × (1 + inflation rate) a. 1.04 × 1.0 = 1.04; nominal rate = 4% b. 1.04 × 1.04 = 1.0816; nominal rate = 8.16% c. 1.04 × 1.06 = 1.1024; nominal rate = 10.24% 64. real interest rate = 1 + nominal interest rate 1 + inflation rate − 1 a. 1.08/1 – 1 = .080 = 8.0% b. 1.08/1.03 – 1 = .0485 = 4.85% c. 1.08/1.06 – 1 = .0189 = 1.89% 65. a. 100/(1.08)3 = $79.38 present value b. 100/(1.03)3 = $91.51 real value 5-17 c. real interest rate = 1 + nominal interest rate 1 + inflation rate − 1 = .04854, or 4.854% d. 91.51/(1.04854)3 = $79.38 66. Internet. Expected results: Students gain experience working with real data to calculate nominal rates of growth and converting them to real growth rates. According to the company profile, Thomson Reuters Corporation provides intelligent information for businesses and professionals in the financial, legal, tax and accounting, scientific, healthcare, and media markets worldwide. Based on the annual income statement, the compound annual growth rates (CAGR) over the period of Dec. 07 to Dec. 09 will be calculated as follow (numbers are in thousands): CAGR(sales) = (12,997/7,296)1/3 – 1 = 0.2122 = 21.22% CAGR(net income) = (867/4,056)1/3 – 1 = –0.4021 = –40.21% Inflation Calculator can be found at http://www.bls.gov/data/inflation_calculator.htm Enter $100 for 2007, then calculate the value of the $100 for 2010. The result will be $105.35 for 2010. Now, compound average annual inflation rate can be calculated as (105.35/100)1/3 – 1 = 0.0175 = 1.75%. Source: http://www.bls.gov/data/inflation_calculator.htm Real CAGR(sales) = (1 + 0.2122)/(1 + 0.0175) – 1 = 0.1914 = 19.14% Real CAGR(net income) = (1 – 0.4021)/(1 + 0.0175) –1 = –0.4124 = –41.24% 5-18 67. Internet: Inflation and Investment Returns a. Expected results: Students gain some perspective on inflation rates over the past century. The inflation calculator provides a comparison between dollar amounts in two different years considering the inflation rate. Enter $5,000 for 1914 and 2010 for comparison, click CALCULATE and you will get $97,250.00. Source: http://www.bankofcanada.ca/en/rates/inflation_calc.html b. The Inflation Calculator (above figure) shows the Average Annual Rate of Inflation equal to 3.14%. c. Tip: As we write this, the button for the Investment calculator is at the top right side of the inflation calculator screen. Expected Results: As we write this, the calculator assumes annual payments. The number of years is equal to the selected year minus the current year. For the example below, the number of years assumed is 4 (= 2014 – 2010). For the numbers below, the real rate of interest is 1.04/1.0314 - 1 = .008338181, or .8338181%. After-inflation value of investment (in 4 years) = $100,000/1.03144 = $88,367.28089 Total interest earned (nominal): = $100,000 ×FVIF(4.0%,4) – $100,000 = $16,985.86 Interest earned after inflation can also be calculated as: nominal 5-19 interest/inflation factor = $16,985.86/(1.0314)4 = $15,009.94262 Total future value: = $100,000 ×FVIF(.8338181%,4) = $103,377.22 Source: http://www.bankofcanada.ca/en/rates/investment.html d. To determine the interest rate used in the calculator, plug in a future target of $100,000. The investment needed today is $96,733.11. The interest rate, r, must solve the following equation: $96.733.11 = $100,000 × PVIF(r, 4) Using a hand-held calculator, with PV = (-) 96,733.11, FV=100,000, n = 4, PMT=0, calculate r = 0.8338181%. We get the same real rate of interest. Therefore, the future target must also be a real amount, the amount in terms of today’s dollars. For example, if you want $150,000 in 2014’s dollars, that equates to $150,000/(1.0314)4 = $132,550.92 in 2010 dollars. If the real rate of interest is 0.8338181%, you need to invest $128,220.63 today, in 2010. Source: http://www.bankofcanada.ca/en/rates/investment.html 5-20 68. a. Since the nominal cash flows are expected to grow with inflation, the expected annual real cash flow is $100,000. The real rate is 1.08/1.03 – 1 = 4.85%. So the present value is: $100,000 × annuity factor(4.85%, 5 years) = $434,749 b. If cash flow is level in nominal terms, use the 8% nominal interest rate to discount. The annuity factor is now 3.99271 and the cash flow stream is worth only $399,271. 69. a. $1 million will have a real value of $1 million/(1.03)45 = $264,439. b. At a real rate of 2%, this can support a real annuity of $228,107/annuity factor(2%, 20 years). = $264,439/16.3514 = $16,172 To solve this on a calculator, input n = 20, i = 2, PV = 264,439, FV = 0, and compute PMT. 70. According to the Rule of 72, at an interest rate of 8%, it will take 72/8 = 9 years for your money to double. For it to quadruple, your money must double, and then double again. This will take approximately 18 years. 71. (1.23)12 – 1 = 10.99. Prices increased by 1,099 percent per year. 72. Using the perpetuity formula, the 4% consol will sell for £4/.06 = £66.67. The 2 1/2% consol will sell for £2.50/.06 = £41.67. 73. The savings calculator can be reached directly from the following link: http://www.financialpost.com/personal-finance/calculators/savings/index.html Total value after 30 years without any savings = $1,000 x (1.0630) = $5,743.49 In the savings calculator enter Years to Save = 30, Total Cost of Expense = $0, Current Savings = $1,000, Deposit Amount = $0, Compounded Annual Rate of Return = 6%, and click Calculate, you will get the same result. Total value with after 30 years with $200 savings per month: (1 + monthly rate)12 –1 = .06  monthly rate = .004867551 Number of periods = 30 x 12 = 360 months Assuming the payments are made at the start of each period: Total Value = $5,743.49 + $200 x FVIFA(360, .4867551%) x (1.004867551) 5-21 = $5,743.49 + $195,851.31 = $201,594.80 In the calculator enter $200 as Deposit Amount and choose Monthly Deposit Frequency, click calculate and you will get the same result with some rounding errors. 74. Expected Result: Using "Buy or Lease" calculator from www.smartmoney.com, enter $20,000 as the Price of Car, $350 as Down Payment, $350 as Monthly payment of lease, 36 months as Lease term, 10% Rate of return, and $10,000 as the value of the car at the end of the lease. The calculator will calculate the value of alternative investments at the end of lease term of $11,510 which is more than the value of the car at the end of the lease ($10,000). Thus, lease is a better option. 75. a. $30,000 × annuity factor(10%, 15 years) = $228,182 b. Fin the annual payment, PMT, such that PMT × future value annuity factor(10%, 30 years) = 228,182. Using the calculator, PV = 0, n=30, i= 10%, FV= (-) 228,182. Compute PMT = 1,387. You must save $1,387 annually. c. 1.00 × (1.04)30 = $3.24 d. We repeat part (a) using the real rate of 1.10/1.04 – 1 = .0577 or 5.77% The retirement goal in real terms is $30,000 × annuity factor(5.77%, 15 years) = $295,797 e. The future value of your 30-year saving stream must equal this value. So we solve for payment (PMT) in the following equation PMT × future value annuity factor(5.77%, 30 years) = $295,797 PMT × 75.930 = $295,797 PMT = 3,896 You must save $3896 per year in real terms. This value is much higher than the answer to (b) because the rate at which purchasing power grows is less than the nominal interest rate, 10%. f. If the real amount saved is $3,896 and prices rise at 4 percent per year, then the amount saved at the end of one year in nominal terms will be $3,896 × 1.04 = $4,052. The thirtieth year will require nominal savings of 3,896 × (1.04)30 = $12,636. 5-22 76. In the 113 years since the capture of Ned Kelley, from 1880 to 1993, one dollar invested in the bank would have grown to be $35.14 (= 1 × (1.032)113 ). By contrast, that same dollar invested in the Australian stock market would have grown to $28, 431.22 (= $1 × (1.095)113). My clients are reasonable people but believe that $1 is a ridiculously low amount. Given the 3% annual inflation, the real value in 1993 of $1 paid in 1880 is only $0.035 (= $1/(1.03)113). Surely the efforts of the trackers is worth more than 3 ½ cents! We think a reasonable payment is $14,233 each, half way between the value of $1 invested in the bank and the value of a $1 invested in the stock market. 77. The interest rate per three months is 12%/4 = 3%. So the value of the perpetuity is $100/.03 = $3,333. 78. FV = PV × (1 + r1) × (1 + r2) = 1 × (1.08) × (1.10) = $1.188 PV = (1 + r1)FV×(1 + r2) = (1.08) ×1 (1.10) = $0.8418 79. You earned compound interest of 8% for 8 years and 6% for 13 years. Your $1000 has grown to 1000 × (1.08)8 × (1.06)13 = $3,947.90. 80. The answer to this question can be found in various ways. The key is to pick a common point in time to measure all cash flows. Here we pick today as the common reference point. Monthly interest rate = 1.061/12 – 1 = .004868 = .4868%. Present value today of the funds need for boat: 150,000/(1.06)4 = 118,814 Funds need for monthly expenses (this is an annuity due) = (2200 + 2200×annuity factor(.4868%, 23 months))/ (1.06)5 = 37,333.29 Funds need for emergencies = 45,000/(1.06)5 = 33,626.62 Total funds needed = 118,814 + 37,333.29 + 33,626.62 = 189,774 The present value of the savings stream must equal the present value of the expenditures: PMT × annuity factor (.4868%,60 months) = 189,774 The monthly savings must be $3,654.9. (Expect slight variations due to rounding) 5-23 81. Interest rate on parents’ car loan = .024/12 = .002 = .2% Monthly car repayment: PMT × annuity factor (.2%, 48) = 4,000 Using the calculator to find PMT = $87.48 Monthly opportunity cost of funds = (1.06)1/12 – 1 = .004868 = .4868% Summary of Car Costs: Month 0 1 2 3 … 47 48 Car payment 87.48 87.48 87.48 … 87.48 87.48 Operating cost 200 200 200 200 … 200 0 Total costs 200 287.48 287.48 287.48 … 287.48 87.48 Present value of car costs = 200 + 287.48 × annuity factor (.4868%, 47 months) + 87.48/(1.004868)1/48 = $12,320.5 You have to decide whether to charge your friends at the beginning or the end of each month. In this calculation, we assume that your friends will pay you at the start of each month. The total amount of money needed each month to cover the car costs, given a 6% EAR is: PMT × annuity due factor (.4868%, 48) = 12,320.5 Switch your calculator to the annuity due setting, and solve for PMT =$287 [n=48, i=.4868, FV=0, PV= (-)12,320.5] If your three friends are to cover the cost of the car, they each must pay $287/3 = $95.67 a month. If you share in the cost, dividing it four ways, you each pay $287/4 = $71.75. Does it make financial sense to buy the car? Given that the cost of a monthly bus pass is $80, it does not make sense to charge your friends much more than $80 per month, unless the bus trip to school is extremely long relative to the time taken in the car. Likewise, you too will not want to pay much more than $80 a month either. Of course, you will have access to the car at times when your friends do not. It depends on the value of the convenience of having access to the car. If you charge yourself $80 a month, then you should ask your friends for $69 a month. Perhaps that might be viewed as a fair trade-off. We have not considered the impact of inflation on the costs of operating the car. The car payments won’t change with inflation but the operating costs will. You can redo the analysis assuming a 2% annual rate of inflation and see how much higher 5-24 must be the monthly charge. One final note: You may want to consider the benefits for the air quality of taking public transit to school. We have not factored into our analysis the economic costs to society of air pollution from the car. 82. a. Assume there are 26 events per year (52 weekends/2). In 5 years, you attend 26×5 = 130 events. The bi-weekly interest rate is (1.09)1/26 – 1 = .00332 = .332% (assuming that 9% rate is an effective annual rate). Cost of Renting Cost of renting a van per weekend = $100 Mileage charge per weekend = 200 km × $.5 = $100 Fuel costs per weekend = 200 km × $.08 = $16 Total cost = $216 per weekend PV of cost of renting = 216 × annuity factor(.332%, 130 events) = $22,775.41 Cost of Owning Weekend operating costs = 200 km × ($.25 + $.08) = $66 PV of weekend operating costs = 66 × annuity factor(.332%, 130 events) = $6,959.15 Cost today of buying van = $20,000 Expected selling price in 5 years = (1 - .1)5 × 20,000 = 11,809.8 PV of selling price in 5 years = 11,809.8/(1.09)5 = 7,675.6 PV of insurance (assume insurance is paid in advance) = 1200 × annuity due factor(9%, 5 years) = 5,087.7 Total cost of owning = 6,959.15+ 20,000 - 7,675.6 + 5,087.7 = $24,371.25 Extra cost of purchasing the van rather than renting = $24,371.25- $22,775.41= $1,595.84 Although the total cost is higher, the van is available to drive at other times. If costs of another vehicle can be saved, then it makes sense to buy. Otherwise, it is cheaper to rent, if inflation is not considered. b. We use the principle of discounting real cash flows at the real discount rate. Assume all cash flows are in current year dollars, including the expected resale value of the car. The real discount rate is (1.09/1.03) – 1 = .05825 effective annual rate. The bi-weekly equivalent rate is (1.05825)1/26 – 1 = .00218 = .218%. Recalculate the present value of the costs at the real discount rate: Cost of Renting PV of cost of renting = 216 × annuity factor(.218%, 130 events) = $24,428.73 5-25 Cost of Owning PV of weekend operating costs = 66 × annuity factor(.218%, 130 events) = $7,464.33 Cost today of buying van = $20,000 PV of selling price in 5 years = 11,809.8/(1.05825)5 = $8,898.2 PV of insurance (assume insurance is paid in advance) = 1200 × annuity due factor(5.825%, 5 years) = $5,374.8 Total cost of owning = 7,464.33+ 20,000 - 8,898.2 + 5,374.8 = $23,940.93 Now the cost of owning is less than the cost of renting. Why? Because the cost of buying of the van does not inflate but the selling price does. 83. a. You can either discount real cash flows at the real discount rate or nominal cash flows at the nominal discount rate. We use real cash flows and discount rate. Real discount rate = 1.06 1.04 - 1 = .01923, or 1.923% PV college costs = 1 (1.01923)10 × 10,000 × annuity due factor(1.923%, 4 years) = $32,138.7 The family makes 10 payments, starting today. Using the annuity due formula, the annual payment is PMT × annuity due factor(1.923%, 10 years) = $32,138.7 PMT = $2,889.89 b. If the family waits one year, they have 9 years to accumulate the required funds. The annual payment is PMT × annuity due factor(1.923%, 9 years) = $32,138.7 PMT = $3,242.65 They must save $3,242.65 - $2,889.89 = $352.76 more each year if they delay the start of their savings program for one year. 5-26 84. The first cash flow, C1, is $35,000. Assume it will be received at the end of the first year. Using the formula for the present value of a perpetually growing stream of cash flows, P0 = r C1 - g g = r - C1 P0 = .08 - 35000 580000 = .08 - .0603 = .0197 = 1.97% 85. a. The maximum price is equal to the present value of all of the cash flows that can be generated by the property: P0 = - 1.5 1.08 - 1.5 1.082 + 1 1.083 × .4 × annuity factor(8%,50 years) = $ 1.21 million If you pay $1.21 million, you will earn 8% return on your investment. If you pay less than $1.21 million, you will earn more than 8%. We are assuming after 50 years no further development will take place. b. If the annual cash flows grow at 1.5% per annum, the value of the land increases. The maximum you should be willing to pay is the present value of the all the future cash flows that can be generated: P0 = - 1.5 1.08 - 1.5 1.082 + 1 1.083 × C1 r - g × [1 - ( 1 + g 1 + r ) T] = - 1.5 1.08 - 1.5 1.082 + 1 1.083 × .4 .08 - .015 × [1 - ( 1.015 1.08 ) 50] = - 1.389 – 1.286 + 5.878 = $3.203 million With 1.5% growth rate of the cash inflows, the present value of all of the cash flows increases to $3.203 million. This is the maximum you should be willing to pay. Again, we’ve assumed that the land has no use beyond year 50. You might disagree with that assumption. However, the present value of a cash flow 50 years away is quite small. 86. Annual real interest rate used for discounting and investing savings: 1.06/1.03 -1 = .029126, or 2.9126% Monthly real interest rate = (1.029126)1/12 - 1 = .002395, or 0.2395% a. How much will the Smiths need to have saved by the time they retire? 5-27 Calculate the present value at the end of Year 35 of all the cash flows incurred during retirement. (1) Retirement income Monthly retirement income (real) = $45,000/12 = $3,750 per month Number of months of retirement = 12/year × 20 years = 240 months PV(real retirement income as of end of Yr 35) = $3,750 × PVIFA(.2395%, 240 months) = $683,918 (2) Bequest to son PV(real value of bequest as of end of Yr 35) = 500,000 × PVIF(.2395%, 240) = $281,602 (3) Value of house Real growth rate = 1.04/1.03 - 1 = .0097087 = .97087% Expected value of house in 55 years = 250,000 × FVIF(.97087%, 55) = 425,329 PV(real value of house as of end of Yr 35) = 425,329 × PVIF(.2395%, 240) = $239,547 Total amount needed for retirement, as of end of Year 35 = 683,918 +281,602 - 239,547 = $725,973 Assume that the Smiths save an equal amount at the end of each of the 12 × 35 (420) months before their retirement. They will need to save PMT such that: $725,973 = PMT × FVIFA(.2395%,420) PMT = $1,004.41 per month In addition to saving for their retirement, the Smiths must save for their child's education. If you assume that they will save all the needed funds just as the child starts university, they will need savings of: Total required savings (real) at the end of Year 8 = 10,000 + 10,000 × PVIFA(2.9126%, 3 years) = 38,333.71 To reach this goal, they must save PMT per month from Year 1 to the end of Year 8: 38,333.71 = PMT × FVIFA(.2395%, 96 months) PMT = 355.64/ month (real dollars) Monthly savings required (real dollars) Years 1 to 8 Real retirement savings = 1,004.41 Real university savings = 355.64 Total 1,360.05 Years 9 to 35 Real retirement savings = 1,004.41 5-28 ed Note: If you assume that the Smiths will continue to save for their child's education while the child is at university, the monthly amount needed will be slightly smaller. To answer the question under this assumption, for each of the $10,000 find the monthly payment needed to be saved, recognizing that each amount is due one year later. For example, the monthly payment to be saved for the $10,000 needed for the first year of university, at the start of Year 8, is: 10,000 = PMT × FVIFA(.2395%, 96 months) or PMT = $92.77. For the second year of university, Year 9, the monthly payment needed is: 10,000 = PMT × FVIFA(.2395%, 108 months) or PMT = $81.24. Repeat for the third and fourth years of university (payments are $72.03 and $64.20, respectively). Then add these amounts together, keeping track of when the payments stop. Years 1 to 8 Real retirement savings = 1,004.41 Real university savings = 310.54 (= 92.77 + 81.24 + 72.03 + 64.5) Total 1,314.95 Year 9 Real retirement savings = 1,004.41 Real university savings = 217.77 (= 81.24 + 72.03 + 64.5) Total 1,222.18 Year 10 Real retirement savings = 1,004.41 Real university savings = 136.53 (= 72.03 + 64.5) Total 1,140.94 Year 11 Real retirement savings = 1,004.41 Real university savings = 64.5 Total 1,068.91 Then for Years 12 - 35, same as above. b. The nominal mortgage interest rate = .07/12 = 0.00583, or .583% The nominal monthly mortgage payment is: 200,000 = PMT × PVIFA(.583%, 12×20) PMT = 1,550.11 The last month of the mortgage is at the end of Year 20 Real payments Real retirement savings = 1,004.41 Real mortgage payment = 858.26 (= 1,550.11/(1.03)20 ) Total payments 1,862.67 5-29 d Nominal payments Nominal retirement savings = 1,814.08 (= 1,004.41 × (1.03)20 ) Nominal mortgage payment = 1,550.11 Total payments 3,364.19 c. The last month before retirement is 35 years from today. The only payment is the monthly retirement savings: Real value = 1,004.41 Nominal value = 1,004.41 × (1.03)35 = $2,826.27 87. Internet: RESP Calculator Note to instructors: You may wish to give students various profiles of families with children to compare the RESP savings requirements. Expected results: Students have experience thinking about the impact of inflation and interest rates on funds needed. Also, the students will see a fairly well thought out online financial planning tool. 88. Internet: Mortgage and Loan Calculators The purpose of this problem is to give students an opportunity to test their understanding of the time value of money concepts while exploring applications of financial tools on the internet. a. Checking the answer to Problem 20 with loan calculator Tips: The annual interest rate used must be the APR. Banks must give interest rates as APRs (the per period rate × number of payments per year) but sometime also report the EAR. Expected results: For the $8,000 48-month loan with a 10% APR, the personal loan calculator says the monthly payment will be $202.90. This is the same number found when we answered the problem ourselves. b. Using a mortgage calculator Tips: All Canadian mortgage interest rates are reported with semi-annual compounding. Thus 6% annual interest means 6%, compounded semi-annually or 3% every 6 months. Expected results: Using the HSBC mortgage calculator, enter $200,000 as the mortgage amount. The annual interest rate is 6% and the number of years to repay the mortgage is 25 years, the conventional amortization period for Canadian 5-30 mortgages. Payments type is monthly. The calculator says the monthly payment for the $200,000 mortgage with the 6% stated interest is $1,279.61. This matches exactly the number found by following the procedure presented in the text: PV = (-)200,000, n = 12 × 25 = 300, i = (1 + .06/2)1/6 – 1 = .49038622%, FV = 0, compute PMT. It is important to carry many decimal places. What about the other payment periods? In each case, the HSBC calculator recalculates the mortgage payment using a different number of payments per year but keeps the amortization constant. In our case, the amortization stays at 25 years. With semi-monthly payments, 24 payments are made each year (2 × 12 months). With bi-weekly, 26 payments are made each year (26 = 52 weeks/2) and 52 payments are made with the weekly option. In part (c), we look at another way to calculate mortgage payments. Weekly Mortgage Payments: Using the HSBC calculator, keep all the information the same except click on the weekly button (or pick 52 payments per year). The computed weekly mortgage payment is $294.74 Checking the HSBC weekly mortgage calculation: PV= (-) 200,000 n = 52 × 25 = 1300 weeks i = (1 + .06/2)1/26 – 1 = .11375235% (find the weekly interest rate equivalent to 3% every 6 months) Compute PMT =$294.74 . This exactly matches the weekly payment at HSBC. Bi-Weekly Mortgage Payments: The same procedure is followed to get the bi-weekly payment. This time, n = 26 × 25 = 650 bi-weekly payments (there are 26 bi-weeks in a year), i = (1.03)1/13 – 1 = .2276341% and PMT = $589.81 . Again, matches HSBC’s amount. Semi-Monthly Mortgage Payments: n = 24 × 25 = 600 payments, i = (1.03)1/12 – 1 = .00246627 and PMT = $639.02 , matching the HSBC amount. c. Comparing mortgage calculators The TD website gives the same monthly mortgage amount. However, for different payment periods, things get more interesting (or confusing, depending on your spirit of adventure). Two general approaches are taken in the variations on the monthly mortgage. The first is to simply change the number of payments per year (e.g., weekly), keep the amortization at 25 years and calculate the new mortgage payment, as was done with the HSBC mortgage calculator. The other approach is to keep the total monthly payment constant but pay it in installments (often called “rapid” or “accelerated” mortgage payments). 5-31 The TD bi-weekly mortgage payment is $590.59. For HSBC, the bi-weekly payment is $589.81. TD's bi-weekly payment is calculated by multiply the monthly payment by 12 and dividing by 26: 12 × $1,279.61/26 = $590.59 The second approach gives accelerated or rapid payment mortgages. The per- period payment is the monthly payment divide by the number of payment periods in the month: 2 for bi-weekly rapid/accelerated, 4 for weekly (or weekly rapid). Now instead of calculating PMT, you can calculate n, the number of periods until the mortgage is paid off. Since for each you pay more money each year, compared to the regular monthly payment, the mortgage is paid off more quickly. Note that banks are free to call these payment options whatever they wish – you will find that some calculators call “weekly” mortgages that are really “weekly rapid”. Do not be fooled by the names. Bi-Weekly Rapid or Accelerated Mortgage: Take the monthly payment and divide it by 2 to get semi-monthly payments of 1,279.61/2 = $639.81. Now instead of calculating PMT, you can calculate n, the number of periods before the mortgage is paid off. The bi-weekly interest rate is .2276341%, PV = (-) 200,000, PMT = 639.81, compute n = 546.81 semi-monthly periods, or 546.8/26 = 21.03 years. This works because you actually pay more each year: 26 bi-weeks × 639.81/bi-weeks = $16,635.06 per year compared with 12 months × 1,279.61/month = $15,355.32 per year. d. U.S. Mortgages Tip: In the US, interest is compounded monthly. So 6% interest means (.06/12) = .005 per month. Expected results: At the www.smartmoney.com, the monthly US mortgage payment is $1,289. The difference boils down to the fact that 6%, compounded monthly is a higher interest rate than 6%, compounded semi-annually: US: EAR of 6%, monthly = (1.005)12 – 1 = .0616778, or 6.168% Canada: EAR of 6%, semi-annually = (1.03)2 – 1 = .0609, or 6.09% You can make a U.S. mortgage equivalent to a Canadian mortgage by carefully ensuring that the mortgages have the same interest rate per payment period. 5-32 Solution to Minicase for Chapter 5 How much can Mr. Road spend each year? First let's see what happens if we ignore inflation. 1. Account for Canada Pension (CPP) and Old Age Security (OAS) income of $750 per month, or $9,000 annually. 2. Account for the income from the savings account. Because Mr. Road does not want to run down the balance of this account, he can spend only the interest income, or .05 × $12,000 = $600 annually. 3. Compute the annual consumption available from his investment account. We find the 20-year annuity with present value equal to the value in the account: Present Value = annual payment × 20-year annuity factor at 9% interest rate $180,000 = annual payment × 9.129 Annual payment = $180,000/9.129 = $19,717 Notice that the investment account provides annual income of $19,717, which is more than the annual interest from the account (.09× 180,000 = $16,200). This is because Mr. Road plans to run the account down to zero by the end of his life. So Mr. Road can spend $19,717 + $600 + $9,000 = $29,317 a year, comfortably above his current living expenses, which are $2,000 a month or $24,000 annually. The problem of course is inflation. We have mixed up real and nominal flows. The CPP and OAS payments are tied to the consumer price index and therefore are level in real terms. But the annuity of $19,717 a year from the investment account and the $600 interest from the savings account are fixed in nominal terms, and therefore the purchasing power of these flows will steadily decline. For example, let's look out 15 years. At 4 percent inflation, prices will increase by a factor of (1.04)15 = 1.80. Income in 15 years will therefore be as follows: 5-33 Income source Nominal income Real income CPP and OAS $16,200 $ 9,000 (indexed to CPI; fixed in real terms at $9000) Savings account 600 333 Investment account 19,717 10,954 (fixed nominal annuity) Total income $36,517 $20,287 Once we recognize inflation, we see that, in 15 years, income from the investment account will buy only a bit more than one-half of the goods it buys today. Obviously Mr. Road needs to spend less today and put more aside for the future. Rather than spending a constant nominal amount out of his savings, he would probably be better off spending a constant real amount. Since we are interested in level real expenditures, we must use the real interest rate to calculate the 20-year annuity that can be provided by the $180,000. The real interest rate is 4.8% (because 1 + real interest rate = 1.09/1.04 = 1.048). We therefore calculate the real sum that can be spent out of savings as $14,200 [n = 20; i = 4.8%; PV = (–)180,000; FV = 0; compute PMT]. Thus Mr. Road's investment account can generate a real income of $14,200 a year. The real value of CPP and OAS is fixed at $9,000. Finally, if we assume that Mr. Road wishes to maintain the real value of his savings account at $12,000, then he will have to increase the balance of the account in line with inflation, that is, by 4% each year. Since the nominal interest rate on the account is 5%, only the first 1% of interest earnings on the account, or $120 real dollars, are available for spending each year. The other 4% of earnings must be re-invested. So total real income is $14,200 + $9,000 + $120 = $23,320. To keep pace with inflation Mr. Road will need to spend 4 percent more of his savings each year. After one year of inflation, he will spend 1.04 × $23,320 = $24,253; after two years he will spend (1.04)2 × $23,320 = $25,223, and so on. The picture 15 years out looks like this: 5-34 Income source Nominal income Real income CPP and OAS $16,200 $ 9,000 Net income from savings account 216 120 (i.e., net of reinvested interest) Investment account 25,560 14,200 (fixed real annuity) Total income $41,976 $23,320 Mr. Road's income and expenditure will nearly double in 15 years but his real income and expenditure are unchanged at $23,320. This may be bad news for Mr. Road since his living expenses are $24,000. Do you advise him to prune his living expenses? Perhaps he should put part of his nest egg in junk bonds which offer higher promised interest rates, or into the stock market, which has generated higher returns on average than investment in bonds. These higher returns might support a higher real annuity — but is Mr. Road prepared to bear the extra risks? Should Mr. Road consume more today and risk having to sell his house if his savings are run down late in life? These issues make the planning problem even more difficult. It is clear, however, that one cannot plan for retirement without considering inflation. 5-35 Solution Manual for Fundamentals of Corporate Finance Richard A. Brealey, Stewart C. Myers, Alan J. Marcus, Elizabeth Maynes, Devashis Mitra 9780071320573, 9781259272011